rex weyler and bill gannon - the story of harmony

Rex Weyler and Bill Gannon

Justonic Tuning Inc.

Acknowledgements

The Authors wish to thank the following people for their kind help: Gordon Rock, Joel

Solomon, Dave Rees, Olwen Rees, Ann Rees, Steve Berry, Debera Barager, Mark

Deutschmann, Pierre deTrey, Carol Newell, Norine MacDonald, Bob Anderson, Jeff

Haley, Linda Gannon, Oliver Gannon, Peter Gannon, Shivaun Gannon, Patty Hervey,

Yasha Spong, Yvonne Kipp, Andrew Davis, Kaye Moss, Cynthia Ann Culkin, Bruce

Weyler, Doug Weyler, Joanne Weyler, Lisa Weyler, Juliet Eastmen, Maureen Bracewell,

Norman Gibbons, Denise Gibbons, Thomas Langley, Shelly Kantrow, Frank Gigliotti,

Anne L. Coulombe, Erwin Liem, Ed Gallagher, Art Lilly, Peter Jensen, Steve Dame,

Debbie Dame, Tom Jeffries, Robert Cribbs, Fathi Saleh, Paul Smith, Walter Beebe, Jay

Katz, Lome Kellett, Paul Horn, Scott Wilkinson, Robert Barstow, Margaret Taylor,

David Darling, Erv Wilson, Ben Johnson, John Chalmers, Robert Rich, John Loffink,

Kent Ormiston, Chuck Jonkey, Ronald Wells, Rick Ingrasci, Gordy Ryan, Jack Weyler,

Ibolya Weyler, Steve Finlay, Lynn Leboe, Paul Barriscale, Brent Flink, Darren Little,

Laura Green, Lionel Wilson, Dave Wade, Jean Gascon, Mark Hasselbach, Helen Bonny,

Rick Bodmer, John Gibbon, Wayne Silby, Stephen James Taylor, Tom Campbell,

Stanley Burke, Bill Weaver, Ron Markley, Dave Nedding, Jennifer Malloy, Haresh

Bakshi, David Mastrandrea, and Bruce Jacobson. Many others have helped, some whom

we may not even know; to everyone, thank you.

STORY OF IlARMONY

Contents

Chapter One: Showdown in Jena . . . . . . . . . . . . . . . . . . . . . .. 1

Chapter Two: The first musical scale , , , , . " 17

Chapter Three: Pythagoras: music & numbers 23

Chapter Four: Harmonists east and west 29

Chapter Five: The stream divides , 37

Chapter Six: A just diatonic major scale . . . . . . . . . . . . . . . .. 43

Chapter Seven: The Arabian contribution. . . . . . . . . . . . . . .. 49

Chapter Eight: Keyboards and polyphony in Europe. . . . . . .. 53

Chapter Nine: As sharp as the ear will endure. . . . . . . . . . . .. 59

Chapter Ten: The art and science of sound , , . , , , , . 65

Chapter Eleven: The age of pianos 73

Chapter Twelve: A paradigm entrenched 81

Chapter Thirteen: The new Harmonists . . . . . . . . . . .. . 87

Chapter Fourteen: Justonic . . . . . . . . . . . . . . . . . . . . . . . . . .. 99

Chapter Fifteen: Perfect tuning: so what? 109

Appendix A: Harmonic overtone series 117

Appendix B: Derivation and use of just musical scales 119

Index: 143

Notes: 153

The Story of Harmony

1

Showdown in Jena

Itwas on a bright autumn day in 1706 that a great tuning showdown tookplace in the German town of Jena, in the hill region of Thuringia. Two sets ofchurch organ pipes, two renowned music masters, and two musical

paradigms clashed on that day. The themes and issues of this clash had beenbrewing for four millennia and reverberate even today.

This great musical debate involved certain material facts about the nature ofsound, and therefore could not be entirely subjective, and yet posed aestheticquestions, so could not be solved with pure rationality or science. Indeed thesequestions defied systemization then, as they do now. World views come and go,aesthetic tastes move in and out of fashion, and yet the nature of sound holds firm.

2 The Story of Harmony

These were essential questions, being debated no doubt by choristers waiting alongthe choir loft balustrade of St. Michael's church in Jena, questions as critical toany art or time as to music at the turn of the 18th century, for they touched at thevery heart of artistic creation, namely: the material application of nature's gifts tothe artist's vision.

The Gothic church of St. Michael had been completed in 1556, ahallenkirche with side aisles that rose nearly as high as the central nave. Thechurch sat at the center of town, and on the main street, inside the fortified wallsof lena on the river Saale. Perhaps some impatient choristers, waiting for themasters to tune the organ pipes, climbed the white stone church tower, where theycould see the silver water curl north on its way to the Elbe, which in turn flowsthrough the vast Mecklenburgh plain to Hamburg, and on to the North Sea.Perhaps they wandered into the dirt streets of Jena, where they would hear thesinging pipes echo in the green, rolling Thuringia foothills, and turning would seewhere the river rose to its source in the Bavarian peaks of Frankenwald andFichtelgebirge 90 kilometers to the south. This was the "Green Heart" ofGermany. Forests and lakes surrounded lena, and vineyards covered the nearbyhills. To the northeast, a faint haze of smoke and dust marked the sky above Halleand Leipzig, seats of regional power and learning.

lena had just weathered a brutal 17th century. One hundred years earlier agreat flood known as the Thuringia Deluge had wreaked havoc in the city. Hardlyhad the citizens collected themselves and their few remaining possessions whenthey were subjected to repeated outbreaks of bubonic plague. The survivors offlood and Black Death were then greeted by the murderous onslaughts of pillagingsoldiers from both the Protestant and Catholic armies of the Thirty Years War(1618-1648). Many buildings in lena still showed the scars of war. Hardly afamily in lena, or in all of Prussia, had escaped suffering and loss. With peacecame the harsh taxes imposed by local princes consolidating their power andfinancing their private armies, sumptuous palaces, lavish feasts, festivals, andhunting expeditions.

Justonic Tuning Inc. 3

Nevertheless, those who endured were about to achieve perhaps the greatestGerman academic and cultural achievements in architecture, philosophy,literature, and music. A great cultural awakening was afoot on the Continent, andthe town of Jena, and the people of Thuringia, would- play a key role.

The fortified town of Jena in the region of Thuringia in 1652.The Church of St. Michael stands in the center of town,

site of the great Bach/Neidhardt tuning contest.The view is from the northwest with the River Saale in background.


The university at Jena, founded in 1548, was becoming an important centerof academic study. Its reputation had been greatly enhanced when the learnedErhard Weigel became Professor of Mathematics in 1653. Weigel considered thestudy of music an essential part of a liberal education, and likely bad a hand inbringing to Jena the renowned organist and choir master Johann Nicholas Bach.

Nicholas Bach, elder cousin of the 21-year-old prodigy Johann Sebastian,had arrived in Jena in 1690, and had become cantor, or organist and choirmaster,at S1. Michael's six years later. He possessed a profound musical knowledge,including a deep understanding of musical instrument theory and construction. Hehad designed and built the unique Lautenklavier, achieving a lute timbre withpianoforte action. He had come to influence both the academic and musical life ofJena, which gradually became a center for progressive musical thought under hisguidance. As a master tuner, skilled in all the current tuning systems for choirs,keyboard, and lute, he was one of the participants in the tuning contest on this dayin Jena. )

Today, for most western music, we use a single tuning system, equaltemperament, in which all semitones are of equal size, and modulation among keysis unrestricted. Some a cappella choral groups and chamber ensembles use formsof pure harmonic tuning, or just intonation, and certain progressive composers useextended just intonation. Although most popular music relies on equal temperedtuning, there exists today a movement among western musicians calling out formore flexibility and precision in tuning.

At the beginning of the 18th century the situation was somewhat reversed.Musicians in Europe used a variety of tuning systems, and the equal temperamentwe use today was only a theory. No truly equal temperament was achieved untilthe 20th century because of the difficulty of tuning such a system by ear. Lutetuning in 1706 was a quasi-equal temperament because frets demand such asystem. Choral groups of the time might sing in pure just intonation whenunaccompanied, singing both pure thirds and pure fifths, but such a systemrequired that notes shift pitch based on the context in which they were sung.


Keyboard tuning at this time was either meantone or well temperament, systemswith slightly sharp thirds and flat fifths, but in which each key had a uniquecharacter because the sizes of these intervals were different.

A master like Nicholas Bach was well acquainted with all these systems, thenuances, advantages, and compromises of each. Choir masters like Nicholas Bachknew the beauty and power of a pure triad, with thirds, fifths and octaves allblending in heavenly perfection. But they knew also that to achieve this perfectionon keyboard instruments was nearly impossible, and that to do so seriously limitedmodulation. The more classical solution at the time was meantone temperament,the many varieties of which used some pure thirds, some slightly sharp thirds andsixths, and flat fifths. Meantone temperaments dated from the 16th century,replacing the earlier so-called ''Pythagorean'' tunings in which the fifths andfourths were pure, and the thirds quite sharp.

The physical problem was a simple law of acoustics: a series of twelve pureharmonic fifths did not land precisely on an octave. Likewise, three pure harmonicthirds did not make an octave. In fact, no series of a single pure harmonic intervalwill precisely match up at any point with another series of a single pure harmonicinterval. A string of thirds will never land on the same point as a string of fifths, oroctaves, or wholetones. The discrepancies were known as "commas." Three purethirds are flat of an octave by a comma almost a quartertone in size. If fifths weretuned pure, the resulting third would be out by about 1/5 of a semitone, an intervalknown as the comma of Didymus, a fact known since at least the first century AD.

Unaccompanied singers could make the necessary adjustments as theymodulated, but to make a free modulation system work with the fixed tones of akeyboard instrument, the major thirds had to be sharpened, and the fifths flattened.Indeed, it was not for the sake of singers, nor the demands of harmony, nor eventhe demands of modulation that drove the centuries-long experiments withtempering the pure harmonies. Rather, the motivation that drove Nicholas Bach,many before him. and many after, was the convenience offered by the two familiesof musical instruments with fixed tones, instruments employing frets or digital


keys, namely lutes and organs. Eighty years earlier, the great French theoristMarin Mersenne had stated that the just intervals were "the easiest to sing andmost natural," and that "if it were as easy to mark the just consonances as theimperfect ones, there is no doubt that the performers would do it." He pointed out,however, that tempering "is done for the reason of convenience," and added that"nature has no regard for our convenience."?

The driving force in music theory of the eighteenth century was simply thediatonic keyboard. At this time, German musicians were unsurpassed in the fieldof keyboard music, and the church organ was esteemed above all otherinstruments. The three leading composers for the church organ in the early 18thcentury were Dietrich Buxtehude (c. 1673-1707), the expatriate Dane who hadspent most of his working life in Germany as organist and choir master at theMarienkirche in Lubeck; Georg Bohm (1661-1733) from Thuringia, and organistat Luneburg; and Johann Pachelbel (1653-1706) who had written his now famousCanon in the year J.S. Bach was born, and who had just recently passed away inhis birthplace of Nuremberg. This great flowering of keyboard music was centeredhere in northern Germany where the modern style keyboard itself had been born.

About 100 kilometers northwest of Jena was the town of Halberstadt. Here,two and a half centuries earlier, on February 23, 1361, organ builder NicholasFaber had completed the now famous Halberstadt organ. The organ featured threekeyboards for various modal scales and accidentals. The upper of the threekeyboards featured a diatonic major scale (Do, Re, Mi ...) in C, with five"accidentals" slightly raised. and behind the main keys. At the time perhapsnothing much was thought of this arrangement of digitals. There were manymodes, many tunings, and many keyboard designs in use. There were keyboards ofseven tones per octave, twelve tones, seventeen, nineteen, and many others, allarranged in various modes of various scales, and supporting certain naturalharmonics and certain temperaments.

At the time of the Halberstadt organ, and later, on the day of the tuningshowdown in Jena, there was no unequivocally correct or proper way to cast a


scale and make music. Naturally, there were advantages and shortcomings to all ofthis. One of the advantages of these many systems was the use of naturalharmonics for singers trained in and sensitive to precise harmony. A disadvantagewas that all instruments could not necessarily play with each other. Lutes, forexample, were often tuned in an approximately equal temperament required by thefrets, keyboards were tuned to varieties of meantone or well temperaments, and acappella choral groups performed in pure harmonic scales.

Inthe search for a unified and simple system, instrument builders innorthern Germany and Prussia had settled on the Halberstadt keyboard with itsseven diatonic major digitals and its five accidentals. The major scale in C hadnow, by custom, been represented by white keys, and the five remaining chromaticnotes by black keys, still raised and to the rear. In the streets of Jena, in the fall of1706, no one knew that this keyboard would dominate musical perception in theWest for the next four hundred years. No one knew that the winner of the day'scontest would have the most to lose, and that the loser would, at least for severalcenturies, win. The singers who were there to judge the tunings had only onepurpose, and that was music. They were there to judge the purely musical benefitsof each system.

"Keyboard temperament" was in a transition phase from meantone to welltemperament. Meantone was always tuned to a key, or tonality, usually C. In theearliest traditional manner, the E was tuned as a just third (five-fourths thefrequency of C), and D was tuned as the "mean" halfway between C and E. Thewholetone, D, was also the "mean" or mid-point between the just Pythagoreanwhole tone and the flatter "minor" wholetone necessary to achieve good harmoniesin certain modulations. From this, meantone temperament later took its name,although in the 16th century it was simply called "keyboard tuning." The systemresulted in two wholetone scales on C and C#, but the thirds varied in size, thefifths were flat, and certain thirds and sixths were unusable "wolf' notes fromwhich musicians stayed away.


In 1523 Pietro Aaron had described in Thoscanello De La musica' themeantone temperament of his time. The major thirds C-E and E-G# were made"sonorous and just," but the fifths (C-G and G-D) were "a little flat," and A wastuned to make the fifths (D-A and A-E) equally flat. Then C# and F# were tunedas pure thirds of A and D. This tuning resulted in a variety of major and minorthirds. Some minor thirds were nearly just, and the flatter minor thirds were nearwhat today we would call blues minor thirds, or "septimal" thirds based on thepure seventh harmonic. Aaron's tuning had eight pure or just major thirds, andfour "wolf'thirds almost a quartertone sharp. The fifths were slightly narrow, andthe fourths slightly wide.

By the time of the tuning contest in Jena, meantone had evolved to the pointthat the thirds were smoothly and progressively sharpened so that they were nearjust in the common or "natural" keys, (C, F, G, A, Bb etc.), but were still discordsin the "distant" keys (C#, 0#, etc.). The fifths were flattened as much as "the earwill bear." The advantage of this tuning was that triads in the more common keyswere very good, and some modulation was possible. Each key in this tuning had itsown character since the semitone sizes were varied. Therefore, a piece of musicwas written for a particular key, and transposing to a new key could render thepiece unacceptable. Within a piece of music modulation was restricted by the wolfintervals.

These restrictions led tuners to experiment with temperaments in an attemptto create a tolerable tuning system that would allow free modulation. The acousticcompromise was to slightly widen the thirds in the common keys. These newtunings were known as «welt temperaments," which Johann Sebastian Bach latermade famous with the 48 pieces ofhis Well Tempered Clavier. In 1691 AndreasWerckmeister, the organist at Quedlinburg, near Halberstadt, publishedMusicalische Temperatur, a treatise on musical tunings including meantone andwell temperament.' He described a well tempered system that included eight purefifths, and thirds that were progressively sharper as one moved through the keys.The thirds of "distant" keys were sharp, but "tolerable.''16 Well temperament had


achieved the goal of being able to support free modulation to all keys. The thirdswere slightly wider than meantone thirds in the common keys, and increased insize until the largest thirds (usually in B, F#, and C#) were about 115of asemitone wide, about the size of a Pythagorean third resulting from a series ofpure fifths. Nevertheless, the orderly progression in the size of thirds wasconsidered an aesthetic advantage. Since the semitones varied in size, each key stillhad its own unique character, as with meantone, so as musicians modulated theywould take advantage of the unique character of each key. Although it waspossible to transpose a piece from one key to another, the results would certainlysound different in the new key. Nevertheless, the meantone wolves had beeneliminated, albeit at the cost of more compromised thirds and fifths, and freemodulation was possible.

It was a variety of this well temperament tuning that choir master NicholasBach tuned by ear in the Church of St. Michael in Jena on this day in 1706. On hisshoulders he carried the tradition of harmonic singing discovered and enriched bythe ancients of every culture on the earth; as refined by the Chinese, theBabylonians, Egyptians, Greeks, and the nomadic Semite tribes; the sacredharmonies chanted in the Gregorian Liturgy, the corporal harmonics sung inAthenian theaters, through to the radical dominant sevenths of Monteverdi'sItalian opera; these harmonies that were not quite achievable on the variouskeyboard systems without compromising free modulation; these mysteriousharmonies that could be sung, but would not yield to any mechanical system; thesesweet harmonies that everyone could hear, but no one in 1706 could quite explain.

His adversary on this day was one of his brightest students at the university,the young, precocious Johann Georg Neidhardt (1685-1739), whose first book hadjust been published, Die beste und leichteste Temperature des Monochordi (TheBest and Easiest Temperament of the Monochord). Neidhardt, 21 years old, hadread all the theorists, from Ptolemy to Mersenne and Werckmeister. He advocatedthe system of twelve equal semitones, a system that had been a theory for over athousand years, The system was impossible to tune by ear without knowing and


listening for the precise number of beats in the mis-tuned intervals, a skill notpracticed in the 18th century. However, mathematicians had made discoveries thatpromised to make such a tuning realizable on keyboard instruments.

Neidhardt represented the emerging science of mechanics, the rationalisttradition of Descartes and Kepler, the astounding calculus ofLeibniz, and themechanical laws of Newton. The age of science was in full swing. Only twenty-two years earlier, in 1684, Gottfried Wilhelm Leibniz (1646-1716) had publishedhis system of the infinitesimal calculus in Leipzig, a discovery that would havegreat impact on music. Newton, who published his version of the calculus inLondon in 1687, had been working on this new mathematics for a number ofyears, and it was because of this, as well as the fact that Leibniz was known tohave been in communication with certain individuals in Newton's circle, that thetwo became embroiled in a rather undignified academic controversy over who wasthe true discoverer of calculus. In any case, the new mathematics of plottingvalues on exponential curves had immediate application for instrument makerstrying to define equal semitones along the frequency curve of the octave.

The theory itself was first given voice in about 330 B.C. by a student ofAristotle, Aristoxenus of Tarentum. Aristoxenus had been a follower of thePythagorean Harmonist school, but had been so baffled by the Pythagoreancomma, by the fact that 12 consecutive pure fifths landed sharp of seven octaves,that he proposed distributing the comma among the 12 fifths to make them equalto an octave. Seven hundred years later Ho Tcheng-tien (AD 370-447) gaveapproximate string lengths for a scale of 12 equal semitones. His mathematicalderivations do not survive, but his maximum deviation from 20th century equaltemperament was less than III 0 of a semitone, and it is possible that he achievedhis results by trial and error.' In 1596 Chinese prince Chu Tsai-yu gave stringlengths for equal semitone temperament which were correct to nine decimal places.Without the aid of calculus this mathematical feat required extracting the 12th rootof numbers containing as many as 108 zeros. French theorist Marin Mersenne had


given accurate approximations of equal semitone fret positions for the lute in1636.

At Quedlinburg, Andreas Werckmeister, who had written on the currentpractices of meantone and wen temperaments, had also investigated the equalsemitone temperament, and had used the new calculus to actually plot out precisestring lengths and fret placements, marking the intervals that no ear had preciselyfound in five millennia of musical investigation. Neidhardt studied Werckmeister,wrote his book, and is credited in history as the co-founder of the formula for theequally tempered semitone: the number, which multiplied by itself twelve times,equaled two. The equal tempered wholetone was the number which multiplied byitself 12 times, equaled four. Instrument makers could thus divide an octave intotwelve equal parts, and map those equal parts to the frets of a lute or the digitalkeys of the organ.

Werckmeister and Neidhardt held in their hands the new technology thatwas going to make playing music more convenient. Everyone knew that thesetwelve precise semitones compromised the purity of harmony. Over this fact therewas no debate. Everyone knew the equal tempered thirds were sharp, the fifths justslightly flat. Singers knew that innatural harmony all semitones are not createdequally, that the distance between a major and minor third, for example, is not atall the same as the distance between the major third and the fourth. The singersknew that in modulating by a series of pure fifths, one would never land on theoriginal tonality, and that to do so required the little comma jumps. Twelve equalsemitones were the wave of the future, not because of their musical merits, butbecause of their convenience. Most musicians of the 18th century were dead setagainst such a system. Scottish theorist Alexander Malcolm, born in 1685, thesame year as J.S. Bach, wrote ''That tho" the Octave may be divided into 12 equalSemitones ... 'tis impossible that such a Scale could express any true Musick."Malcolm added that the temperaments derived "not from the Nature of the Systemof Musick itself, but the Accident of limiting it to fixt Sounds.,,8


And thus the terms for the showdown in lena were set. The traditions ofharmonic singing were being challenged by the latest mathematical musictechnology. At the center of the conflict was the 12-semitone keyboard, with itseminent functionality and its inflexible tones. The musical issue was thetranslation of harmonic tradition to the fixed tones of the keyboard. Nicholas Bachrepresented the current practice of maintaining as much natural harmoniousness aspossible in the common keys, and retaining the character of individual keys, whilestill allowing free modulation, Neidhardt represented the final and completecompromise of pure harmony in favor of the convenience of keyboards and frets,the new era of music in which every semitone and every key would be identical.

The choristers lingering in the church or outside on the street would haveheard Nicholas Bach tuning his set of organ pipes. We do not know what standardpitch he might have used as a starting point. German church organ pitch at thattime ranged from A=393 Hz in Strassburg to A=495 Hz in Holstein, a differenceof more than four semitones. The tuning fork had not yet been invented, and tunersused their own local standard or perhaps their own best sense of pitch. In any case,Nicholas Bach probably began with a middle C somewhere between 235 and 300Hz. From there he would have tuned a pure octave below, and from there a fifth,G, that was about 5% ofa semitone flat of the pure harmonic. Then he would havetuned another fifth, D above that G, also flat by a similar amount, a pure octave Dbelow, and then another slightly flat fifth, the A above, and again the E above that.The flatness of these fifths he would have judged by ear and experience, knowingthat the flatter these fifths were tuned, the more harmonious would be the thirdsand sixths. He would then have carefully tested certain intervals such as the sixthbetween C and A, the tenth between the lower C and the E, the third betweenmiddle C and E, and the minor triad A-C-E. In each case he would have checked.the color effects of various chords and intervals, and made slight adjustments tobring these intervals into the best possible relationship based on his musicalexperience and aesthetic judgment. He would have made sure that the diatonicmajor thirds were close to pure. After tuning the entire keyboard, he would have


checked the potential wolf notes, such as G# - Eb, to make sure they weretolerable. He would then have tested all the major triads progressing by fourths orfifths to determine if the key colorings changed in a gradual and even manner. Theresult would have been the highest and most refined version of well temperament.

Neidhardt had calculated the precise organ stops using the 12th-root-of-2formula. His job was much easier, more precise, and entirely objective, requiringno listening to intervals nor aesthetic judgments. Neidhardt himself admitted in hiswritings that "equal temperament carries with it its comfort and its discomfort"No one doubted that this was the case. The advantages were more practical thanaesthetic. The predominate musical advantage was the ability to transpose amusical piece to any key without changing its nature. Modulations were exactlyeven, without variety and key coloring. The major thirds were all equally sharp,and the fifths all equally flat. This was a tuning that was indeed in step with theemerging scientific paradigm.

At last the choir was called in and assembled in the loft. They sang aprogram of material in several popular keys. They modulated through complexpieces, sang the common harmonies of the more simple hymns, and performedeach piece in both tunings. Musicians, students, and the curious denizens of Jenacrowded into the church, drawn by the beauty of the musie as well as by thepromise of a good contest.

And the conclusion? On this particular day in 1706 the traditional welltemperament of choir master Nicholas Bach won. The choristers claimed to preferwell temperament, with the orderly key colorings, even in the distant keys. Singinga variety of thirds posed no particular problem, and the relatively sweet triads inthe common keys were considered a great advantage.

In fact, equal temperament would not entirely insinuate itself into mostwestern music for another 200 years, but to some extent the die had been cast. Thegrowing popularity of keyboard and fretted instruments would eventually makeequal temperament the most practical tuning for instrument makers, although notthe most universally loved among musicians.


Three years after the tuning showdown in Jena, in 1709, Italian harpsichordmaker Bartolommeo Cristofori made four gravtcembali col piano e forte, the firstpianos. Keyboard manufacturers clearly favored the simplicity of the equaltempered scale. The tuning fork was invented in 1711 by King George's RoyalTrumpeter John Shore, partially in response to the need of piano tuners who foundthey could not tune the equal tempered scale precisely by ear because of theimperfect intervals.

Nevertheless, thirty years after equal temperament was calculated byWerckmeister and Neidhardt, both Handel and Bach continued to play in eitherjust tuning, in meantone, or in well temperament. Handel often played a split-keyorgan with 17 or 19 tones per octave to allow for alternatives to the wolf notes.Bach would often retune his clavichord between pieces when he changed the key.He was very adept at this, but many musicians using fixed tone instruments foundthis to be a considerable handicap.

There is a misconception that Bach supported equal temperament becauseofhis famous 1722 collection, Das wohltemperierte Klavier (The Well TemperedClavier). Bach's clavichord was not equally tempered, but well tempered, like thetitle of the work says. The 48 pieces, two ineach major and minor key, werewritten to show the character of each key in this temperament, the effect beingcompletely lost in equal temperament. One can experience this by tuning akeyboard to well temperament, and then transposing Bach's Prelude and Fugue inC-major to C#-major, and vice-versa. The results will clearly show what themaster was up to, and that the pieces were written for the nuances of each keywith its particular coloring. Bach's theoretical framework was for pure intonation,and his music was written to be played in pure intonation, either by altering thetuning of the keyboard for each piece, or by using a flexible temperament thatallowed pure tones in most popular keys.

As instrument manufacturers pushed equal temperament, they were metwith considerable resistance by musicians prior to the 20th century. In 1749 Dr.Robert Smith in England referred to equal temperament as an "inharmonious

Justomc Tuning Inc. 15

system of 12 semitones, [producing a] harmony extremely coarse anddisagreeable. ,,10

In the 19th century William Pole complained in his book The Philosophy ofMusic that "The modem practice of tuning all organs to equal temperament hasbeen a fearful detriment to their quality of tone. Under the old tuning an organmade harmonious and attractive music ... Now, the harsh thirds, applied to thewhole instrument indiscriminately, give it a cacophonous and repulsive effect."

Herman Helmholtz observed that "When I go from my justly-intonedharmonium to a grand pianoforte, every note of the latter sounds false anddisturbing." and added that "These are unpleasant symptoms for the furtherdevelopment of art. The mechanism of instruments and attention to theirconvenience, threaten to lord it over the natural requirements of the ear, and todestroy once more the principle upon which modern musical art is founded. 11

A leading modem authority on historical tunings, Owen H. Jorgensen, haswritten that "temperament developed because of the mechanical impracticability ofconstructing instruments with more than thirteen key-levers to the octave whichalso caused increased performing difficulties. Tempering was easier." He adds that"the acoustical truth is that inhistory as one quality was gained, another quality orvirtue was sacrificed and lost. The gain for equal temperament is a homogenizedneutral gray coloring that is completely dependable ...," and that ''Western musicnow exists under the dictatorship of this one homogenized temperament. ,,12

The fact is, in Jena in 1706, nothing at all was settled. Today we are facedwith the tuning question just as surely as it was faced three hundred or threethousand years ago. The nature of sound is unchanging, and the challenge isalways the same: to understand sound; to apply the nature of sound to instruments)and to use those instruments to make music.

We are now entering a new era of music because we have a whole newgeneration of musical tools. Just as keyboards and frets were the new musicaltechnology of the Renaissance, synthesizers and digital processing are the newmusical technologies of the future. These: technologies do not replace the older


technologies, they simply add new capabilities. It is worth remembering thatkeyboards and frets were once the new digital instruments, because they digitizedpitch selection at a time when string and wind instruments had complete freedomof pitch selection. Keyboards and frets did not replace the old musical instruments,but they added new capabilities. For three centuries, keyboards have drivenwestern music. The digital revolution will change this to some extent, but we donot yet know how all these changes will occur.

Pitch selection, however, is one change that is already well under way. Wenow have the capability to map a wide selection of pitches to a single key. This isthe breakthrough that promises to liberate keyboards and fretted instruments fromthe limits of their fixed tones.

But let us go back again to the very beginning of musical understanding, sothat we can see how our conventions were developed. It is helpful to understandhow sound works, why instruments were designed to support the nature of puresound, and how certain musical systems were designed to make the most of theinstrwnents available at the time.

The showdown in Jena has not even yet been settled, but has merely passedfrom one generation of musicians to the next. Now, at the dawn of the 21stcentury, we are greeted once again by these fundamental musical issues. Thefollowing historical survey will give the reader some understanding of andappreciation for the musical tuning dilemma that has faced musicians from ancientChina and Greece, to Bach and Mozart, and still to our present day.


2

The first musical scale

The earliest human music was almost certainly singing or chanting anddrumming, a part of human culture that predates any archeological record.

The oldest evidence of human music instrumentation is a flute fragment with fourvery precisely drilled holes in the thigh bone of a bear, discovered in Slovenia, anddating from 82,000 to 43,000 years ago, at the time of the Neanderthals, beforemodern Homo sapiens. According to Bonnie Blackwell at Queens College in NewYork, the fragment "looks very similar to the bird bone flutes of much laterperiods. ,,13

The hunting bow was developed some 20,000 years ago, and may have beenthe first string instrument, plucked to vibrate the string, and bent to change thepitch. The earliest wind instrument was probably a form of the didgeridoo fromMelanesia, a simple hollow log manipulated by the lips and vocal cavity. Bambooflutes were common in China 6,000 years ago, and Egyptian musiciansaccompanied the voice with harps, flutes, lyres, and even an early double clarinet.

18 1'heStory oj Harmony

Four thousand years ago there were trumpets in Derunark, and the nomadic Hittitetribes carried lutes and other stringed instruments.

A flute fragment that is 42,000 to 82,000 years old. The flute wasmade by Neanderthal musicians from the thigh bone of a bear.

Music is as old as any human art or craft. Knowledge of pure harmonicsis at least 4,500 years old, perhaps much older.

Most early human literature is song, the history of tribes passed fromgeneration to generation. The English Beowulf, the ncar eastern Gilgamesh, andthe Hitidu Ramayana are all songs, the words of which have been passed down,the original music lost. Song in this early context was the cultural thread by whichmoral and practical teachings were communicated. The elements of verse, rhyme,rhythm, and melody were all memory aids that insured the accurate transmission


of cultural knowledge. The earliest extant sources of the Old Testament of theBible contain musical notation for the singing of verses. In the old Anglo-Saxonpoem Wtdsith, the Song of the Wandering Minstrel, the poet declares: "WhenScilling and I with clear voice raised the song before our victorious lord, loud tothe harp the words sounded in harmony ... ,,14

We do not really know how ancient the art of harmony might be. We thinkof early music as monophonic, consisting of a single melodic line withoutharmonic embellishment. However, the historical record is incomplete, and itseems likely that over thousands of years of singing, humans experimented with allsorts of harmony, unisons, octaves, fifths, fourths, thirds and sixths long beforethese relationships were given a name. It is certain, however, that the earliestrecorded knowledge of human sound indicates an understanding of a fundamentaltone to which other tones were related. We tend to think of harmony evolving aftermelodic forms, but the historical record indicates that the earliest melodic scaleswere derived from a recognition of the harmonic qualities of sounds. The earliestknown scales all reflect the harmonic relationships of octaves, fifths, and fourths.These early musicians discovered that certain tones, when sounded together,created pleasing sounds, and these relationships became known as harmony.Therefore, in a very real way, melody evolved from harmony, although the processwas more likely auditory and instinctual, rather than theoretical. Sound's inherentharmonies were simply strung out into scales and melodies.

Today we know something of the physical structure of sound, and we knowthat these musical relationships are entirely natural. We know that fifths andoctaves and thirds are all supported in the harmonic series (or overtone series) ofafundamental tone. A pure harmonic fifth is as much a part of nature' as sounditself. Early singers discovered these relationships, eventually discovered theproperties of these relationships, and later gave them names. For a more detailedexplanation of how scales evolved from the harmonic series, see the appendix, TheDerivation and Use of Harmonic Scales. Every musician ought to be aware of theharmonic series, since it is the foundation of music. It is as simple as 1,2; 3. A


tone is sounded because something vibrates, a string, a vocal chord, a column ofair in a hollow tube. When a string vibrates at a given frequency, say 100 cyclesper second, it also has a secondary vibration at 200 cycles per second, and theseries continues at 300,400,500, and so forth. The higher harmonics tend to growweaker, but on a good cello string, for example, one can pick out the first 12 or 14overtones.

In the history of human music, the "octave" is the most ubiquitous of allharmonic relationships. We take it for granted, but in its nature it is no differentfrom other harmonic relationships. An octave is the second harmonic after thefundamental tone, and it vibrates at twice the frequency of the fundamental tone. Ifthe fundamental tone is one, the octave has a relationship of two- to-one, which wewrite:

2/1

This relationship was discovered by all the earliest cultures who developedmusical scales. The next relationship they discovered was what we now call the"fifth," the third harmonic. This harmonic has a 3-to-l relationship with thefundamental, and a 3-to-2 relationship with the octave, This 3/2 is a pure fifth, apure G in the key of C, etc,

The earliest known musical scale was derived from these natural harmonics,a five-tone scale from China attributed to Ling Lun, musician in the court oflegendary emperor Huang- Ti in 2,700 B.C, 15 We cannot know if Ling Lun heardovertones, but even ifhe did, he had no real understanding of the harmonic series.Nevertheless, he clearly understood the concept of a fundamental tone and knewthat the tones that sounded pleasing in relation to that fundamental tone hadprecise numerical relationships with it. He built his scale on a series of 3/2harmonics. Ling Lun fashioned a set of bamboo pipes, or "Iii" measuring 81, 72,64,54,48, and 40,5 units, recording the earliest human understanding ofharmonic sound, The pipes were closed at one end, and a player would sound them


by blowing across the open end. His pipe of 81 units is the fundamental tone, ortonic, and is exactly 3/2 times the length of the pipe of 54 units. By continuing tomultiply by 3 and divide by 2, Ling Lun calculated the lengths of the other pipes,and discovered the series of tones that form this early pentatonic (5-tone) scale.

This natural pentatonic scale, based on the harmonic of the fifth (3/2), wasalso discovered or learned in ancient Java, Sumatra, New Guinea, North America,Japan, Babylon, and Greece, and can still be heard in modem oriental music andGaelic airs. Inour system of notation, in the key of C, this scale is:

C,D,E,G,A,C

These are, of course, the tonic, a whole-tone, a third, a fifth, a sixth, and theoctave. Because the scale relies only on 3/2 for all the calculations, both the E andA, the third and sixth. are slightly sharp of modern just intonation. What we call apure musical third is based on the fifth harmonic (5-to-l) of the fundamental, andhas a 5/4 relationship with the tonic. Ling Lun did not yet know this, and his thirdis based on a cycle of pure fifths. It is sharp of the pure third by what later becameknown as the comma of Didymus. The Ling Lun sixth is sharp by the sameamount. The numerical relationships of Ling Lun's scale were as follows:

1/1 9/8 81164 3/2 27/16 2/1

It is unlikely that the sharp third or sixth were ever actually sung by voicesunless they were following accompaniment by an instrument tuned in this manner.Thus we see that the difficulty of creating instruments that could imitate thenatural intonation of the voice is as old as instruments themselves. Any instrumentthat fixes the pitch of the notes, such as Ling Lun's set of pipes, is going toencounter the difficulty of matching the flexible voice. The natural instinct of thevoice is to follow the harmonic series. Ling Lun's sharp third and sixth do not


follow the harmonic series, but rather are derived from the series of pure fifths.Therein lies the rub, which we are still dealing with nearly five millennia later.

Octaves and fifths were known also to the Assyrian-Babylonian culture, andto the Egyptians. The Babylonians may have used an entire 12-tone chromaticscale by 2000 B.C., based on a series of 3/2 fifths, and may have learned it fromthe earlier Sumerians." We do not know if the Mesopotamians discovered theseconcepts themselves, or if the information filtered across the Eurasian steppe, butby 750 B.C. the Babylonians were writing cuneiform notation for hymns using 5-tone, 7-tone, and 12-tone scales. The Greeks would later name certain scale modesafter the nomadic Middle Eastern tribes of this period, the Phrygians and Dorians.How much these peoples actually understood about the fundamentals of music wedo not know, but it is possible that some theoretical discourse circulated amongthe civilizations of the first two millennia B.C. The Greek theorists would borrowfrom and build upon this body of knowledge, and they would make furtheradvances in the tuning of musical scales.


3

Pythagoras: music & numbers

By 700 s.c. the Greeks had developed a diversity of short song styles andscale modes. Itinerant singers, or "Rhapsodes," traveled the countryside

with 5-string and 7-string kitbaras or lyras, the strings tuned to various modes,and the tuning based on pure fifths. In 650 B.C. famed Kithara player Terpanderwrote Compositions for voice and string accompaniment, and is credited as theoriginator of the Greek mixolydian mode. 17

The Greek scholar Pythagoras (c. 540e510 B.C.) traveled to Babylon,Alexandria, and Chaldea, probably learning the numerical proportions of musicalharmony in his travels. 18 We have no first-hand account of what Pythagoraslearned from others and what he may have discovered himself, but he was clearlya brilliant mathematical innovator, and the leading musical theorist of his time;


Pythagoras is credited with inventing the monochord, a single stringstretched over a moveable bridge. By dividing the string into proportional parts,Pythagoras was able to experiment with two-note harmonies, and determine the

- ..... IIi. - -_ ---.-'

(---E_li __ -_li-__ ~l -y

---

The monochord ofPytbagoras (top), and the natural harmonicsof a vibrating string: 111, 2/1, 3/1, and 4/1.


numerical relationship between them. By moving the bridge one-third of the wayalong the string, he divided the string into two parts, one twice as long as theother, This was, of course, the octave, 2/1. Likewise, he could divide the stringinto a 3/2 fifth, and perhaps his own discovery, a 4/3 fourth. ( ''Fourth,'' and"fifth" are modem terms that Pythagoras never used.)

Pythagoras was smitten by the perfection of these numerical relationships.Harmony was not a rational calculation, but was rather a natural observationdiscovered by the acute ears of musicians. Yet these perfect harmonies wereshown to have exact munerical relationships, This was a revelation forPythagoras. He saw a mystical and universal beauty in the perfect numbers of themusical intervals. Aristotle wrote of this in his Metaphysics 300 years later,speaking of the Pythagoreans: "they say that the attributes and ratios of themusical scales were expressible in numbers ... [and] they supposed the elementsof numbers to be the elements of all things, and the whole heaven to be a musicalscale and a number."

pythagoras founded an entire school of learning and a cosmological viewaround the perfection of numbers, a view that spread among the Greek city-statesand lived long after he himself passed away. His students were taught to live apure moral life as well as to seek an understanding of the natural laws of heavenand earth. The perfection of the pure harmonic 3-to~2 musical fifth was at theheart of his philosophy. He postulated that this perfection was a sign of thecreator's hand, and a symbol of the perfection of the universe. Westernmathematics starts with Pythagoras, and for Pythagoras mathematics started withthe perfect musical fifth. We could say then that western mathematics starts withmusic, or at the very least was motivated by music at its inception. Music andnumbers are therefore eternally linked, not due to convention or theory, butbecause of the very nature of sound.

Pythagoras's pentatonic scale was identical to Ling Lun's with the thirdsand sixths derived from a series of fifths. However, Pythagoras took this series outeven farther, and made his most important musical discovery. Pythagoras


discovered that a series of twelve pure fifths almost landed on the same tone as dida series of seven octaves, thus creating - almost - a cycle of fifths. Ironically, hisgreat discovery was neither the fifth, nor the cycle, but the "almost." Pythagorasnoticed that the twelve fifths did not land precisely on the octave, and thisdisturbed him. The tiny discrepancy seemed to shatter the perfection he thought hehad perceived in the harmonic proportions. He had fashioned an entire philosophyaround that perfection, and suddenly his world view had a kink. He calculated theamount that the series of fifths was sharp of seven octaves, and he arrived at thistroubling interval:

531441/524228

This unwieldy fraction, which we now know as the Pythagorean comma, isabout a quarter of a semitone, a little interval with huge implications. PoorPythagoras. He got trapped inside his own paradigm. Had he been able to see thatthis was simply the natural order of things, that a series based on 3 could neverand would never match up with a series based on 2, that is, that a series of fifthswould never coincide with a series of octaves, he could have saved himself muchaggravation, and perhaps saved future musicians a great deal of trouble. ButPythagoras presumed that the twelve fifths should match up at the seventh octave.The perfection of the universe implied that tlris should be so. The nagging littlequarter of a semitone drove him to distraction, and the assumption that it shouldbe eliminated sent music theory on a millennial quest after a scale that would makea cycle of fifths fit inside an octave.

For musicians who may think they don't understand the math. consider this:imagine a waltz rhythm, 3 beats to a bar. Inside each beat, let's have triplets, sonow you have 9 triplet-beats in the bar. Okay? Now imagine a 4/4 rhythm, 4 beatsto a bar, and inside each beat let's sound the back beat (eighth notes), so we have8 half-beats in a bar. Ifyou assumed that 9 was supposed to equal S, you'd have aproblem, because it never will. No matter how many times you multiply by 3 you


will never have a number that is divisible by 2. Never. That's just the way it is.For this same reason, a series of pure fifths will never match up with a series ofpure octaves.

The commas also appear when attempting to modulate in a scale of pureintervals. Singers, it was known even in Pythagoras's time, could change key, andinstantly sing the new intervals by ear, but the fixed strings of the kithara,precisely tuned to one key, had to be retuned for the new key, a cumbersomemaneuver. In modem terms, the problem is this: when singers are singing pureintervals, and then modulate from the key of C (the fundamental tonic representedby the ratio 111) up one step to the key of 0 (the wholetone, 9/8 of e), some of thenew tones in the key ofD will be different. For example. the pure F note in the keyofC is not the same as the F note in the key ofD. In the first case the F is afourth, 4/3 ofC, and in the second case it is a minor third, 6/5 ofD, which is 9/8ofC. Therefore, the F in the key ofD is:

6/5 x 9/8 = 54/40 = 27/20

This is al1 :fifth grade math today, multiplying fractions. The two F tonesare not equal because 4/3 does not equal 27/Z0. The two tones differ by the smallfraction of 81/80, that we already know as the comma of Didymus. Modem equaltemperament splits the difference and says all F notes are the same, Thiscompromise, however, is the destroyer of pure harmony, and Pythagoras did notrest well with it.

Pythagoras and his followers were perhaps fooled by the smallness of thesenecessary adjustments. They erroneously asswned that the tones of different keysshould be interchangeable, since they were so close. It is quite phenomenal that awell trained musician can make these minute adjustments in tuning by ear, and ithas been troubling that fixed-tone instruments cannot.

We will do well to remember that the theorists like Pythagoras and LingLun only calculated and confirmed the musical ear of the singers. They did not

28 The Story a/Harmony

invent the pure intervals out of numbers. 'The numbers were revealed to them whenthey calculated precise methods of tuning the harmonies that singers discoverednaturally. Although the tones used in a particular musical scale may vary fromculture to culture, and from style to style, the mathematical ratios of the tones, theperfect intervals - 211,3/2, etc. - remain constant. It is one of the great marvels ofmusic that such diverse cultural or personal expression can arise from these simplerelationships. Indian ragas, classical sonatas, and modern jazz are all based onthese intervals, regardless of whether or not the tuning of the true intervals hasbeen tempered. In fact, for example, the flat "blues seventh" is a tone that matchesthe seventh harmonic, quite flat of the equal tempered minor seventh. Musiciansjust naturally find these harmonics that are reinforced by the harmonic series ofthe fundamental tone. On the other hand, our modem keyboard :fifth was notdiscovered by any musician's ear. It is simply a slightly.altered pure fifth, and itsmusical meaning is 3-to-2~ the third harmonic of the fundamental tonic.

Pythagoras may have died a discouraged perfectionist, but in his memory aschool ofHarmonists flourished. Greek theater and choral music was at a zenith,with song, poetry and dance all a part of a typical performance.


4

Harmonists east and west

BOth Greek music and music theory flourished in the third and fourth centuryB.C. At the theater, the chorus was accompanied by the stringed kithara and

lyra, and also by the aulos, an oboe with a double reed and up to 15 holes. Theaulos often had two pipes, one being a drone (the fundamental 111) and the secondplaying the melodies. Thus we see that harmony was indeed a part of the earlyGreek music that we think of as "monophonic." It is true that the primarystructure of this music was melody, but harmony was understood, sung, andplayed.

Pythagoras had added an eighth string to the kithara to assist in modulationand changing mode. In457 B.C. Phrynis won the music award at the GreekPanathcnaic competition with a kithara to which he had added a ninth string tofurther facilitate changing mode. However, when Timotheus of Sparta added threemore strings to make twelve, he evidently stepped beyond the bounds of goodtaste, for he was viciously mocked by the poet Pherecrates for introducing "weirdmusic," and unceremoniously booted out of Sparta. 19 From this little incident we


see that the cultural inertia resisting innovation in musical instruments is nothingnew. The "traditionalists" always forget that their tradition was once a novelty.Timotheus later settled in Athens where his innovations were tolerated, and wherehe received encouragement from another innovator, the great dramatist Euripides.

At this time a group of Pythagoreans began to call themselves the"Harmonists." In Greek mythology Harmonia was the daughter of Venus andMars. Cupid was her brother. She had a son and four daughters with her husbandCadmus, but her grandmother, Juno, mother of Mars, despised Venus and took it

F-1 F.. -".'.'" '" ). ..... ··........,·,·H

. 1

\'( 7 '\ ,

I

II

-, I )'\. ./

•• t I. . . . . . ... •t ..~- ... \

f..

.1

A Greek kithara, Pythagoras added an eighth string to accommodate variousscales and modes, Phrynis added a ninth and won the Panathenaic music

award, but when Timotheus added three more strings to make twelve he waskicked out of Sparta for making "weird music."

out on poor Harmonia by persecuting her children. Harmonia and Cadmus were so


distraught by this that they asked to be sent away to the Elysian fields, restingplace of the virtuous. Harmonia personified order, perfection, and peacefulness.The Harmonists took her name to symbolize those qualities in the pure harmoniesof music.

The greatest scholar among the Harmonists was Archytas from the Greekcolony in Tarentum, Italy. He was the first in history to notice that the interval ofthe third that singers naturally sang was not the sharp Pythagorean / Ling LUllthird derived by the series of fifths (81/64) but was rather the pure harmonic tonerepresented by the relationship 5/4. Archytas was not aware of the harmonicseries, but his pure third was the fifth harmonic in that series." The singersinstinctively sang this harmonic tone unless forced sharp by an instrument tuned tothe Pythagorean third. Archytas noticed this, calculated the relative string lengthson a monochord, and recorded for history the pure just intonation third.

Archytas did not stop there with his observations and experiments, and lateradded a tone based on the seventh harmonic, an alternative wholetone about aquarter of a semitone sharp of the Pythagorean 9/8, a tone represented by the ratio8/7. Thus, he is credited as the first inhistory to recognize and document thenatural tones based on the fifth and seventh harmonics.

Greek musical scale theory at this time was based on the concept of thetetrachord, the interval of the fourth, and the division of this interval into threesubintervals by the placement of two other tones between the tonic and the fourth.These four tones were called the hypate,parhypate, lichanos, and the mese. Thefirst tone was always the fundamental L'I, and the mese was always the pureharmonic fourth 4/3. The four tones created three intervals, and these were alwaysrepresented as ratios of whole numbers since the Greek Harmonists understoodthis to be the true expression of a harmonic interval. The two internal tones gavethe tetrachord its particular quality. When the three intervals were two wholetonesand a semitone the tetrachord was called diatonic. When the three intervals were aminor third and two semitones the tetrachord was chromatic. A major third andtwo quartertones created an enharmonic tetrachord. There were many different


sizes for a "wholetone," for a "semitone," or "quartertone," and also for either a"major third" or a "minor third." These terms were general, not specific. Forexample, a wholetone could be 9/8, the sharper 817, or the flatter 10/9. A typicaldiatonic tetrachord might be composed of the three intervals:

16/15 10/9 9/8

These are a semitone, a small wholetone, and a Pythagorean wholetone.These three intervals would create a tetrachord scale with the following four tones:

111 16/15 6/5 4/3

These ate the tonic, a semitone, a minor third, and the fourth. Musiciansunfamiliar with seeing intervals and musical tones represented this way may findthe fractions (or ratios) confusing, but it is worth attempting to understand theseharmonic expressions, because they are the very foundation of the musical arts. Ina more familiar language, the four note scale above, in the key of C is:

C C# Eb F

The intervals are then clearly recognized as a semitonc and two tones. Inour modern equal temperament all the semitones and tones are the same size, butthis is not the case in just intonation, or pure harmonic music, and was not thecase with the early Greek Hannonists. The tetrachord above (with various sizes oftones and semitones) is found in Babylonian, Greek, Balinese, East Indian, Arabic,and African music. For musicians interested in deepening their understanding ofthis fundamental scale theory, or expanding their use of scale options, seeDivisions of the Tetrachord by John Chalmers. 21

The Harmonist Archytas was the first to define tetrachords in all threeclassical genera, the diatonic, chromatic, and enharmonic. Octave scales were


constructed by sticking two tetrachords together with a disjunctive tone betweenthem. For example, ifwe took the tetrachord above, added it to itself with awholetone in the middle, we would have:

C C# Eb F - G Ab Bb C

This is the Greek Dorian (Ecclesiastical Phrygian), and also the East IndianHanumat Todi, and the Arabic Ishartum, In modern, western style these toneswould all be equal tempered tones and semitones, but in the other traditions theprecise harmonic intervals can vary. In any case, this is the fundamental form ofscale construction upon which our entire western musical tradition was founded.In terms of just intonation harmonic intervals, the above scale could look like this:

1/1 16/15 6/5 4/3 3/2 8/5 7/4 2/1

The Greek modes were created by shifting the tonic within a given scale. Inthe Greek Dorian mode in C above, if we shift the tonic to the Ab, we have theGreek Lydian, our modem diatonic major scale in Ab.

Archytas was also the first to describe the difference between the arithmeticand harmonic division, or "mean," in the construction of scales. The arithmeticmean was simply the equal division of a string as performed by Pythagoras. Thisdivision is spatially equal but not aurally equal. By dividing the wholetoneharmonically, Archytas introduced the sernitone 16/15 which has stood the test oftime as a popular just intonation semitone of choice. 22 This harmonic semitone isabout a quarter of a semitone sharper than the Pythagorean version derived from a

series offifths.Archytas and the Harmonists can be credited with liberating harmonic

music from the Pythagorean series of fifths. They recognized that the semitone, thethirds, and the sixths were harmonic tones in their own right. By instinctivelyhearing and recognizing the harmonies of the higher harmonics, they set pure


harmonic music on its way. Archytas himself must have had an extraordinary ear.He literally picked these pure harmonic tones out of the air without any traditionor aid to guide him, and some of his enharmonic tetrachords suggest a keen earthat was able to discriminate among a variety of tiny intervals.

Composer Harry Partch has commented that "In a healthy culture differingmusical philosophies would be coexistent, not mutually exclusive," and certainlythe era of the Greek Harmonists witnessed this sort of musical openness andcuriosity. Both Plato and Aristotle extrapolated harmonic theory and attempted to

apply the sense of order and perfection to human morality and governance.Perhaps they carried the harmonic concepts too far, but in any case we can seehow powerful was the influence of early music theory on mathematics, philosophy,and political theory.

Meanwhile, in the far east, the harmonic tradition of Ling Lun had evolvedalong lines very similar to the developments in Greece. Chinese musicians had,like Archytas, recognized the tones based on the fifth and seventh harmonics.During the fourth century B.C. a bronze kin (small koto), called the "scholar'slute," was tuned as follows:

III 8/7 615 5/4 4/3 3/2 5/3 2/1

In our nomenclature, in the key of C, this scale would be:

CDEbEFGAC

The harmonically interesting feature here is that the intervals all appear inGreek music at this time, although the scale stylc is entirely distinct from anythingone would see in Greek music. This shows that the harmonic fundamentals areuniversal, but that the musical application is cultural. The distinguishing feature inthe scholar's lute scale is that the tetrachord, the interval of the fourth (4/3), isdivided into four intervals rather than the Greek three. This allows for both the


major and minor third in the same scale. The wholetone (8/7) is supported by theseventh harmonic of the fundamental, and is slightly sharp of the wholetone ofLing Lun I:U1dPythagoras (9/8) which is based on the third harmonic, the pure fifth(3/2).

These harmonic discoveries were made almost simultaneously in both Chinaand Greece, and it is safe to say they were made independently. Thus we see thatthe ears of the musicians inboth the east and west were in complete agreement asto the precise harmonic tones that were considered pleasing. Without knowledge ofthe harmonic series or other physical qualities of sound that would be discoveredlater, these musicians independently found by ear the just intonation ratios of pureharmony.

However, the practical matter of building fixed-tone instruments thatallowed free modulation was about to raise its head, and everything would bethrown into question. The Pythagorean notion that a cycle of:fifths should fit intoan octave would not go away.


5

The stream divides

A ristoxenus was the son of musician Spintharos in the city of Tarentum,home of Archytas a generation earlier. As a student of Aristotle, he learned

the theories of Pythagoras, Arehytas, and other Harmonists. However,Aristoxenus became obsessed with the Pythagorean comma, the amount by whicha series of 12 fifths was sharp of seven octaves. Aristoxenus subscribed to thenotion that this series of fifths should be a cycle, without having to make thecomma adjustment, and that this cycle should fit neatly into the octave. Heproposed distributing the Pythagorean comma among the 12 fifths to make themfit. His proposed solution, however, put all the fifths out of tune, hardly acceptableto the singers of Greek choral music, the fifth being the most prominent of allmusical intervals. The Harmonists were appalled, and as Harry Partch noted twomillennia later, "The war was on."

Aristoxenus set out to fashion a theory that would explain and support hisposition. He was a formidable scholar, author of some 453 published works,including biographies of Pythagoras and Archytas, and his Elements of Harmony


is probably the earliest extant treatise on Greek music theory. 23 In laying thefoundation for his theory of tempering the fifths, Aristoxenus claimed that thetones of the scale had a narrow range of acceptability that the ear would tolerate.This, of course, is entirely SUbjective since the acuteness of the ear varies widelyamong musicians, and the ear might grow accustomed to anything. On the otherhand, the Harmonists argued, a pure harmonic interval is either in tune or it isn't.A cycle of fifths that fit into art octave, however, required that these intervals becompromised, so Aristoxenus held finn to his conviction that the ear would acceptsuch a compromise.

He gave new definitions for fourths and thirds and wholetones. He defined awholetone as the interval between his tempered fourth and tempered fifth, orsimply one-sixth of an octave. A sernitone was a twelfth of an octave, and a fourthwas a semitone and two wholetones. It almost seems that he was describing equaltemperament, although he did not use any such term. Nevertheless, he abandonedharmonic ratios altogether, and described his scales in units that were 1112 ofawholetone. Therefore his tetrachords were simply divided into 30 parts, and thoseparts subdivided into intervals. For example, two of his chromatic tetrachordswere 4,4, and 22 parts, or 6, 6, and 18 parts. He claimed that any interval smallerthan a fourth was a dissonance, and therefore could be of any size. He allowed thewholetone to be divided "as melody admits of half-tones, thirds of tones andquartertones, while undeniably rejecting any interval less than these.,,24 Thus heset a lower limit on the size of musically usable intervals, and a limit on how finelya wholetone could be divided.

Aristoxenus clearly understood the music of his time, and some of histetrachords are fair approximations of those of Archytas. His scholarship wasthorough. Still, his logic seems forced, somewhat circular, and his rationalizationsare sometimes even contradictory. For example, Aristoxenus maintained that theear, not numbers, should determine proper tuning. He faulted the Harmonists forrelying on numbers rather than on the ear. On this point he clearly misinterpretedthe history of harmonic tuning. The intervals of the Harmonists did not evolve


from mathematics. On the contrary, the intervals were recognized by ear; and thenumerical relationships were discovered when theorists from Ling Lun to Archytasapplied these intervals to pipes or strings. The ear clearly favors the simpleharmonic ratios (211, 312, etc.), the octave being the simplest and most obvious ofthese.

Aristoxenus also says that "the subject of our study is the question, inmelody of every kind, what are the natural laws according to which the voice inascending Of descending places the intervals.,,25 Left to itself, the voice places theintervals at the pure harmonics since this is the easiest and most natural. We areagain reminded by Aristoxenus's comment that Greek music was primarilymonophonic, with choirs perhaps doubling at the octave. However, we would bemistaken to assume, therefore, that the Greek musicians and theorists did notunderstand harmony as two tones being sounded simultaneously, for surely theydid. That is what the monochord of Pythagoras and Archytas was all about. Theearliest notation is melodic, so we can only speculate what singers may haveachieved harmonically. It is hard to imagine that with all of the choral singinggoing on, no one ever sang a fifth or third harmony part, or droned a tonic whilethe melody was sung, if not by design, then at least by experiment or even byaccident. It is equally hard to imagine with all those strings being added to thekithara that no one ever plucked more than one at the same time or sang andplayed different parts. Almost certainly they did, and in any case, they knew whata simultaneous fifth or fourth or third sounded like. We tend to think of theevolution of knowledge, in this case musical knowledge, as being linear -monophonic, homophonic, polyphonic, harmonic ~but in fact the evolution ofknowledge is never quite so neat. Clearly, early Greek music was monophonic inexecution, perhaps with the octave doubling (which is itself a harmony, albeit thesimplest possible harmony), but it is equally clear that these early Greek musiciansfully understood more complex simultaneous harmony, learned the musicalintervals from such harmony on their monochords, and built their melodic scalesfrom these natural harmonic principles."


Finally, in some cases, Aristoxenus is just simply wrong. He claimed thatthe Hannonists "fabricated rational principles, asserting that height and depth ofpitch consist in certain numerical ratios and relative rates of vibration - a theoryutterly extraneous to the subject and quite at variance with the phenomena.t'"Here, Aristoxenus goes too far. He attempts to throw out the discoveries of LingLun, pythagoras, Archytas, and others. His attempt to temper the fifths to fit acycle of twelve into an octave is not in itself wrong; it is a subjective approach tomusic, and has some practical value, but to base this approach on the assertionthat the harmonic ratios are "extraneous" and "at variance with the phenomena" ofacoustics is wrong. AU tempered music is a sacrifice of harmony for a gain insimplicity. Furthermore, to base everything on the octave, 2/1, as Aristoxenusdoes, and then reject all other harmonic ratios is entirely inconsistent.

Meanwhile in third century China, musical theorists were struggling withthese same issues. Rather than temper the fifths, King Fang ("King" is a name, nota title) extended Ling Lun' s series of 3/2 fifths looking for the point at which itwould match an octave (an impossibility we know will never happen). Hecalculated the lengths for a series of 60 Iii, bamboo pipes like those of Ling Lun.He observed that the fifty-fourth Iii was only a tiny bit sharp of a higher octave ofthe fundamental. The comma between the two was infinitesimal, about 36-thousandths of a semitone. Western music theorists did not discover this untilNicolas Mercator proposed a similar system in the 17th century, and later the 53-notes-per-octave scale was used by some composers as a good approximation ofjust intonation.

In China, however, the notion of tempering the fifths, although considered,was not practiced. The practice may have even been forbidden by some Chineserulers who considered such a compromise to be risking the wrath of the gods.There was a belief that altering the harmonies violated natural principles andwould lead to social decay. There are stories of emperors traveling the country,requiring local musicians to play for them, to ascertain the stability of the regions.


In Greece. the tempering theories of Aristoxenus met an immediatechallenge by Euclid (c. 300 B.C.). the founder of western geometry. Euclidexposed Aristoxenus's most obvious errors, and asserted correctly that theharmonic ratios were natural, not theoretical. He demonstrated that six wholetones(six 9/85) were sharp of an octave (Theory 9 in his Section of the Canon), andtherefore showed that describing a wholetone as "one-sixth of an octave" wasarbitrary. Euclid showed how to divide an interval geometrically. anddemonstrated that the Aristoxenean ideal of dividing a wholetone into "halves.""thirds," and "quartertones," was impossible; since 9/8 cannot be divided intoaurally equal parts by ratios of rational numbers. Indeed the war was on.

In fairness to Aristoxenus, and in the spirit of co-existing musical systems,his proposed tempering of the fifths was an inevitable theory as later systems oftemperament attest. Aristoxenus erred in discounting the pure harmonic ratios asextraneous. Had he been more precise in his evaluation perhaps he would haveviewed the tempering for what it was, a practical compromise for the benefit offixed-tone instruments.

It was during this same time, in Alexandria, that another musical innovationwas made that would influence music even until our time. Ktesibios of Alexandriadesigned and built the first keyboard instrument, a pipe organ called a hydraulos.The wind for the pipes was supplied by hydraulic pressure from water enclosed ina container that was fixed with a hand pump.

Today. music can be said to be divided into two separate streams. On theone hand there is keyboard and fret based music which has traditionally requiredtemperament, most notably equal temperament. On the other hand, there still existstoday pure harmonic music as performed by a cappella choirs, barbershopquartets, certain chamber groups. string and brass ensembles, and as written bymany modem composers. These two streams of music are isolated from each otherbecause they use different fundamental tuning systems. The third century B.C. canbe seen as the point at which the streams began to diverge. The theories ofAristoxenus, and the invention of the keyboard set music on two separate courses,


and although many musicians since have temporarily succeeded inmerging thetwo streams, they remain asunder. The 21st century may see these streams mergeonce and for all.


6

A just diatonic major scale

The last three centuries B.C. experienced an information explosion, whichfrom the perspective ofthe time was quite dramatic. The silk road became

the conduit of learning among the Chinese, Indian, Persian, Arabian, Babylonian,Egyptian and Greek cultures. Science and the arts benefited and flourished.Geographers and astronomers shared knowledge and theories. Forms of writingand the recording of those ideas also flourished. An obscure astronomer in Samos,Aristarchus (c.270 B.C.), actually hypothesized correctly that the earth movedaround the sun, although he found no believers, and his theories were ignored for1,800 years until the time of Copernicus, another example of how knowledge isnot at all a linear progression.

Music and music theory benefited from this intellectual commerce, and thecenter of western music began to drift across the eastern Mediterranean fromAthens to Alexandria, Syria, Byzantium, and Persia. Eratosthenes (276-194 B.c.),director of the Library of Alexandria, known as "pentathlos," an all-around,versatile scholar, devised a method of extracting prime numbers (1, 2, 3, 5, 7, 11,


13, 17 ... ), wrote history and literary criticism, measured the circumference of theearth with great accuracy, and advanced the musical theories of Archytas. Hecontributed to the understanding of how acoustics suggest the prime numbers andthe harmonic ratios. His contribution to scale theory was to substitute theharmonic minor third (615) for the flatter and less singable Pythagorean version(32/27) derived from the series of fourths.

Musical notation was now widely used. The oldest extant manuscripts ofthe Old Testament, dating from around 165 B.C., contain 25 notationa1 signs forthe melodic phrases and cadences of Jewish ritual chants. This notation may haveevolved from earlier Egyptian demotic characters. Similar musical notation hadbeen developed in Greece, suggesting a common origin in Egypt. Musical notationhad also been developed in Han China, and from there had traveled the silk roadinto the Middle East.

In the first century AD the directorship at the Library of Alexandria passedto the scholar Didymus, who advanced the Harmonist tradition. He suggested achromatic tetrachord (C, C#, D, F) that was more harmonious than earlierversions:

111 16/15 10/9 413

The interval between the wholetone (l0/9) and the fourth is a pure harmonicminor third. Didymus also constructed enharmonic tetrachords and scales inwhichintervals smaller that a semitone were used. However the actual use of theseenharmonic scales was going out of fashion by this time, in favor of diatonic andchromatic scales. Harmonic theory continued to revolve around the harmonicratios, and the theories of Aristoxenus had claimed no champions. The name ofDidymus is forever linked to the comma 81/80, which he noted was the differencebetween the two wholetones that were used in scales, the Pythagorean 9/8 and theflatter 1019 used since the time of Archytas. The comma of Didymus, is also the


distance between the Pythagorean third and the pure harmonic third of 5/4 whichsingers naturally used.

Claudius Ptolemaeus, or Ptolemy, (AD, 90-168) was a geographer,astronomer and musician in Alexandria a generation after Didymus. He wasknown among the Greek intelligentsia as megiste, the greatest, and among theArabic scholars as Almagest, the Great One. He gave the positions of over athousand stars, devised a system of longitude and latitude for geographicpositions, and made maps of the world which were used by Columbus thirteencenturies later. He catalogued all of the tuning systems known at his time,including the tetrachords of Archytas, Aristoxenus, Didymus, Eratosthenes,Babylonian, Arabic, other obscure versions, and his own. He attacked Aristoxenusopenly for discounting the harmonic ratios, and for his arbitrary definitions oftones, semitones and other intervals. He was considered by Marin Mersenne(1635) as "the most knowledgeable of all those who have taught us Greekmusic.,,28

Ptolemy's Harmonics provides us with a very clear picture of the state ofharmonic theory in his day. He states flatly that ''the aim of the musician consistsin keeping the law of canon, or of the harmonic rule, which does not repel thesenses at aU and which are accepted by the greater part of mankind. ,,29Countering Aristoxenus, he states that the natura] harmonic ratios display a"beautiful order, and that it is not by chance." He also faults some of thePythagoreans for being too fixated on numbers, and says that the ear and therational ratios are in agreement. He gives one of the earliest classifications of theconsonances, placing the octaves in the first category, fifths and fourths next,followed by wholetones and thirds. This classification of levels of consonance isnecessarily subjective, and the debate is to continue even until the 20th century.

Ptolemy advanced the tradition of Greek scales and modes, and eventuallyarrived at our modem diatonic scale. The Greek diatonic tetrachord is constructedfrom a semitone and two wholetones (C, C#, Eb, F), regardless of the precisetuning of these intervals, and there were many versions of the tuning. Ptolemy's


version used 16/15 for the semitone, 9/8 for the first wholetone, and 10/9 for thesecond. These harmonic ratios share the characteristic that the higher number islarger than the smaller munber by 1. This feature of harmonic ratios (called"superparticular ratios") was considered by Ptolemy to be characteristic of themost pleasing scales. The tetrachord resulting from these interval ratios wouldhave been tuned as:

1/1 16/15 6/5 4/3

Adding a second tetrachord of the same construction with disjunctive toneinbetween yields a version of the Greek Dorian in pure harmonic intervals.Transposing to the Lydian mode of this Dorian scale gives a scale that includes thesixth (5/3) and the major seventh (15/8) which were Ptolemy'S innovation:

1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1.

This scale is the diatonic major scale that we know as do, re, mi, fa, sol, la,ti, do, the white keys of the piano. Our modem tuning is tempered, althoughPtolemy's tuning was pure harmonic. This ratio tuning of the major scale, the"Ptolemaic Sequence," is the first in recorded history to reduce all the intervals totheir lowest natural ratios, achieving perfect just intonation.

Another innovation of Ptolemy's appears in an alternative version of thediatonic tetrachord. Rather than the 16/15 semitone, he used the 12/11 tone whichis about a quartertone sharper. Ptolemy may have heard a similar three-quartertone in Arabic music, but his specific tuning introduces the eleventh harmonic ofthe fundamental, and thereby implies the seventh harmonic as well. These higherprime number harmonics add new tones to the palette. The non-prime higherharmonics do not add new tones since, for example, the fourth harmonic is simplyan octave of the second harmonic, and the sixth harmonic is an octave of the thirdharmonic. (See the table of harmonics in the appendix).


Ptolemy was the first to record the observation that when two strings ofdiffering thickness are stretched to the same tension their pitches have the sameratio as their two circumferences. To bring the pitches into unison, the fatter stringmust be stretched tighter by the same proportion. These observations were helpfulin the development of stringed instruments, and were later expanded upon byMersenne in the seventeenth century.

Ptolemy was a great scholar, but not infallible. He disregarded Aristarchus,maintaining that the earth was the center of the universe, and he underestimatedthe earth's circumference. This latter error helped convince Columbus that theocean passage to India was achievable, and led to Europe's arrival in the westernhemisphere. Still, in the end, Ptolemy made significant advances in musical scaletheory, settling the just intonation diatonic scales, establishing the fact that themusician's ear and the harmonic ratios were in agreement, and expanding theresources to the higher harmonics. His work greatly influenced Arabic andEuropean music of the Middle Ages.


7

The Arabian contribution

As with early Chinese and Greek musicians, Arabicmusic was based on _series of 3/2 fifths and 4/3 fourths; the fifths and fourths being inversions of

each other. As early as the Sassanides Persian dynasty (226 A.D.), Arabicmusicians were aware of the commas between the tones of this system and otherpure harmonic tones. They had read and. translated Aristotle, Ptolemy, and otherGreek scholars, and were therefore up to date with Greek music theory. Theiroctave scale construction was, like that of the Greeks, based on tetrachords linkedtogether with a disjunctive tone.

However, Arabic music took its own course in the Middle Ages, creatingstylistic distinctions that persist to the present. Whereas European music driftedtoward diatonic scales of tones and semitones, Arabic music embraced theenharmonic or quartertone intervals. Although communication existed both eastand west of the Middle East, we do not know how much theory the Arabicmusicians developed themselves, and what may have come from Greece and

50 1he Story oj Harmony

China. We have already seen that the fundamental understanding of harmonicintervals was universal.

In China, during the fourth century A.D., musician and theorist Ho Tcheng-tien (c.370-447) calculated history's first numerical approximation of equaltemperament. He gave string lengths for 12-tone equal temperament in three basepitches, and his maximum deviation from exact equal temperament was about atenth of a semitone. His mathematical derivations do not survive, and it is possiblethat he achieved his result by ear and by trial and error." In practice, the Chinesedeclined to use this equal temperament, and if knowledge of the system traveled tothe Middle East, Arabic musicians also declined.

Early in the eighth century Zalzal the Lutist revised the Pythagorean systemas applied to frets for the lute. Zalzal was a composer and performer more so thana theorist, and he made his adjustments by ear." Lute frets were, at this time;made from chord tied about the neck of the instnrment, and Zalzal moved his fretsto suit his ear. Like Ptolemy, he introduced the eleventh harmonic intervals, but hecarried this much farther, using the three-quarter tone 21111, the quartertone-diminished fifth 16/11, and the quartertone-diminished sixth 18/11. All three ofthese tones are still used in 20th century Arabic music. However his mostcelebrated contribution was the neutral third, a quartertone between the major andminor thirds, 27/22, known as the Wasta of Zalzal in Arabic tradition. Thisquartertone third is still used today, and is a distinctive feature of Arabic music.Some Arabic compositions use all three thirds - major, minor, and neutral - in asingle piece.

In the tenth century Al-Farabi gave directions for a 22-tone scale. The firstseventeen tones were calculated on a series of fourths, and he ended up withseveral unwieldy ratios, such as a slightly flat :fifth of 32768/19683 and a majorseventh of 409612184, slightly flat of the more harmonic 15/8. It is unlikely anysingers would sing such tones unless guided by instruments tuned in this manner.He did construct some theoretical tetrachords that employed the seventh harmonictones such as the large wholetone, 817. He included Zalzal's neutral third and


other eleventh harmonic tones in his 22-note scale. and added a very sharp majorseventh, 64/33.

He then advanced to the next prime harmonic, the 17th harmonic, andincluded the semitone 18117. This interval is almost an equal-tempered semitone,about one percent of a semitone flat, and was therefore sometimes used as theinterval for lute frets; a series of 12 of these semitones would almost make anoctave, and with the slightest of fudging would do so. Al-Farabi also calculated aforty-tone arithmetic division of the string known as the Tanbur of Baghdad.

In his book AI Musika Al Kabir, he defined the four main Arabic regulartetrachord types as Agam, or major (tone - tone - semitone); Nahawand, or minor(tone - semitonc - tone); Rast (tone - 3/4-tone - 314-tone); and Higaz (semitone-minor third - scmitone). 32 Any two of these tetrachords (in any particular tuning)can be joined by a disjunctive tone to create a scale. From any such scale, allmodes or transpositions of the tonic are possible, creating a wealth of scale optionsstill used in Arabic music today.

As polyphonic music was developing in Europe, Arabic musicianscontinued in a homophonic tradition, and built up this sophisticated system ofmodes. In the thirteenth century Safiyu-d-Din devised tables oftetrachords andtheir inversions. He also gave divisions ofpentachords (the interval of the fifth),which suggests origins independent of the Greeks, possibly the Babylonians orChinese. His Zirafkend Bouzourk pentachord is:

III 14/13 716 615 27/20 312

In modem nomenclature, in the key of C, this would be:

C C# Ebb Eb F G

The Ebb is what we might call the "blues third," a very flat minor third, butnot quite a quartertone, formed on the seventh harmonic. The C# would sound a

52 The story oj Harmony

bit sharp to modem ears; it is a linear division of the Ebb in the manner ofDidymus. The F would sound as a disturbingly sharp fourth? more of atheoretically derived tone, than a practical vocal tone, but the other tones are aUquite singable, and the scale clearly has an Arabic feel.

In the fourteenth century, Mahmud Shirazi and Abdul Kadir formalized theMaqamat, or set of modes, as still used today. The four regular tetrachordspatched together create 84 unique Maqam. The modem tuning is consistent withthat set out by Al Farabi. Some the modes use the leading tone, the major seventhleading to the octave, a feature that was also becoming popular in Europeanmusic. Arabic composers also learned to employ irregular tetrachords such asenharmonic versions with two quartertones and a major third, and developed thepractice of changing scale or mode within a piece of music. What Arabic musiclacked in polyphonic development, it gained in scale variety. Arabic lutc tunings,particularly the 18/17 near-equal semitone, would later appear in Europe.


8

Keyboards and polyphony in Europe

Throughout the Middle Ages in Europe, as polyphonic music inexorablymatured, musicians and theorists struggled to sort out the harmonic and

acoustic issues that they were only beginning to understand. Harry Partch pointedout that the four central harmonic ideas that would dominate discussion for thenext thousand years had already been articulated by the time of Ptolemy, and theywere: (1) the harmonic proportion, (2) the arithmetic proportion, (3) thePythagorean series offifths, and (4) the Aristoxenean tempering."

Not until the late ninth century (Regino, Abbot of Prum, De harmonicainstitutionei and early tenth century (Hucbald, De harmonica) do we have clearreference to simultaneous harmony as a feature of musical performance. Asdiscussed earlier, we know that musicians and theorists had heard simultaneousharmony much earlier, on the monochord and other instruments, and that melodicscales were constructed from the tones revealed by such harmony, but prior to thedevelopment of the early parallel organum in the ninth and tenth centuries we haveno record of harmonic musical performance other than perhaps octave doubling.


In the l lth century, Guido d' Arezzo introduced solmization for the eightnotes of the diatonic major scale, that is; "ut, re, mi, fa, sol, Ia, ti, do," althoughhis own hexachord for singers did not include either the major seventh nor theoctave. At that time, only three intervals were considered consonant, the octaves,fifths, and fourths plus the octaves of these (11ths, 12ths, and double octaves).Guido wrote that "these three intervals blend in organum congenially and smoothly... hence they are called •symphonies,' that is, compatible unions of notes ...,,34

The exclusion of the major third could be explained by the possibility thatthe tuning waS still Pythagorean. and therefore the thirds were sharp of theharmonic. The sharp Pythagorean third will beat - we know now - not only withthe fundamental's fifth harmonic, but with the combination tones created by thesesimultaneous tones. Therefore, if the thirds were being sung or played inPythagorean tuning the sonorous effect would certainly be more rough than thepure 3/2 fifths in resonance with the harmonic and the subsequent combinationtones. All of this reasoning was unknown in the 11th century since the discoveryof the harmonic series and combination tones was still seven centuries away.However, it is also possible (and even likely since they had been known for wellover a thousand years) that some singers were singing pure 5/4 thirds, and thatthese dyads were just simply not recognized yet as consonances.

However, by the close of the 11th century thirds were appearing in theharmony parts of free and contrary organum. The keyboard instruments were alsobeginning to have an impact on musical styles. By the end of the tenth century theWinchester Monastery in England had an organ with 400 pipes. It was also aboutthis time that a technological advance in balanced levers linking keys to hammersmade keyboard instruments more practical. By 1250 the portatio, a small, portableorgan, had been designed and built. The new and better keyboards allowed forrapid polyphonic playing. Hence by the mid twelfth century melismatic organumcould feature several notes in the harmony part against a single melody Line.In the12th century John of Garland, Franco of Cologne, and other writers wereadmitting both the major and minor thirds as "imperfect consonances." These were


most likely pure harmonic major (5/4) and minor (6/5) thirds. Franco of Colognealso mentions the major and minor sixths as "imperfect dissonances," and thesemitone, tritone, and sevenths as "perfect dissonances, ,,35 showing an acuteawareness of the harmonic qualities of these intervals.

We get confirmation that singers were singing pure thirds, rather thanPythagorean thirds, from the English Monk Walter Odington (c. 1240-1280). Hesays that singers in the faux bourdon vocal tradition intuitively used the pure ratiointervals and not the Pythagorean intervals.i" Ibis would require altering somemelodic intervals to accommodate the vertical harmony, and indicates that singerswere making these melodic adjustments in favor of pure simultaneous harmonies.Odington includes the major sixths as "discordant concords" and also mentions themajor chord, the earliest record of this triad being mentioned.

Throughout the Middle Ages, there was a secular music tradition as wen asecclesiastical and scholarly ones, and among these there was a cross influence insong styles during the development of polyphonic music in Europe. Celtic.Germanic and English minstrels; French troubadours; German minnesingers andMeistersingers; and the trouvere poets of Charlemagne'S court all likelycontributed to the rise of polyphonic music through stylistic experimentation.There may well have been more experimentation with polyphony among someminstrels than among certain church musicians since Pope John xxn was stillforbidding counterpoint in church music as late as the mid-fourteenth century. Thetext ofbis Papal decree of 1322 makes it clear that secular music was influencingpolyphonic development. Inhis tirade he bemoans the fact that "Certain disciplesof the new school ... truncate the melodies with hoquets, they deprave them withdiscants, sometimes even they stuff them with upper parts made out of secularsongs.',37 The pope forbids taking such liberties, but allows "occasionally, andespecially upon feast days," the use of octaves, fifths and fourths) so long as theydo not distract from the melodies. Thirds and sixths were not to be sung in thechurch.


Chinese musicians bad, like the Europeans, discovered the pure thirds. TheTang Dynasty court orchestra in 619 A.D. included over a hundred instruments(flutes, gongs, bells, kotos, and other string instruments) tuned to pure harmonicratios, including the 5/4 harmonic third. Chinese opera began at this time,predating European opera by a thousand years. By the ninth century, the Persiansand Chinese had both developed bowed instruments, again predating Europe bysome six centuries. The erhu, developed during the Sung Dynasty (960-1279),with two strings, the bow clasped between them, and a sound box covered in snakeskin, is still used today.

In Europe, by the 14th century, as counterpoint styles were developing, boththe major and minor sixths were considered as «imperfect consonances." InArsnova, Philippe de Vitry (1291-1361) includes the sixths and states that both themajor third and sixth are more consonant that the minors. Four part canons andmotets were already well established, and Guillaume de Machaut wrote a four partMass. In singing these pieces, however, musicians came face to face with thechallenges of pure harmony.

Singers knew that when they modulated through a piece of music, notes ofthe melodic scale would have to change pitch to balance the harmony. Thischanging of the musical key was not a problem when the voice wasunaccompanied. However, when keyboards or fretted instruments accompanied thevoice problems did arise because of the fixed tones. The fixed-tone instrumentscouJd only be tuned to play in one key. Some keyboard makers inserted the extranotes needed between the keys of a single scale. These "split-key" keyboardswould sometimes contain 17, 19, or even 31 tones per octave instead of 12.However; this was only a partial solution, allowing musicians to play pure tones insome, but not all, keys. These systems were also cumbersome to build, andcomplicated to play, and this put pressure on musicians to compromise pureharmony for the sake of these instruments.

Then in 1361, Nicholas Faber completed the Halberstadt organ with itsdiatonic major scale on the lower digitals and the chromatic accidentaIs on the


raised digitals of the upper keyboard. This humble little keyboard would come todominate western musical thought for the next six hundred years, right up untilour time. The tuning of this instrument was likely Pythagorean, and possiblyfeatured some tempering of the fifths.

The first modern style keyboard, built by Nicholas Faber in 1361,one of three keyboards for the organ in Halberstadt, with the diatonic

scale in front and raised accidentals behind.(From Michael Praetorius's Syntagma musicum, 1618).

Organ players had already experimented with tempering a justly tunedinstrument by simply lopping off some pipes and extending others. Once thirdsand fifths were routinely played together in triads, and once fixed-tone 12-notes-per-octave keyboards were established, some method of tempering seemed


inevitable. However, the equal tempered tuning that would, in the 19th and 20thcenturies, be wedded to this diatonic major keyboard had not yet been realized.The dream of Aristoxenus, of fitting twelve fifths into an octave, was still only atheory.

The stage was thus set for the 18th century showdown in Jena which wasreally just a single skirmish in a protracted battle between acoustic perfection andpractical convenience. The particulars of the musical and acoustic issues may havebeen complicated, but the musical choices were simple. Musicians playingpolyphonic and harmonic music from the Renaissance to the 20th century couldchoose among the following:

(1) play pure harmonic music, but don't use fixed-tone instruments;(2) play pure harmonic music, but add extra digitals to the fixed-tone

instruments to allow for enharmonic shifts of some notes;(3) temper the notes in favor or relatively pure triads in the common keys,

and accept some unusable wolf intervals in the distant keys; or(4) equally temper all the notes, and accept the compromise of all

harmonic intervals, except the octaves.It was not until the end of the 20th century that a new option would present

itself, as instruments were able to take advantage of electronic and computertechnology:

(5) build keyed or fretted instruments in which the tones of all notes canchange as needed.

Debates over the first four of these options would dominate intonationtheory and music instrument design for the next 500 years following thecompletion of the organ at Halberstadt.


9

As sharp as the ear will endure

The growing complexity of music from the Renaissance motets and hymns ofJohn Dunstable and Guillaume Dufay, to the Italian operas of Monteverdi,

culminating in the brilliance of J.S. Bach, coupled with the growing popularity ofkeyboards, seemed to make some sort of temperament inevitable. The 15ththrough the 17th century witnessed a long running debate on the relative merits ofpure harmony and tempered harmony, and what form such temperament shouldtake. All of this, as we have seen in Chapter 1, led to the tuning showdown of1706 between Bach's cousin and the young man credited with the co-discovery ofthe equal tempered solution, Johann Georg Neidhardt.

We know from Franchino Gafori's Practica musica in 1496 that organistscommonly tempered the fifths and thirds. The organ at St. Martin's cathedral inLucca had split keys with separate digitals for D# and Eb, and for G# and Ab. Welearn from Bartolomeus Ramis de Pareja's Musica Practica in 1482 that the breakhad been made with Pythagorean thirds that Ramis claims were "tiresome for


singers ."38 And just as the singers were bringing the thirds into harmonic tune, thekeyboards were forcing them back out.

The first certain voice of meantone temperament was that of Arnolt Schlick(1455-1525), the blind organist for Count Palatine at Heidelberg. In his Spiegelder Orgelmacher und Organisten in 1511 he gave a tuning formula that was anearly form of meant one. He also mentioned the practicality of split-key organs as asolution. Twelve years later Pietro Aaron in Venice described his meantonetemperament with "sonorous and just" thirds, and fifths "a little flat."

Then, in Brescia, Italy, Giovanni Maria Lanfranco gave tuning rules thatapproached equal temperament in which the fifths are tuned so flat "that the ear isnot well pleased with them," and the thirds "as sharp as can be endured.'?" Thisdescription can hardly be interpreted as an endorsement for the system, but it doesgive us some idea of how sensitive ears first reacted to the tempered intervals.

In the meantime. the performing musicians were charging ahead withstylistic advances, variations, ornamentations, diminution, augmentation,modulation, inversions and retrograde motion, all made possible by the relativesimplicity of keyboards. It was about this time, mid-16th century, that HenricusGlareanus misnamed all the Greek modes when assigning names to the 12ecclesiastic modes. It was also the age when violins, previously considered crudefolk music instruments, were becoming accepted among serious composers andperformers.

In 1555 Don Nicola Vincentino built a harpsichord-like instrument that hecalled an Archicembalo with 31 notes per octave arranged on six ranks of keys.The first rank consisted of the 7 white keys of the diatonic major scale, on thesecond rank were the five black keys, and above those seven alternative blackkeys, forming the 19-tone octave:

C, C#, Db, D, D#, Eb, E, E#, F, F#, Gh, G, G#, Ah, A, A#, Bb, B, B#, C


Ranks four and five were "enharmonic" tones with the purpose of revivingancient Greek scales employing quartertones. The final rank repeated the diatonicscale of the first. Vincentino failed to interest his contemporaries in bringing backthe Greek enharmonic genera, but the 19-tone scale gained some support, and wasrevived a generation later by Michael Praetorius in Germany.

Franco de Salinas (1513-1590), another blind organist, thoroughlyexamined the growing tuning problem, and investigated, by ear obviously, all theknown or hypothesized solutions for 12-fixed-tooe instruments which, it isinteresting to note, he calls "artificial instruments. ,.40 He set out the rules formeantone which had already been formulated by Schlick and Aaron before him.He noted that for the thirds and sixths to be "made sweeter," the fifths had to beflattened. For fretted instruments he proposed equal temperament, at least intheory, but not by giving precise formulation. He proposed for the placing of fretson viols that "The octave must be divided into twelve parts equally proportional."And finally, he proposed a 24-tone enharmonic matrix based on just intonationwhich allowed some, but not all, pure major and minor triads."

The theoretical battles of this era are best witnessed, however, in the greatclash between Gioseffo Zarlino, choir master at the Basilica San Marco in Venice,and Vincenzo Galilei of Florence, a former student of Zarlino's, a lute player, andthe father of the more famous Galileo Galilei. Zarlino maintained that the naturalvoices of the singers demonstrate the true intervals, and that those intervals for thediatonic major scale are represented by the Ptolemaic Sequence (1/1,9/8,5/4,4/3,3/2,5/3, 15/8,2/1). He reconfirms that singers did not sing Pythagorean thirds. Hepointed out that arithmetic and harmonic proportions could coexist in the samemusical schema, and he showed how the number series 1-2-3-4-5-6 created boththe major and minor tonalities. Zarlino designed keyboards of 17 and 19 tones peroctave, but also accepted temperament for keyboards and lutes, with theunderstanding that such temperament compromised the consonance of intervalsand triads. He described meantone temperament, and the Euclidean geometricconstruction used to calculate the mean proportional.

62 The Story oj Harmony

Zarlino pointed out that "every composition, counterpoint, or harmony iscomposed principally of consonances. Nevertheless, for greater beauty and charmdissonances are used," and that "the ear not only endures them but derives greatpleasure and delight from them ... ,>42 Here Zarlino was referring to pure harmonic

The lute, with its fixed frets, limited the flexibility of the player,and demanded some form of temperament.

(From Marin Mersenne's Book of Instruments, 1636).


intervals, not tempered intervals. He was a choir master with a keen ear, andalthough he investigated all the options, he clearly favored the pure harmonies thatthe voices intuitively and naturally found.

Vincenzo Galilei attacked the theories of his former teacher in his 1580Diaiogo della musica antica et della moderna. He claimed that the voice does notteach the true intervals, but rather the voice is taught correct intervals by singingto instruments properly tuned by correct theory. Vmcenzo Galilei's version ofcorrect theory was that of Aristoxenus, a series of tempered :fifths that would fitinto an octave. He maintained that art does not simply follow nature, but improvesupon nature, and he devised a system for tuning the lute that he claimed createdtwelve equal semitones. To begin, he divides the string length into 18 parts, andthe first part from the nut he marks as the first fret. (This is the arithmeticproportion, and gives an 13/17 semitone, flat of the Ptolemaic 16115.) The lengthfrom this first fret to the bridge he again divides into 18 equal parts and marks thefirst part as the second fret. This procedure he continues until he has marked off12 semitones of proportionally equal size (18/17). "This brings me to the midpointof the whole string; the first and lower octave thereof! find I have divided intotwelve equal semitones and six tones, as said by Aristoxenus.,>43

Vinccnzo must have been overwhelmed with excitement at his discovery.Unfortunately, his math did not quite add up, and his "midpoint" was flat of thetrue midpoint or octave of the string. His sernitone is the same 18/17 semitoneused by Arabic lutists, but it is slightly flat of a truly equal semitone, and thiserror is compounded over the twelve divisions. Vincenzo's octave is flat by aboutone-eighth of a semitone. Zarlino, clearly the better mathematician in addition tohis keen ear, wasted no time in publicly replying to Vincenzo and pointing out hiserror. He published a "Musical Supplement" to his earlier Dialogo, setting therecord straight, and reasserting his position that the natural voice and the trueharmonic ratios are in agreement.

Commenting on this controversy in the nineteenth century, Germanscientist, musician, and authority on acoustics Hermann Helmholtz gave Zarlino


credit for reintroducing "the correct intonation," and added that "Singers were thenpracticed with a degree of care of which we have at present no conception. We caneven now see from the Italian music of the fifteenth and sixteenth centuries thatthey were calculated for most perfect intonation of the chords, and that their wholeeffect is destroyed as soon as this intonation is executed with insufficientprecision. ,,44

Their argument over art, nature, and the perfect tuning system raged onuntil Zarlino's death in 1590, and thereafter was taken up by others. A generationlater, Johannes Kepler, who inadvertently discovered the laws of planetary motionwhile trying to show that the distances of the planets conformed to the harmonicseries, corrected Vincenzo's semitone and calculated the correct string lengths tomake 12 equal semitones equal one octave. His calculations were published in hisHarmonices mundi of 1619.

Everyone was getting into the act. Galileo, son ofVincenzo, who hadalready seen the moons of Jupiter through his telescope, began investigating thelaws governing the vibration of strings. And French mathematician andphilosopher Rene Descartes gets credit for being the first voice in history toacknowledge the presence of the harmonic overtones. Descartes wrote his onlywork on music, Compendium Musicae, at the age of 22, shortly after graduatingfrom law school and joining the French army so he could travel and think. Hestated that 'We never hear any sound without its upper octave." The work wasonly circulated in manuscript during Descartes' life, and was published in 1650,the year he died. Inhis Abrege de fa Musique, Descartes observes "Of the twoterms required to form a consonance, the lower ... in some way includes the other.This is manifest on the strings of the lute. When one of these is plucked> those anoctave or a fifth higher vibrate and sound by themselves."

Knowledge of the overtone series, or harmonic series; would cbangeeverything. It gave 18th century Harmonists a physical foundation for theharmonic intervals.

Justomc Tuning Inc. 65

10

The artand science of sound

The greatest irony of the entire history of harmonic theory is that just asscience was beginning to glimpse the physical foundations of pure harmony,

instrument makers were giving up on it. The mechanical limitations of fixed-toneinstruments seemed, in the 17th and 18th centuries, to preclude pure bannonicintervals other than octaves, for keyboards and fretted instruments. There wasoften a conflict between the keyboards using mean tone or well temperament, thefretted instruments using approximately equal temperament, and vocal musicwritten for pure harmony. The keyboards and guitars clashed, and neither couldplay with the justly tuned vocal music.

The musicians, however, had not given up. Handel, Bach, and Mozart allwrote music with the intention of pure intonation, particulatly vocal music. Theirinstrumental music was intended for intonation as pure as the instruments couldachieve, Bach's Well Tempered Clavier being an example. Handel, as we know,performed on a split-key instrument, and Bach would often retune the clavichord


between pieces. It is clear that musicians of the 17th and 18th centuries still had anear for pure harmony,

In the 17th century, Marin Mersenne heard the harmonic overtone series inthe sound of the trumpet and realized that he was hearing the major triad in the

One of the many keyboards designed by French theorist Marin Mersenne,who insisted that pure harmony should not be sacrificed to convenience.

(From Mersenne's Harmonics, 1648).

first five harmonics: a tonic, the octave, the fifth, another octave, and the third. Herealized that the harmonic series continued, and that this harmonic triad wasfollowed by another fifth and then the seventh harmonic which formed a naturaldominant seventh harmony. Because of this awareness of the seventh harmonic,


Mersenne considered the 7/6 minor third (a flat, or sub-minor third, which we nowcall a "blues third;') to be a consonant interval.

Although Mersenne gave instructions for the main systems of temperamentfor his time, including quasi-equal temperament, he concluded that "it does notfollow that the system of Aristoxenus, in which the fifth contains seven-twelfths ofthe octave, is more perfect than that in which it is pure." He designed severalkeyboards to address the problem keyed instruments. In his Harmonie Universellein 1636 he faults both Zarlino and Vincenzo for arguing over the hierarchy of thevoice versus theory, reminding his readers that the voice, the ear, and harmonictheory agree. In reference to the Pythagorean third, which Francisco de Salinashad said was acceptable in keyboard tuning, Mersenne says «Actually, the trueratio of the major third, which is 5 to 4, is much easier and much sweeter." Inreference to the pure harmonic fifth (3/2), fourth (4/3), andthird (5/4) he says that"there will be no person but does not acknowledge these consonances to be veryexact, and that the ratios correspond perfectly to experience. ,>45

Mersenne also hung weights from brass strings and determined that thefrequency is not only inversely proportional to string length (known earlier), butthat frequency is also proportional to the square root of the tension, and inverselyproportional to the square root of the string thickness.

Instrument makers and musicians continued to experiment, bolstered by thewave of new knowledge that was becoming available. In 1630 Giovanni BattistaDoni built a double lyra after ancient Greek models, but with added strings to

achieve extended harmonic scales. He also built a three-manual keyboard whichgave the Dorian, Phrygian and Lydian Greek modes in just intonation intervals. In1640 Nicolaus Ramarinus constructed a keyboard based on the cycle of 53 fifths,first suggested by King Fang two thousand years earlier. In 1670 NicolasMercator advocated the same system. This 53-notes-to-the-octave system of purefifths gives very close approximations to just intonation, but naturally presentssome manufacturing and performance challenges. The system lived a long andhealthy life, well into the 19th century.


Descartes died in 1650, and his Compendium Musicae was published,revealing to a wider audience his introduction of harmonic overtones. AntonioStradivari built his first violin in 1666. Violins could now achieve a volumesufficient to take a major role in symphony hall orchestration. Since intonation isflexible on the violin, it could successfully accompany pure harmonic singing.

In 1685, the year before Bach was born, Gottfried Leibniz published hispapers on calculus (followed by Newton two years later), providing a means ofcalculating more precise string lengths for equal temperament. The first churchorgan tuned to equal temperament was probably that designed and built by ArtSchnitger at the Church ofSt. Jacobi in Hamburg, in 1688.46 Three years later,Andreas Werckmeister published his formula for calculating equal temperament inhis Musikalische Temperature. The seven white, five black Halberstadt keyboardhad become the standard. J.S. Bach was three years old, standing on his tiptoes topeck out diatonic improvisations on the family clavichord.

After Descartes and Mersenne, soon everyone was hearing overtones. It wasbecoming common knowledge among the musical elite that sound was a vibration,and that the major triad and the dominant seventh chord were supported by thefirst seven harmonic overtones. In 1700 Joseph Sauveur published d 'acoustique etde musique, measuring and explaining the vibrations of musical tones, andproviding experimental evidence of the overtone series. And here, at the dawn ofthe 18th century, our historical irony is in full flower as precise equal temperamentand the laws of harmonics were being simultaneously discovered and understoodat exactly the same time. In the first decade of the 18th century, Bach wrote hisfirst cantata, Denn Du wirst meine Seele, Neidhardt and Johann Nicholas Bachhad their great tuning showdown in Jena, and in Italy harpsichord makerBartolornmeo Cristofori made a new, heftier keyboard instrument that he called a"gravicembali col piano eforte;" the first piano.

The knowledge of the harmonic overtones lead musicians to listen ever moreacutely to the sounds they were playing, and this lead to a completely newdiscovery about the nature of acoustics. In Italy Giuseppe Tartini, solo violinist in


Padua, and chamber musician to Count Kinsky at Prague, began to hear additionaltones that were not in the overtone series. He called them ''terzi suoni," or thirdtones, and these were later called Tartini's tones. These mysterious tones areformed by the difference between two other tones, and today we call them"differential tones." A high tone and a low tone sounded loudly and continuouslytogether create a third tone in the middle. This acoustic phenomenon wasindependently discovered by German organist Sorge, and later in 1863 Germanacoustician Hermann Helmholtz discovered "summation tones," which are thirdtones created by the addition of two tones, such that two lower tones will create athird higher tone. Helmholtz gave the name "combination tones" to this wholefamily of phenomena.

Tartini was also one of the first musicians to recognize that the pureharmonic seventh (harmonic proportion 7/4, quite flat of the equal temperedversion), was the true minor-seventh based on the harmonic series, and that "thisharmonic seventh is not dissonant, but consonant." He adds that this pure minor-seventh "has no need either of preparation or of resolution: it may equally wellascend or descend, provided that its intonation be true. ,>47

The discovery of harmonic overtones and combination tones gave musiciansa partial understanding of why pure harmony sounded so good. In a pure harmonictriad, the overtone series of the three fundamental tones match up and blend in thehigher registers, and the combination tones created not only by the fundamentals,but by the natural overtones, also tend to match up and blend. On the other hand,in a tempered triad, all the upper harmonics clash, and the combination tones alsoclash. The natural combination tones formed the pure fourth (4/3) and the pureminor third (6/5). Inversions ofnatural harmonics gave the major and minorsixths. Every tone in the just chromatic scale is accounted for in these naturaltones. The great composers found that their carefully worked out harmony partswere imitating the pure harmonies of nature. Later, this understanding of why pureharmony sounds good was filled in by Hermann Helmholtz, Harry Partch, andothers.

70 The Story ofHarmony

Jean-Philippe Rameau, whose Treatise on Harmony iTraite de l'harmoniereduite a des pnncipes naturels) introduced our modem concept of chordprogressions and guided several generations of composers, built his theoriesaround the collective acoustic knowledge that he learned from reading Zarlino,Descartes, Mersenne and others. When he first published in 1722 he had not yetread Sauveur, and did not understand the harmonic series, but he updated andcorrected his work in his 1726 "Nouveau systeme. " He gives voice to the notion oftonality when he states that "melody arises from harmony," and that "The sourceof harmony ... [is the] harmonic center to which all the other sounds should berelated." He adds that "Whether this fundamental sound be implied, inverted,supposed, or borrowed, reason and the ear are in such good agreement on thispoint that no exception can be found." 48 When Rameau spoke of harmony, heintended pure harmony, pure 3/2 fifths, 5/4 thirds, and so forth. These perfectharmonic proportions were, to Rameau, "simple, familiar, precise, true, andaccurate." He found a foundation for minor tonality in the harmonic overtones 10,12, and 15.

J.S. Bach's Well Tempered Clavier was written in the same year, 1722,that Rameau' s first book was published, and it is clear that musicians andtheorists of this era tended to think of harmony as pure harmony to whateverextent was possible on the instruments they played. Handel introduced the ideathat ending on a minor triad was acceptable, but his intention was a pure harmonicminor triad for voices. The tempered minor thirds of meant one, well temperament,or equal temperament, sound harsh compared to the pure minor third, andHandel's acceptance of the minor in a final cadence was predicated on theharmonic proportion 6/5. The keyboard minor thirds caused a problem; and this isone reason Handel played split-key keyboards.

The popu1ar opinion of equal temperament at the time was summed up byDr. Robert Smith of England in his Harmonics, 1749, when he described thetemperament as "that inharmonious system of 12 semitones [producing a]harmony extremely coarse and disagreeable .' >49


Science continued to advance the understanding of acoustics and the natureof harmony. Swiss mathematician Leonard Euler, Swiss physicist DanielBernoulli, and French mathematician Joseph Louis Lagrange worked out theequations for sound waves in horns. Bernoulli and Euler worked out theories toexplain the acoustics of vibrating strings, and Bernoulli described how the larynxacts as a vibrational source of sound waves. The next generation of scientists,Michael Faraday, Jean Fourier, and Georg Ohm documented the nature ofharmonic motion, complex waveforms, and resonance.

Returning for a moment to the illustrious Bach clan, it is not likely that theyoung Johann Sebastian Bach was in Jena on the day of the tuning contestbetween his cousin Nicholas and the student Neidhardt. At that time the 21-year-old prodigy was more likely in Arnstadt, 40 kilometers west of Jena, where he hadbeen hired in 1704 to put the new church organ through its paces. The youngBach's reputation must already have been exceptional, because considerabletechnical knowledge would have been required for such an assignment as testingail the parameters of a new church organ. He so impressed the Arnstadt elders thatthey transferred their organist and choir master to another church and hired JohannSebastian.

Although his technical skills were beyond question, his ability to manageand discipline the choristers was lacking, since he was nearly their age, and givento some insubordination himself. One clash with his charges deteriorated into aknife fight in which western music could have suffered an immense loss.Furthermore, his tendency toward improvisation with the chorale standards, not tomention his radical "Easter cantata No. 15,~'had infuriated the Arnstadtestablishment. He was soon thereafter granted a four week leave. He reportedlywalked (likely hitching rides on carts or wagons) the 350 kilometers north to visitDietrich Buxtehude, Danish composer and renowned organist at the Marienkirchein the Baltic port of Lubeck. There he was exposed to Buxtehude's "free" choraland orchestral works, an experience that would influence his own creativity andimpact upon western music for centuries to come.


The one month leave became two, and three, and more. The authorities atArnstadt had had enough, and in any case, Johann Sebastian's interests had driftedelsewhere. His artistic interests tended toward manifesting his own creativepassions, improvising, writing and performing music. His personal passions hadturned to his cousin Maria Barbara Bach. On the fateful day in lena, Bach waslikely courting Maria, whom he shortly thereafter married. Although this youngprodigy was known locally as a skilled organist, the crowds in lena in 1706 had noidea that the absent young cousin of the choir master Nicholas was to becomealmost synonymous with western music, one of Europe's greatest composers, farovershadowing all but a fcw others, all of his contemporaries, his family's othermusical geniuses, and the renowned Johann Nicholas who was painstakinglytuning the organ pipes in the St. Michael's church in Jena.

As the eighteenth century closed, the piano was gaining in prominence; andbeginning to replace the violin as the driving force in orchestral music, and even insome chamber music. For a century the violin had been the leading instrument ofboth orchestral and chamber music, and the dominance of pianos changed theintonation landscape. Hermann Helmholtz mentions in The Sensation of Tonesthat "when quartets are played by finely-cultivated artists, it is impossible to detectany false consonances ... practiced violinists with a delicate sense of harmony,know how to stop the tones they want to hear, and hence do not submit to the rulesof an imperfect school." The piano began to insinuate "keyboard" temperament onaU the other instruments because the tempered intervals sound even worse whenheard with pure intervals. Since the piano player could Dot alter his or her tones,the pure tone players were forced to succumb to the new temperaments.

True equal temperament was still a century away, but quasi-equaltemperament began to make inroads into music. Helmholtz later commented thatMozart "is master of the sweetest possible harmoniousness, where he desires it,but he is almost the last of such masters." The two musical streams - pureharmonic music on the one hand, and keyboard based music on the other - hadnow clearly drifted apart.


11

The age of pianos

The keyboard, their particular temperaments, and the standardization of thediatonic and equal-semitone chromatic scales gave something very valuable

to western music, namely simplicity and ease of play. Music is boundless, so byorganizing some intervals, making them all comfortably accessible, and makingmodulation unrestricted, the modem keyboard helped musicians sort out the vastpotential of music into a manageable system.

Beethoven brazenly exploited the power of this 12-tone system. His musicis filled with speed, spirit, energy, and drama. The roughness of equaltemperament is somewhat masked with this style of music, particularly with fastmoving chords, and Beethoven made the most of it, producing effects no onebefore him had ever attempted.

However, Owen Jorgensen, authority on historical tunings, points out that"For every gain intemperament history, there has been a corresponding loss." 50

Temperament is a compromise of harmonic purity, sacrificing the full power anddelicacy of the pure harmonic proportions for simplicity in construction and playof keyboard and fretted instruments. Musicians of the 19th century knew this.

74 The Story oj Harmony

Today, most musicians grow up and are trained from inside the equal temperedparadigm, and may be only marginally aware that there is such a thing as a pureharmonic interval that is distinct from a tempered interval, or that there are infinitescale choices beyond our diatonic and chromatic default choices.

Jorgensen points out that pure equal temperament was not practiced onpianos before 1885. However, quasi equal temperament was used, and theintention was to achieve equal temperament within the limits of availabletechnology. In fact, since tuning equal temperament by ear, by counting the beatsof the mistuned intervals, had not yet been developed, tuners of equal temperamentin the 19th century relied on devices such as the tuning fork tonometer invented byJohann Heinrich Scheibler in 1835. When speaking of equal temperament in the19th century, we are speaking of quasi-equal temperament.

The 1. G. Pleyel piano factory was established in Paris in 1808. AlfredJames Hipkins introduced equal temperament to the James Broadwood pianofactory in England in 1846, although James Broadwood himself had introduced theidea as early as 1811.51 Broadwood introduced the importance of the strikingplace on the piano string, a parameter that influences the tone since the strikingpoint may dampen certain harmonics of the string. Broadwood's solution was tostrike the string somewhere between one-seventh and one-ninth of the length. Thishad the effect of dampening the seventh harmonic which clashes with the minorsevenths of equal temperament. In 1852 the Exeter Hall organ was tuned to equaltemperament, and two years later English organ makers Gray & Davidson andWalker & Willis made equal tempered organs. The Henry Steinway & Sons pianofactory was established in New York in 1853, and Baldwin Piano & Organcompany was founded a decade later.

Francois Fetis introduced the term "tonality" in 1835 to describe theprinciple of musical intervals having a relationship to a fundamental, and he notedthat the music of different cultures and scale styles placed different emphasis onthis tonality. Fetis also pointed out that the ear can grow accustomed to almost any


tuning system, and he tells a story that emphasizes how far habitual hearing canlead one away from the harmonic proportional intervals.

Fetis was the teacher of the young composer Nicolas Jacques Lernmens.Lemmens had been born in the village of Zoerle-Parwijs, the district of Campine,in the Belgian province of Limbourg, on a frozen January morning in 1823. As ayouth he learned music on a harpsichord that was badly out of tune due to theharsh weather of the region and the fact that no tuner lived in this rural district.Nearby was the abbey of Everbode, and there the organ was also mistuned andbroken. An organ builder was summoned to repair the Everbode organ, and bychance this man stopped to visit the Lemmens family. Young Nicolas performedfor the guest, and when the tuner heard the dreadful state of the instrument, heoffered to tune it.

Later in life Lemmens recounted the story for his teacher Fetis, He toldhow, when the instrument was first tuned, he experienced "the most disagreeablesensations.vf Growing up with no other. musical reference, his ear had adapted tothe arbitrary intervals of the mistuned harpsichord, and it took him a long time tofeel comfortable with the intervals of the tuned instrument. Nicolas Lemmens laterbecame one of the great organ masters ofbis time, a teacher at the BrusselsConservatory, and composer of symphonies, choral works, and keyboard pieces.He was certainly a musician of considerable skill, with a good ear, and yet as ayouth his ear had completely adapted to an entirely arbitrary tuning. The storyreminds us that our certainty about what tuning system we prefer can be biasedsimply by our habitual way of'bearing, a good reminder for 20th centurymusicians who have grown up with equal temperament.

Many music theorists and musicians of the 19th century contributed to thecounter-current in the evolution of tuning. Equal temperament was establishingitself, but by no means exclusively. In 1812 the Glover sisters in Norwich,England devised the "movable doh" system for teaching singing, using "doh" asthe keytone, and teaching the diatonic major relationships to this fundamental as"re, mi, fa, sol, la, and ti." They were so successful in teaching young children to


sing that their system was later introduced into the English schools. Thirty yearslater the minister John Curwen visited the school where Sarah Glover was teachingthe system, then known as the "Sol-fa system of singing." He was so impressedthat he joined with the Glover sisters to form the Society of Tonic Sol-faists, andhe later published the singing system as Singing for Schools and Congregations.By 1862 there were some 150,000 Sol-faists in England. The movement alsospread to the continent.

One advantage to young singers using the Sol-fa system is that all songs inall keys use the same symbol to express a given interval. That is, the interval of athird from the tonic is "mi." Since this never changes, a student more easily makesthe association between the sound of the harmonic or melodic interval and thesymbol. Curwen advanced the system as a purely just intonation system with thehelp of instrument builder Perronet Thompson who, in 1864, built an "enharmonicorgan" with 40 tones to the octave on three ranks of digitals.

Advances were also made in the understanding of acoustics. MichaelFaraday advanced the theory of resonance? the phenomenon of one vibrating bodysetting a second vibrating body in motion. Jean Baptiste Fourier describedcomplex wave forms as additions of simpler waves. Today, "Fouriertransformation" allows additive synthesizers to create complex musical timbresfrom sets of simple oscillations.

French scientist Jules Lissajous (1822-1880), hooked a large pendulum tohis ceiling. Vincenzo Galilei' s son Galileo had earlier determined that a swingingpendulum keeps perfect time. Lissajous surmised that sound was some kind of awaveform in the air, and he imagined that sound had regular beats like those of theswinging pendulum. To the pendulum he attached a bag in which he made a smallhole, in the bag he placed sand, and under the pendulum he placed a long scroll ofpaper. He set the pendulum swinging in one direction, and pulled the paper acrossits path in a perpendicular direction. The pendulum, the thought, represented thevibration of sound, and the moving paper represented its movement through theair. Through the small hole in the bag the sand spilled back and forth across the


paper; but as the paper moved the sand formed a wavy line, a line we now call a"sine wave. " Lissajous predicted that this was the visual image of a singlefrequency of sound, He was correct, of course, but his theory was not confirmeduntil the invention of the oscilloscope in the early 20th century.

More impressively, however, Lissajous conducted a more elaborateexperiment entirely in his mind. He imagined that two sound waves would interferewith each other, and would therefore create more unusual wave forms. Heimagined that two imaginary pendulums swinging at different rates could swingacross each other's path, and he imagined the resulting images they might makewith sand on the paper below. With this thought experiment, Lissajous predictedthat two tones in perfect harmony would create certain well-defined patterns.Again he was correct, and later pro-ven so by oscilloscope images. Each harmonicinterval has its own unique signature interference pattern, and these we now call'Lissajous curves."

Just as the wave of equal temperament was about to wash over the musicworld, German scientist Hermann Helmholtz, beganhis studies of acoustics.Helmholtz was a great general scientist, unfettered by specialization. Between1845 and 1894 he wrote over a hundred definitive papers and books on theconservation of energy, propagation of nerve impulses, the theory of complexcolors, a review of Goethe's scientific work, optics, muscle movement, weather,electrodynamics, magnetism, the sensation of hearing, the physics of sound, andaesthetics of music. Music was his passion, he was an 'accomplished musician, andhis instrument of choice was a harmonium tuned to just intonation.

He began his study of acoustics in 1852 and wrote On the Nature ofHuman sense perception. He built a "vibrating microscope" using a piece of glassthat oscillated in one direction and a tuning fork that oscillated in theperpendicular direction, connected both to sources of pure tone, and found thegeometric patterns of sound predicted by Lissajous. He published On combinationtones in 1856, followed by On Musical temperament, and On the motion of thestrings of a violin. He reviewed the worle of every music theorist from pythagoras


to Neidhardt and Lissajous, and was arguably the most knowledgeable person inhuman history on the complete physics, physiology and aesthetics of sound. Hismagnum opus on the subject, On the Sensations of Tone, was published in 1862,is still in print, and has been translated into every modem language.

The design for Henry Poole's lOO-tone per octave organ was published in1867, but was never built. Similar designs by other researchers, includingHelmholtz, proved too cumbersome to be practical for musicians. (From

Helmholtz/Ellis, 1885)

After exhaustive accounts of the physics and physiology of sound, and themathematics of harmony, Helmholtz at last engaged the historical dialogue ofintonation theory and aesthetics. "There is nothing in the nature of music itself todetermine the pitch of the tonic of any composition, II states Helmholtz. "[It is]


necessary for musicians to have free command over the pitch of the tonic. Forsingers these transpositions offer no difficulties ...But the matter becomes muchmore difficult for musical instruments ... [that] only possess tones of certain definitedegrees of pitch, 1153 the keyboards and fretted instruments.

"The justly-intoned chords," said Helmholtz, "...possess a full and saturatedharmoniousness; they flow on, with a full stream, calm and smooth, withouttremor or beat. Equally-tempered or Pythagorean chords sound beside them rough,dull, trembling, restless. The difference is so marked that every one, whether he ismusically cultivated or not, observes it at once. ,,54 He pointed out that chordinversions, modulations, the contrast between consonance and dissonance, andcontrast between major and minor chords "are much more decided andconspicuous" in just tuning. Therefore the effect of these musical techniquesbecomes "much more expressive." On the other hand "when the intonation ofconsonant chords ceased to be perfect. ..the differences between their variousinversions and positions were, as a consequence, nearly obliterated."

"In a consonant triad every tone is equally sensitive to false intonation,"Helmholtz observed, "...and the bad effect of the tempered triads dependsespecially on the imperfect Thirds. II He outlined the four species of beats heard ina tempered major triad due to impure intervals, and remarked that they were"always quite audible," and that they "strike the ear as a marked roughness ...Thebeats arising from the Thirds ...are decidedly disturbing in the middle positions,even in quick time, and essentially injure the calmness of the triad. II Helmholtz, thescientist and musician, at once appreciated both the technical and aesthetic natureof pure sound. "When I go from my justly-intoned harmonium," he observed, 'to agrand pianoforte, every note of the latter sounds false and disturbing. "

Helmholtz lamented that "These are unpleasant symptoms for the furtherdevelopment of art. The mechanism of instruments and attention to theirconvenience, threaten to lord it over the natural requirements of the ear, and todestroy once more the principle upon which modem musical art is founded." Inconclusion he said that "after all, I do not know that it was so necessary to


sacrifice correctness of intonation to the convenience of musical instruments. Assoon as violinists have resolved to play every scale in just intonation, which canscarcely occasion any difficulty, the other orchestral instruments will have to suitthemselves to the correct intonation of the violins. Horns and trumpets havealready naturally just intonation. n

A solution to the problem, in other words, was not out of reach. Helmholtzand others in the late 19th century proposed solutions. Henry Poole built a justintonation pipe organ, and designed (but never built) a keyboard of 100 tones peroctave. Perronet Thompson's 40-notes-per-octave organ included a device forcorrecting the changes caused by temperature on intonation. He claimed to havetaught a blind organist to play the instrument in six days "thereby settling thequestion of the practicability of just intonation on keyed instruments." ColinBrown at Andersonian University in Glasgow designed a 40-tones·per~tavevoice harmonium, and built three such instruments. One of these was later playedby 20th century composer Harry Partch who found it "easy to play and itsintervals and triads a delight to the ear. ,,55 R.H.M. Bosanquet applied the 53-tonedivision to an enharmonic organ, and founded a company with T.A. Jennings tooffer custom built organs with up to 84 tones per octave. These inventions werenot complete solutions, and they posed problems of construction and play, but theyheld promise. Nevertheless, when Helmholtz died in 1894 the general acceptanceof equal temperament slowed further research. The secrets of harmony hadbecome visible, but the science of building instruments to play pure harmonyfaltered as equal temperament dominated western music.


12

A paradigm entrenched

The two streams of music - harmonic and tempered - bad now wandered alongquite distinct paths. Not only keyboards, but guitars had become popular.

The C, F. Martin company, founded in 1833, and Gibson Guitar Company,founded in 1896, were turning out high quality acoustic guitars, and theinstruments were popular. Keyboards and frets sent the tempered stream along itsown path. Certain a cappella choirs and chamber groups continued to performharmonic music, but from the perspective of 1900, the future of music, at least inthe west, looked like equal temperament.

By 1937 the Encyclopedia of Music and Musicians confirmed the currentparadigm by claiming that tuning by pure intervals was "strictly impossible," that"the mechanical obstacles to pure tuning in keyed instruments are insurmountable,and the question of some compromise is hence necessary.':" In 1950, LlewelynLloyd wrote in Intervals, Scales, and Temperaments that "The present supremacyof equal temperament will remain unassailable until someone invents a reallypracticable means of playing ... alternative notes ... by a single key."S7

The main currents of musical theory and composition of this century have


accepted the compromise deemed inevitable. The parallel chords, wholetonescales, and serial techniques of composers like Debussy. Ravel, Stravinsky, andSchonberg carried Beethoven's instrumental music into ever new subtleties ofexpression. The chords of Schonberg and Scriabin built on fourths. tookadvantage of equal temperament, and moved music toward an atonal aesthetic.These composers fully embraced tempered harmony, in part because their musicwas a rejection of traditional harmony and tonality all together, but equally sobecause that was the intonation available on modem instruments. Inmuch modemmusic dissonance is featured, and a sense of tonal center is obliterated.

In Problems of Harmony Schonberg introduced twelve-tone and serialtechniques. His "emancipation of the dissonance," was a movement to put allintervals of the cbromatic scale on equal aesthetic footing. Schonberg was wellaware of pure harmonics, and saw the major scale as the addition of the tones ofthe tonic, dominant and subdominant major triads. He wrote that "we actually tosome extent hear and to some extent feel this relationship in every sounding tone."He equated the other chromatic tones also with higher harmonics, saying "if wenote the more distant overtones (up to the 13th) ... we find the chromatic scale. ,,58

In fact, as Henry Cowell later pointed out, this is not exactly correct. For example,Schonberg states that the 13th overtone of G is Eb, but the 13th overtone is a 13/8interval that is almost a quartertone sharp ofEb, a "neutral sixth" that has beenused in certain Arabic and other scales. This tone has no western chromaticcounterpart.

Cowell knew this, and felt that the natural evolution of chord developmentwas not 12-tone chromatic, but rather would follow the harmonic series, "from theseventh overtone upwards." He adds that "There seems to be need of such asystem to further the understanding of contemporary material, which has had DO

adequate theoretical coordination. ,,59

Paul Hiodemith faced the same contradiction. He attempted to explain thewestern chromatic scale by the overtone series that did not entirely support it, yethe understood that tonality itself was a natural phenomenon. He wrote in A


Composer's World that «In music we cannot escape ... tonality. The intervalswhich constitute the building material of melodies and harmonies fall into tonalgroupments, necessitated by their own physical structure and without ourconsent." He calls tonality "a very subtle form of gravitation," and adds that"some composers who have the ambition to eliminate tonality, succeed to a certaindegree in depriving the listener of the benefits of gravitation. To he sure they donot, contrary to their conviction, eliminate tonality.'.60 Hindemith may have comeup with the best definition of equal temperament: "a compromise which ispresented to us by the keyboard as an aid in mastering the tonal world, and thenpretends to be that world itself. ,>61

The aleatory (chance) music of John Cage and Karlheinz Stockhausen tookthis revolution against tonality to ever further Jimits. Cage's experiments withrandom sound, prepared piano, and even complete silence in 4 '33" had nothing atall to do with harmonic proportions. It was sound Dada. He was not onlyliberating dissonance, but liberating noise and even silence, as musical elements.

Blues and jazz forms were developed in the U.S., from the early 19thcentury, and although this music evolved within the equal tempered stream, itproduced some interesting intonation experiments. Equal temperament gave jazzplayers the freedom to roam unrestricted among tonal centers, and the musictended toward chord sequences with a widening definition of chromatic chordalresources. Musicians like Louis Armstrong, Charlie Parker, Sarah Vaughan, andothers were delivering music into new realms of sound, speed, and rhythm. LaterMiles Davis, John Coletrane, and a new generation of jazz musicians advanced theform, playing up and down through fast chordal cycles, leaving the melody forspontaneous inspiration, integrating Afro-Caribbean polyrhythms and Latin-Oriental off-beat melody lines. All of this was helped along by the ease of equaltemperament. Horn players like Davis and Coletrane, however, instinctivelyshaped the intonation to fit the mood of a passage. Later, in the 19705, flutist PaulHom recorded over his own echo inside the Pyramids and the Taj Mahal, working


with the natural resonances of these sites as well as the natural harmonics of theflute tones.

String bass players, like singers and hom players have the freedom to placethe intonation where they want to because they aren't limited by frets. However, abass player with a good natural ear for harmony must be careful not to clash withthe lower notes of the piano. This is because the string bass player mayinstinctively want to stop the note at the natural harmonic, something the pianocannot do. Jerry Coker, a former saxophonist with Woody Herman, Stan Kenton,and others, mentions this problem in his book on jazz improvisation. "If thepianist's left-hand base note coincides with the root played by the bass player,there is an intonation problem." In this case, the bass player either has to stayaway from this note, or play the piano intonation. Most modem musicians, ofcourse, learn to play the piano intonation since the piano cannot change.

Blues musicians also wandered instinctively from equal temperament. The"blue notes" heard in some of this music tend toward the seventh harmonic whichis obliterated in equal temperament. In "Septimal Harmony for the Blues," DudleyDuncan analyzes W. C. Handy's St. Louis Blues, in G, and concludes that " ...thesinger or horn player uses inflection of the melody or countermelody for color (inways probably beyond the scope of any simple theory), [but] for simple harmonythe blue notes are merely the just intervals 7/6, 7/5, and 7/4.',(j2 These are the blueminor third, augmented fourth, and minor seventh. The blue Eb in the key of G,nominally an augmented fifth, Duncan suggests is, in the blues treatment, thequartertone 14/9 which forms a blues third above the fourth.

Guitar players can find the pure seventh harmonic of a guitar string bylightly touching the string about halfway between the second and third fret as thestring is sounded. On the bass E string this is a D note. Compare this note to thehigh octave D at the twelfth fret of the D string tuned two fretted fourths from E.These two "D's" are clearly not in tune with each other. The harmonic D is muchflatter" and this is the blues seventh. The tempered flat-seven, found with the frets,is the worst of all the tempered intervals when compared to its natural harmonic.


The blues singers and hom players simply used their instinct and good ears to findthe pure harmonics that were found by the Hannonists of ancient times.

Guitar players know that when they tune perfect harmonics from the bassE-string to the treble E-string, the two E's are not in tune with each other. Bymatching the fourth harmonic of the bass E string to the third harmonic of the Astring, etc., and by matching the fifth harmonic of the G string to the fourthharmonic of the B string, a guitarist ascends by four pure fourths and a pure third,but this lands flat of two octaves. The interval by which this is flat is the commaof Didymus that the reader may remember from the first five chapters, adiscrepancy that has been known by musicians for two thousand years. It is about1/5 of a semitone. The ear hears this, and with a flexible instrument, the ear caneasily guide the necessary intonation changes. Frets however, demand equaltemperament, which disguises these natural discrepancies by spreading them outacross the octave and polluting every interval. Noticing the intonation andplacement of these harmonics on a guitar is a good way to hear, and even see, theproblem with tempered. intervals.

The fifth harmonic of the G string is a pure major third of G, a B note.When you play the B harmonic on the G string, you are dividing the string intofive vibrating parts by touching the string at one-fifth its length, just slightly flat ofthe fourth fret. Play the B harmonic on the G string of a guitar. Now press downand hear the fretted B on the G string. These two B's are noticeably out of tunewith each other. You can see how far behind the fret your finger is when you getthe ringing B harmonic. On a Martin D-35 the fourth fret is more than liS-inchsharp of the harmonic. They say in framing a bam that 118 inch tolerance isacceptable. To the delicate art of harmony, it is a problem.

Modern popular music, particularly electric rock guitar playing, is based onfast and colorful chord changes, rather than on the subtleties of pure harmony.Helmholtz predicted this evolution of instrumental music a century ago when heacknowledged that "in rapid passages ...the evils of tempered intonation are butlittle apparent," and added, "We might, indeed, raise the question whether


instrumental music had not rather been forced into rapidity of movement by thisvery tempered intonation, which did not allow us to feel the full harmoniousness ofslow chords to the same extent as is possible from well-trained singers, andinstruments had consequently been forced to renounce this branch of music. ,,63

Fast, chord-based music is by no means aesthetically inferior, and indeedgreat music has been created by artists in the blues, jazz, and rock genres. Theissue for musicians is one of musical options. The simplicity of equal temperamentcreated new options, but also foreclosed certain other options. As musicians, dowe merely accept those limitations, or do we chose to investigate all the musicaloptions that could enhance our ability to make expressive music? The questionwill have different answers for different artists, but asking the question is theissue, and seeing the options is the opportunity for modem musicians to expandtheir creative resources.

Composer, producer, and music innovator Brian Eno mentioned thislimitation of the 12-tone keyboard in a March, 1995 interview in Keyboardmagazine. "The keyboard," said Eno, "gives you distinct islands rather than acontinuous set of pitch possibilities. That's a disadvantage for keyboards."?' SteveO'Keefe, editor of the Piano newsletter was even more blunt in saying "The pianois one of the most frustrating instruments on God's green planet. .. .If you want toplay the piano you are locked into that vision, unable to use anything outside ofwhat is provided.?" The guitar, of course, has the same problem. "Frets areslightly out of tune," wrote John Schneider, in Acoustic Guitar magazine in 1994,"enough to produce a warbling effect on every chord you play and to sabotage youevery time you try to tune. ,06<) These musicians are addressing the aestheticlimitations of their instruments, a healthy move for any artist.


13

The new Hannonists

The second stream of modem music, pure harmonic-based music, seemed atrickle at the dawn of the 20th century. Some choral groups and string

ensembles kept the 18th century traditions alive, but the seeds of HermannHelmholtz, Perronet Thompson, and other innovative instrument designers of the19th century fell on fairly dry ground. As we will see, however, these seeds wouldsurvive and take root once the weather changed.

One modem Harmonist link from the 19th century was Julian Carillo(1875-1965) of Mexico, a solo violinist who experimented with 24-tone, 48-tone,and 96-tone equal temperaments in an effort to find an equal tempered solutionthat also served the harmonic proportions. He called his system "el Sonido Trece,"the 13th Sound, and formed the "13th Sound Ensemble" which performed hismusic such as Preludio a Cristobal Colon for harp-zither, octavina, cello,trumpet, and soprano voice. He modified a guitar to achieve quartertones, andbuilt many of his own instruments. Near the end ofhis life, in 1962, he completedMissa de Larestauracion for Pope John xxm, written in quartertones.


Ferruccio Busoni (1866-1924) was another link. He transcribed Bach andLiszt; edited the Well Tempered Clavier; wrote nwnerous operas, choral pieces,violin concertos and chamber pieces including Fantasia contrappuntisttca; andexperimented with 36-tone equal temperament. He wrote ''We have divided theoctave into twelve equidistant degrees, because we had to manage somehow, andhave constructed our instruments in such a way that we can never get in or aboveor below or between them. Keyboard instruments, in particular, have sothoroughly schooled our ears that we are no longer capable of hearing anythingelse - incapable of hearing except through this impure medium. Yet nature createdan infinite gradation - injinite!'~7

English musicologist Kathleen Schlesinger made a great contribution to theHannonist school with her 1939 work The Greek Aulos. Schlesinger maintainsthat the evenly spaced holes in the Greek flute suggest that Greek music was basedon the arithmetic progression, divisions of strings (or flutes) into 11, 12, 13, 14,15, etc. equal parts (not equal temperament). Whether or not she is correct aboutGreek musical performance, her system of equal divisions gives harmonicproportions and allow her system to achieve versions of all the diatonic Greekmodes. For example, the Mixolydian scale uses a 14-division of the string, theDorian an l l-division, and so forth. The Greek enharmonic and chromatic scalesare achieved in the Schlesinger system by doubling the number of divisions.Schlesinger advanced the 11th and 13 harmonics which generate natural harmonicquartertones, and her work inspired American Harry Partch who visited her inLondon in 1935.

Henry Cowell, who corrected some of Schonberg's approximations withregard to the harmonic series (see chapter 12), also inspired the AmericanHannonist movement. He encouraged experimental composers of all persuasionsby publishing their work through his New Music publishing company. He alsoinvented a keyboard percussion instrument, the Rhythmicorn, with innovativeinstrument designer Leon Theremin. His work with Theremin was early evidence


of a third stream in 20th century music, a stream that eventually would helpreunite keyboard music and pure harmonic music: electronic musical instruments.

The "radio valve," or diode tube had been invented by John AmbroseFleming in 1901, and five years later the first primitive electronic instrument, theTelharmonium, was built by Thaddeus Cahill. The Telhannonium was a 200-tonemachine in which a rotating cam generated the tones of the scale as selected by akeyboard. The signal could be sent over the telephone line, and thus the name. Themachine itself, however, was a monster which Cahill moved to New York City onseven railway flatcars. At about this same time the newly invented tape recorderled to experiments in "musique concrete" by George Antheil and Edgard Varese.

Varese also used a "theremin" invented by Leon Theremin in 1927. Thetheremin is a vacuum tube that sends a signal up an antenna. The instrument isplayed by a musician who moves both hands along the antenna thereby generatingvarious frequencies. The instrument was still in use in the 1970's by groups suchas the Beach Boys (Good Vibrations), and Led Zeppelin (Whole Lotta Love).

By 1934 Laurens Hammond had invented and built the first fully integratedelectronic keyboard, a refined Telharmonium utilizing a synchronous electric clockmotor, toothed gears, two upper keyboards and a foot-pedal keyboard, the firstHammond organ. Near the gears were magnets wound with wire which caused themagnetic field to fluctuate as the teeth passed. Each coil output a sine wave, andthese were added together to adjust the timbre using manual drawbars. This wasearly additive synthesis. The tuning gears were set to the ratio 196/185, anapproximation of the equal tempered semitone. Equal temperament had beenturned into a harmonic proportion, albeit a rather cumbersome one. This eventforeshadows the ability of electronic instruments to achieve variable tuning, but in1934 Laurens Hammond had not yet imagined this potential.

It was at this same time that the 17-year-old Lou Harrison became a studentof Henry Cowell. The Harmonist stream was swelling. Cowell introducedHarrison to international cultural music as well as to great composers like Vareseand Schonberg, and taught him counterpoint and composition. In 1946 he


conducted Charles Ives' Third Symphony at Carnegie Hall. Harrison developed alifelong passion for cultural tuning systems and pure harmonic intonation.Harrison's famous line, "Just intonation is the best intonation" later became theslogan of the Just Intonation Network in San Francisco.

In 1987 the editor of 1/1, the Network's newsletter, David Doty,interviewed Harrison who said "If unpolluted by other instruments, stringorchestras, left to their own devices, play quite well in just intonation. ,>68 By thetime of this interview, the Harmonist stream had swollen to a steady flow, and LouHarrison was one of the mentors. "Just those tiny, tiny differences, sometimesbetween a just and a tempered interval, interms musical, makes all thedifference," he said. ''The real intervals of music are very beautiful, there's nodoubt about it, and moving through them is beautiful ... when you can havesomething real, why mess everything up?" Jazz pianist Keith Jarrett recordedHarrison's Piano Concerto for well tempered piano, and violinist Lucy Stoltzmanjoined him on Harrison's Suite for Violin, Piano, and Small Orchestra. Therecording was released by New World Records in 1989.

An influence on Harrison, and on the entire revival of Hannonist thinking inthe 20th century was composer Harry Partch (1901-1974), a pioneer who set outon his own at the age of 22 to learn how to make the music that his ear heard andhis mind conceived. He educated himself in the Harmonist tradition, devised ascale of 43 tones per octave, conceived a notation system to conununicate hisideas, wrote music for this set ofhannonic intervals, built his own just intonationinstruments to achieve the intervals, and performed the pieces himself and with hisfriends and musical associates. As if this was not enough for one lifetime, Partchset out his theories in a book that became one of the guiding lights of 20th centurypure harmonic music. He began the book in 1925, completed a draft three yearslater, and published the final version, Genesis of a MUSiC, in 1948.

Partch called his system "monophony" to emphasize the relationship of alltones to a fundamental consonance represented by the number" 1" or theproportion Ill. "A tone, inmusic," Partch explains, "is not a hermit, divorced


from the society of its fellows. It is always a relation to another tone, heard orimplied." Partch outlines four basic fundamental concepts of pure harmonicmusic:

(1) The scale of musical intervals begins with 111, absolute consonance, andprogresses through more complex proportions - 2/1, 3/2,4/3,5/4, etc. -which become more dissonant as the proportional nwnbers get larger.

(2) Every harmonic proportion has a dual identity, as a higher harmonicidentity in relation to its tonic, and as a tonic itself.

(3) These tonalities represent the nature of sound and an immutable faculty ofthe human ear to perceive its qualities.

(4) Human use of consonance in music has proceeded from the unison, to thesimple harmonic intervals, to the more complex, toward the «infinitude ofdissonance. "

Partch's 43 tones per octave are based on all the tones generated by the first11 harmonics. He defines the "3" identities - the fifth and the fourth as "power"intervals; the thirds and sixths (5/4, 815, 6/5, 5/3, 1119, 18/11, 917, etc.) whichincludes approximate quartertones and other intervals not found in a 12-tonechromatic scale, are "emotional" intervals; the seconds (817, 9/8, 10/9,81/80, etc.)and the sevenths (15/8, 9/5, 7/4, etc.) are "approach" intervals, and the intervalsbetween the fourth and the fifth, the quasi-tritone intervals are "suspense." Onsome of his instruments he lays out these intervals in a "tonality diamond" thatrecognizes each interval's dual identity. "The ear," says Partch "does not budgefor an instant from its demand for a modicum of consonance in harmonic musicnor enjoy being bilked by near consonances which it is told to hear asconsonances. "

Music to Partch is "corporal" rather than "abstract," arising from spokencadences, song, recited or intoned poems, chants, dance, movement, and drama.Partch takes up the theme of Zarlino that the human voice naturally sings pure

The Story of Harmony

intervals, and he defends singers who struggle with the intonational ambiguity ofmodem music. "Much of the oft-heard railing against the intonation of singers isscandalously lacking in candor," he says. «As composers and educators we givethem an accompanying instrument - the piano - which is continually at odds with

An octave of Harry Partch's Chromelodeon II keyboardshowing the placement of his 43 harmonic tones.

their instincts. After they have mastered this incongruity we pose them in an acappella choir or before an orchestra, where they are at the mercy of eachintonational whim of concertmasters and conductors, and proceed to criticize themfor their 'bad' intonation." He adds finally that "The great need for a betterinstnunent than the piano in the training of singers and for accompaniment ofsongs is too self-evident to be labored. ,.69

Partch also pointed out that the pure sound of harmony was due to morethan just the harmonic series and the combination tones. The ear, he suggested,


recognizes the simple harmonic proportions -.3/2, 5/4, etc. - because when musicaltones are perfect ratios of each other the sine waves of the tones match up exactlyat certain points. For example, if the pure third (5/4) and the pure fifth (3/2) areplayed together, five cycles (wave lengths) of the first tone fit exactly into sixcycles (wave lengths) of the second tone, thus creating a perfect and pleasingharmony. The wave length is the inverse of the frequency, so the mathematics ofthat simple harmony can be written as:

5 X 4/5 = 20/5 = 46 X 213 = 12/3 = 4

That is, 5 cycles of the 5:4 tone are precisely equal to 6 cycles of the 3:2tone, and they fit into exactly 4 cycles of the root tone. These three tones playedtogether form, of course, the major triad. This perfect fit of sine waves isharmony. On the other hand, the tones of equal temperament do not make anyperfect harmonies (except octaves) because the tempered tones are not perfectfractions, they are what mathematicians call irrational numbers. The 12th root oftwo, for example, is 1.05946 ... a decimal that never ends. So, when the temperedthird (1.25992 ...) and the tempered fifth. (1.49831...) are played together, the sinewaves of the two tones never match up exactly, and the harmony is imprecise.

Genesis of a Music was an underground musical classic, read by manyyoung students, and inspiring them to open their hearing to the delicacies of pureharmony. One of these young students was composer Ben Johnston at theCincinnati Conservatory. Johnston had been influenced by Scriabin and Debussy,but also by Ives, and Carrillo. In high school he had taught a fellow student toplaya Bach piece in just intonation, entirely by instinct and without training. Helater recalled "I could hear very early that the piano was out of tune and she wasin tune. That made me very aware."70 Partch's approach to music resonated withJohnston who traveled to California to work with him. Johnston had a great ear,and was able to intone Partch's intervals, singing on demand a 16/15 interval or a


16/11 interval. When Partch's instruments were slightly out of tune, Johnstonwould notice by comparing his voice intonation. Partch was impressed, and heaccepted Johnston as a student.

Johnston. now Professor Emeritus of Music at the University of Illinois,credits Partch with opening the floodgates of extended just intonation in the west,but Johnston himselfhas been equally influential. 'Western music has based itselfon an acoustical lie," is a now famous remark Johnston made in the notes to theFine Arts Quartet's performance of his String Quartet No.4, on April 28, 1974.He explained this remark in a paper that appeared in the Proceedings of theAmerican Society of University Composers. ''The acceptance of keyboardtemperaments in the sixteenth and seventeenth centuries ... impoverished the logicof harmony and tonality by weakening the perceptibility of consonance anddissonance ... .it took only a little over a century for the exhaustion of the freshpossibilities of the system to become a major aesthetic problem. ,,71

Johnston compares equal tempered music to a movie that is slightly out offocus. The audience, he suggests, may not be consciously aware, but they arestraining to see until the film is brought into focus and everyone notices that theywere straining. The same phenomenon happens with music that is brought intopure harmonic tune. "Music," says Johnston, "is not simply an intellectualexercise; it is a physical response to sound," a point made by Lou Harrison as wellas by Harry Partch.

In the same year that Partch's Genesis of a Music was published, 1948, theRadiodiffusion studio was opened in Paris by Pierre Schaeffer and Pierre Henry,the Radio Cologne studio was opened in Germany by Friedrich Enke, and RadioCorporation of America engineers Herbert Belar and Harry Olsen built the RCAMark sound synthesizer based on the ENIAC computer. A decade later MaxMathews at Bell Labs predicted "the computer will become the ultimate musicalinstrument." In 1964 Cornell University doctoral student Robert Moog introducedhis first Moog synthesizer to the public. The sound synthesizer used a voltage-controlled oscillator conected by plug-in telephone jack patch cords. Electronic


music was a new fact of musical life, and this change took place just as arenaissance of pure harmony was flowering. One of the-first to bring these twomovements together was Wendy Carlos" Carlos's original compositions have beenperformed by the London Philharmonic and Boston Symphony Orchestras, and bythe Kronos Quartet. In 1971 she used synthesizers to create the music for StanleyKubrick's film A Clockwork Orange. She used a new Moog synthesizer to recordSwitched on Bach in 1968, one of the largest selling classical recordings evermade. Although this early Moog was tuned to equal temperament, and evensuffered from pitch drift at that, Carlos saw the potential for developing electronicmusic in the direction of pure harmonics.

InJlISf Imaginings, a composition from the recording Beauty In The Beast,Carlos used a Hewlett-Packard computer to access the tuning tables ofsynthesizers to achieve pure harmonics throughout. In Poem for Bali, shecombines Indonesian and western scales, sounds, and styles. She has alsoexperimented with meantone and other historical temperaments, and believes thesetemperaments that favor certain keys may evolve into new versions with the aid ofdigital synthesizer technology. In "Tuning: At the Crossroads," in the ComputerMusic Journal in 1986, Carlos notes that "computer-controlled synthesis ... hasinherent need to respect these heretofore inescapable limitations [of equaltemperament]." Among the many scales she uses, she describes a scale built fromthe first 27 harmonic overtones, mapped to the twelve tones of the keyboard. Thisscale is:

111 17/16 9/8 19/16 5/4 21/16 11/8 3/2 13/8 27/16 7/4 15/8 111

These harmonic tones are mapped to the 12-tone chromatic keyboard as:

C Db D Eb E F F# G Ab A Bb B C

96 The Story 0/Harmony

In this natural harmonic scale the Eb, F, F#, Ab, and A are all quartertonesthat have no direct counterpart in 12-tone equal temperament. When modulating,Carlos uses an external computer to control her precise tuning. "There is scarcelya more worthwhile venture to pursue as soon as possible," Carlos says, "thanadopting a standard for and then manufacturing at least a limited edition of thesenew keyboards" that allow musicians tuning flexibility.

In 1964, La Monte Young, also influenced by Partch and Lou Harrison,wrote and performed The Well-TunedPiano, a 7-hour piece for justly tunedpiano, emphasizing the seventh harmonic minor third 6/7, and major third 9/7, andtheir inversions as the sixths. His The Second Dream of the High-Tension LineStepdown Transformer is based on C, F, F#, and G built on the harmonics 12, 16,17, and 18. Here the ''F'' is the tonic 111,the F# is 17/16 (the same semitone usedby Wendy Carlos in the scale above), the G is the classic just wholetone 9/8, andthe C is a pure harmonic fifth of 3/2.

La Monte Young influenced Glenn Branca who, in 1983, performed hisSymphony No. 3 inNew York, a large ensemble, rock influenced piece based onhigh harmonics, and their combination tones, "the first 127 intervals of theharmonic series," according to Branca, achieving a result he describes as a "fieldof sound."

In 1978 Dave Smith and Sequential Corp. built the first microprocessor-controlled, programmable keyboard, the Profit-5. At the National Association ofMusic Merchants trade show in June of 1981, Smith introduced the idea of aMusical Instrument Digital Interface (MIDI) protocol. The following year the newprotocol was adopted, a visionary move by the industry because it allowed alldigital electronic instruments to communicate.

By this time, the pure harmonic movement was once again a wide, flowingriver with many contributors from around the world, including Terry Riley andPandit Pran Nath; Arthur H. Benade, whose Fundamentals of Musical Acousticshad been publisbed in 1976; James Tenny; Pauline Oliveros; Joseph Vasser whoexperimented with 19-tone equal temperament; Franz Herf with 72-tone, and


Adriaan Fokker with 3 l-tone systems; David Rayna and Harold Waage, who bothdeveloped computer-enhanced systems; William Alves who wrote on theory;Carter Scholz and Robert Rich who helped establish a tuning standard for theMIDI protocol; and Douglas Keislar, George Kirck, and Larry Polansky whoadvanced computer applications for pure harmonic music. John Chalmers, whopublished the exhaustive study of historic harmonic scales, Divisions of theTetrachord, published the journal Xenharmonikon to help expose musicians tothe work of theorist and composers like Erv Wilson who has advanced theunderstanding and application of historical and international harmonic structuresin music.

In 1985 David Doty, Henry Rosenthal, Larry Polansky and others foundedthe Just Intonation Network in San Francisco, and began publishing Ill, theNetwork newsletter featuring the work of the composers, musicians and writers inthe Harmonist tradition. In 1988, Electronic Musician magazine writer, andexperimental musician, Scott Wilkinson wrote Tuning In, Microtonality inElectronic Music, further exposing electronic musicians to the potential of pureharmonic music. Clearly the Harmonist seeds had taken root in the digital,electronic age, bringing thousands of years of acoustic knowledge and harmonictradition together with advanced musicall instruments. "Lack of access to suitableinstruments has always been an albatross around the neck of micro tonal music,"wrote Douglas Keislar in Computer Music .Journal. Harry Partch had earlierwritten that "to produce music in Just Intonation we must have instruments, andinstruments are no small problem." The time had come for that problem to besolved.

Keyboards gave music simplicity. Pure harmonics give music itsfundamental nature. The question we might now raise is whether we can haveboth> whether we can restore pure intonation to music without sacrificing the easeand simplicity that modem music demands. The technology available forinstrument making has evolved several orders of magnitude since the days ofBeethoven and Helmholtz. It is possible that the nineteenth century acceptance of a


sixteenth century compromise to the two thousand year old tuning problem is notthe end of the story of musical intonation.

Music evolves. Music grows with the spirit of the times, with technology,and with the skill and knowledge of musicians. Early musicians discovered naturalharmonics by eat, and from these patterns ofhannonic resonance, scales andmodes evolved on every continent, in every culture, in a virtually infinite numberof variations. Most musicians are aware that the harmonics of their voice orinstrument strings do not match the tuning of the piano nor the frets of a guitar.Musicians have been led to believe that the problem was insurmountable, and thatthe compromise was therefore inevitable. They have been misled.

With today's computer technology we are presented with an historicopportunity to restore pure harmony to music while retaining aU the advantages oftemperament. We can now open a great door to long ignored musical possibilities.The two streams of music may once again merge and flow as one.


14

Justonic

In1979, when the first microprocessor-controlled musical instruments becameavailable, and the early personal computers were available, Bill Gannon was a

self employed accountant. He was also a former professional bass player from amusical family. Gannon had gained a working knowledge of computers, and hadan interest in the new synthesizers. He noticed, however, that the synthesizers wereusing the tempered scale, and he thought that this was a failure to take advantageof the technology. Surely, he surmised, there must be an electronic/computersolution to the problem of free modulation and pure harmony.

Gannon did some research, investigated historic scales, and shared his workwith his friend and associate Rex Weyler, an amateur musician with somebackground in mathematics and engineering. By way of introduction, these are us,the authors of this book. In 1991 we proposed a solution that would takeadvantage of current technology, restore pure harmony to music, and still retainevery expediency of temperament, that is, the advantages of ease and speed,simple key modulation and chord changes. In 1993 we founded Justonic Tuning


Inc. in Vancouver, British Columbia. We built a prototype of our invention andwrote a patent with the intention of marketing our solution to the music industry.Our patent attorney Jeffrey Haley, a musician himself who helped the us reviseand file our first patent, was optimistic that our solution would work and that ourpatents could be protected. We found a few visionary investors and set to work onbuilding a commercial version, software to drive the system, and a synthesizerwith the tuning resolution to make the most of it.

It seemed to us that most music could be enhanced by pure harmonictuning, pending the availability of a workable just intonation system. For somemodern music, based on dissonance and distortion, the change to pure intonationwill hardly matter. However, for music based 011 melody and harmony, thedifference is significant. A pure chord and a tempered chord are audibly different.A pure chord rings with unmistakable internal resonance. A tempered chordwobbles and beats, and there is no way around it, except to smother the effect inever more chaos. A chord is either inperfect tune or it is not, and no convention orspecial sound effect can change that natural fact.

The acceptance of the tempered scale comes from the erroneous belief thatthe problem is insurmountable. But is it? Inthe nineteenth century theorists triedto solve the problem with multiple keyboards, with foot pedals, and other devices.These mechanical solutions were cumbersome. Yet, it is important to rememberthat the problem is not with the music, the problem arises with the limitations offixed-tone instruments. In fact, keyboard and fretted instruments were the first"digital" instruments, because they digitized pitch selection. Piano keys are evencalled "digitals," and indeed on a piano keyboard the player has the choice of thiskey or that, but cannot access the infinite tones between the two.

The compromise solution, the tempered scale, throwing every tone out,putting all harmonies out of tune, seems dubious now that we have the technologyto overcome the mechanical limitations of earlier instruments. We believed thatwith modern digital signal processing, coprocessing technology, and with our


Justonic system, there was a real solution. The problem, it turns out, was not"insurmountable," only tedious, and too complicated for pre-electronic technology.

The first generation of electronic, digital musical iifstruments had thetempered tones fixed into their circuitry. Later versions included microtuningcapabilities, but even so, the ability to tune a single scale to just intonation,without the ability to modulate keys, is of little use in practical music.

Our vision for the Justonic system was to create an instrument that waseasy for musicians to use, since musicians want to make music, not programcomputers and tinker with gear. For a complete just intonation system to work formusicians it must be unrestricted as to pitch calibration and key, modulationamong keys, scale variations, and other parameters, and it must not encumber theprocess of making music.

Through Haley, our patent attorney, we met another music instrumentinnovator, Steve Dame, founder of Virtual DSP Corp. Dame was working on pitchdetection technology for a guitar-to-MIDI interrace, no small problem. Keyboardsare easy to interface with digital sound because each key can send out its ownnote. A guitar interface, on the other hand, must read the pitch of the string andconvert it to a MIDI note. To do this in real time, accurately, has been a problem,and Steve Dame was onto an innovative solution. He also had the know-how tobuild our synthesizer with the tuning resolution we wanted. By January 1996 wehad our first working model of a commercial instrument.

We asked Gannon's brother Oliver, a world class jazz guitarist, to test thisinstrument. He was skeptical at first, saying, "Yeah, its a good idea, but you guysare going to be swimming up Niagara Falls." However, after a few sessions withthe first prototype, he reported that "I'd like to tell you than I am very impressed.I'm well aware of the intonation trade-offs inherent in the equal tempered system. Irigorously tested every type of chord from C major to F# 13(b9 b5) ...all chordssound better in the Justonic system."

We began to introduce the prototype to other working musicians. Jazzsinger Patty Hervey said, "Before the recent presentation by Justonic I had no idea


that there could be a solution to the inherent problems I have often encountered inusing the familiar tempered scale. The Justonic system ...produces correct,pleasing and in tune chords." Classical and jazz violinist Margaret Taylorcommented, "This is a venue to really play in tune ...1was astonished at the puresound of the simple A7 chord ...In tum, the more sophisticated chords are evenjuicer." Flutist Paul Hom told us, "The Justonic tuning is more mellow, becauseof the overtones, definitely. II We felt we had something that worked for musicians,and we began to complete the user interface for the software.

Musicians suggested to us that the software should work with othermicrotunablc synthesizers, so we set to work to interface with these machines.Unfortunately the microtuning systems of the commercial synthesizers have notyet been standardized under the MIDI protocol although Carter Scholz and RobertRich have written a tuning standard that was adopted. As a result, we have had tocreate the unique interface for each microtunable synthesizer or sound card. Twoother problems limit the current state of the art in these instruments: tuningresolution, and speed. The tuning resolution is not fine enough in some instrumentsto achieve heatless, smooth chords. The speed at which the messages are relayedin some instruments makes real time tuning changes difficult.

Nevertheless, it is possible to interface with microtunable synthesizers, andthe Justonic software does so. Naturally, these instruments will improve theirresolution and speed over time. The Kurzweil K2500 operating system, forexample, now allows notes to cbange pitch while they are sounding, one option ofthe Justonic method.

With the Justonic system, the musician plays in the familiar chromaticcontext although the system is using more than J 50 distinct tones within eachoctave. The Justonic software makes the same tuning adjustments a well trainedchamber music player makes based on the parameters of reference pitch, scale,key, and harmonic structure. The system has been designed to be as flexible aspossible, so the musician can achieve the tuning effect he or she wants.

Jus tonic Tuning Inc. 103

Although the parameters of reference frequency and chromatic scale arefixed in most western music, and the musical key is chosen by the composer ormusician, precise just intonation requires that the frequency of each note varieswith the harmonic structure. The reference pitch, the scale, and the musical key

classical harmonic scale after Ptolemy and Mersenne; note thed tritone based on the 7th harmonic.

The scale editor from the Justonic Pitch Palette software.The user can select from a menu of scales or enter any harmonic

scale, mapping the tones to the 12-tone keyboard.

can all change. Within a key, as the music moves through different tonal centers,the tuning adjustments must be made to each note. The Justonic method refers tothis parameter as the "tuning root."

The tuning root can be sequenced or selected by the Justonic software, butthe discerning musician may also want manual control over this parameter.Musicians who use the Justonic tuning method soon discover not only the natural


beauty of properly tuned chords, but the exquisite power and subtle nuances oftuning root variations.

The default reference frequency of A-440 can be changed at the discretionof the musician. A chromatic scale is selected, allowing harmonic experimentswith world music scales. The scale can also be changed in real time as one plays.This allows musicians to work with music that requires more than twelve degreesto a scale. An example would be some Arabic music that may require a majorthird, a minor third, and a neutral or quartertone third. These different thirds canbe mapped to the same key by changing scale. The musical key may bepreselected, and may change at any time during performance. The choice of atuning root sets the dynamic intonation of the instrument. Notes are selected bythc player in the traditional manner using keyboard, guitar. or other controller.

During performance, the scale. key, and tuning root may be changed in realtime by several means:

Sequenced: key and tuning root instructions on a MIDI sequence track for anyprc-detennined musical piece or program.

Software: For free improvisation Justonic bas written software that recognizesharmonic structure on the fly. and assigns a tuning root. However, ambiguouschords such as Cmaj6 and Am7 may be tuned differently. Since these chords aremeant to perform different musical roles, the fact that they can sound different ismusically advantageous. Inthis way.just intonation restores the subtlety of tonaltechniques obscured by temperament. Musicians will discover that this is a pointof artistic choice, and will want manual control over the tuning root. Therefore:

Manual: Key and tuning root may be selected manually in real time with footpedals, a secondary keyboard, or other switches. The switches may control aUtuning root selections in a piece of music, or may simply override the sequenced orsoftware selection at the discretion of the musician. The software automatic


selection option can be turned off entirely. A single "root player" could select thetuning Toot for all networked instruments in an ensemble.

There is an art to selecting the tuning root. Ifone tunes a standardsynthesizer to a just scale in the key of A, and then plays a Bm chord, the intervalsof this chord are too narrow, and either the B must be flattened or the D and F#must be sharpened. The choice is an aesthetic one. If the melodic priority is tokeep a pure whole tone between an A note in an A chord followed by a B note inthe B chord, then the tuning root for the second chord is B, and the other notes aresharpened. If, however, the melodic priority is to keep a pure fourth between a C#in the first chord and an F# in the second, then the tuning root of the second chordwould be F#, and the B would be flattened. There are other harmonicconsiderations as well, and a correct system must allow for a manual selection oftuning root on the fly.

Musicians will find that the tuning root parameter is a important andexpressive musical element. By ignoring this fundamental feature of music,temperament allowed music to be played without the musician having to listen toand adjust the intonation of each note. Musicians will discover new sources ofcreativity once this intonation awareness is reclaimed

There are an infinite number of scale choices. Another of the seriousaesthetic limitations of tempered music is that we have locked ourselves into onechromatic scale. It's as if painters were forced to throw out all but a few simplecolors from their palette, or if dancers hobbled themselves with leg irons. Musichobbled itself to the limitations of 17th century mechanical technology.

We have virtually ignored the harmonic seventh, the neutral thirds andsevenths of middle eastern music, the alternative minor thirds, and variety ofwholetones. All of this we have relinquished for simplicity. Modem technologyallows us to reclaim these lost musical scale resources. The Justonic systemallows the composer or musician to use any cultural, historical, or newly createdjust scale. This opens the whole world of tonal possibilities to modem musicians.


Pure harmonic blues scales, quartertone scales, Raga scales, and hundredsof other choices are available to us as musical resources. One difficulty musicianshave had in the past with these pure harmonic, or "just" scales is that they requireflexibility with each note if one is to play polyphonically. Arabic music, forexample, has developed with very little harmonic embellishment because of thisproblem. Justonic is working with musicians around the world to solve thisproblem for any harmonic or just scale.

Western musicians looking for something new in music will be delightedwith the tonal possibilities available simply by having more scale choices that areeasily available and simple to apply to familiar harmonic forms. Familiar modes -blues, be-bop, minor, etc. - all sound different in various harmonic scales.

For choral or chamber ensembles who strive for pure harmony, the Justonicsystem will be helpful since the chords may be played in precise just intonation,and each player can hear his or her intonation in harmonic context.

Musicians may want to detune an interval up or down to achieve anynumber of musical effects. The Justonic system allows for these additional pitchshifts, and when the default intonation is pure, the musician can more easily hearthe desired detuning effect.

Musicians deserve access to the full palette of tonal resources. They havebeen denied this access by our reliance on an old technology. Justonic points backto a fundamental purity of musical harmonics, but at the same time, points aheadto the whole, vast potential of modern music. The system does not force any styleor technique upon the musician, but rather makes pure tuning available. When thecomposer or musician chooses dissonance or atonal forms, the system supportsthose choices, but when the composer or musician wants pure harmony, they getit. Effects such as modulation, chord inversions, and simple harmonic parts aremuch more expressive and vibrant with just intonation.

There is no way around it: For music to modulate freely, and for allintervals to remain in perfect harmonic relationship, the actual frequency, or pitch,of all notes must be flexible. A good choral group can do it. When we hear that


sweet harmony our ears perk up, our souls lighten, something resonates inside.There is no cheap substitute. When singing groups like Sweet Honey and the Rockor the Nylons sing a cappella on stage; anyone can hear it. Where does that soundcome from? It comes from real harmony.

"Music stands in a much closer connection with pure sensation than anyother art," Hehnholtz noted in the last century, because II••. in music, thesensations of tone are the material of the art." He added prophetically that "thetheory of hearing is destined to playa much more important part in musicalaesthetics ... music has to look for the foundation of its structure."

The Justonic system is based on that very foundation. The authors feel thatthis breakthrough will perhaps help usher in the new era of pure harmony inmusic. Eventually, we might guess, all serious music will return to the pure andnatural harmonics known to musicians for thousand of years, and the era of allinstruments being out of tune will be over.

108 Tne Story of ttarmony


15

Perfect tuning: so what?

Intonation theory is as much a fundamental part of music as color theory is apart of the visual arts, as grammar is to literature, or as ethics is to law. For

four thousand years musicians have investigated the harmonic nature of sound andapplied their discoveries to the artistic making of music.

The design of musical instruments is also fundamental to music. Instrumentdesign has always followed from harmonic theory> but also must answer technicaldemands. Herein lies the challenge to music. As with any craft, skill, orprofession, there is a constant struggle to apply fundamental knowledge to actualresults. The earliest bamboo pipes of Ling Lun and stretched strings of Byzantine,Egyptian and Greek musicians attempted to mimic the natural harmonic structuresof sound, and were successful within limits. In the medieval period, keyboards andfretted instruments introduced to music a valuable simplification, namely theability to strike a key or fret and achieve a tone of fixed intonation. This simplicityhas benefited music for a thousand years. But there was a price: the naggingcompromise of tempering harmonic intonation.


Now, at the dawn of the 21st century, we have the technological ability toretain the simplicity of keyboards and frets while gaining back the perfectharmonic purity recognized by musicians throughout history. Some musiciansreact to this with enthusiasm and curiosity. However, one might also ask: Sowhat?

For about the last hundred years most western music has been played inequal temperament. We've been told in our music theory texts that the problemwas "insurmountable," and we have adapted. Our ears suffer the bad musicalthirds, and perhaps don't miss the altogether lost seventh harmonic. Our triadssound roughly in tune, especially if the music moves quickly. For those who wantpure harmonics, there are a cappella choirs, string ensembles, barbershopquartets. Why do we need pianos that can pray in pure harmonics? Music is doingfine as it is.

Silent movies were doing fine before the talkies came along. In fact, manylearned commentators claimed that audiences would not care about talking movies."If they want talking they can go to the legitimate theater." Radio was doing finebefore television. In fact, learned commentators claimed that "people don't want tosit down and watch something from over the airwaves. People are happy with theirradios. They can listen as they work or play." Writers wcre certainly doing finewith typewriters, and did not need word processors to create great literature. Somepeople believed CD's would never make it. Who wants to spend twenty bucks on aCD when you can get a record at less that half the cost, and it works just fine onthe turntable that you already have.

On the other hand, a few people throughout history have always advancedart and culture by imagining that "doing fine" does not preclude improvement.Aside from making fortunes, the visionaries also have given tools of great value tothe culture. Sound cinema, television, computers, and CD's are not the half of it.Every tool we have was, at one time, an innovation. Let's face it: Neanderthalswere doing fine before hammers.

Justonic Tuning Inc. III

But what about music? What advantages to music can be derived fromenabling keyboards and guitars to play pure harmonics, and to modulate in realtime while maintaining pure harmonics?

The first and most obvious advantage is that music, as we have learned inthis long history ofhannonic tuning, is divided along two parallel paths. Pureharmonic music is isolated from keyboard-and-guitar-based music. The two cannot join forces because the basic intonation systems are mutually exclusive. Bygiving the keyboard/fretted instruments the option of playing inpure harmonics wesimply allow these two strains of modem music to reach each other. There is noartistic down-side in this; there is only creative opportunity. Allowing moremusicians to work together has to be a positive step. Some will take advantage,some won't.

To begin with, a cappella choral groups and chamber ensembles who aredevoted to pure harmony now have a source of pure harmony for rehearsalpurposes. Previously, there was no accurate source for correct intonation. Choralrehearsals have been guided by the excellent ears of a few top directors. ''Weansingers early from the piano, " says Harvard Choral Director Jameson Marvin.because "When the piano plays, the conductor cannot hear or listen acutely forproblems of intonation=? Dr. Robert Barstow, chair of the Department of Musicat the State University of New York at Oneonta wrote to Justonic that "As achoral director, it is a constant hassle to teach my students to sing in tune when infact they most often sing with an out-of-tune instrument, viz., the piano." Now, bygiving keyboards the ability to achieve perfect harmonics with free modulation, thechoral director, chamber group, or barbershop quartet, has a new tool. Thedirector can play the piece, and the singers can hear their precise parts. Just aswriters gravitated to word processors because of the added ease and speed ofediting? not because it made them write better, these musicians will embrace thenew keyboard technology because of the ease and speed of rehearsal. Theadvantage is simple: save time, save work, and get on with the creativity.


Furthermore, these pure harmony musicians can now include pianos andguitars in actual performance if they wish. This is simply a new artistic option,certainly not required, but clearly valuable. The arts evolve, and the advancementof artistic tools is a major contributor to this evolution. Bach and Mozart wrotemusic to be sung and played in precise tune, and they did the best they could withthe instruments they had. We can now achieve what they could only approximate.If a choral ensemble wants to do a pure harmonic Mozart piece with pianoaccompaniment, we can now achieve that. We thereby gain a new advantage. Wegain no advantage by limiting ourselves to seventeenth century technology inmUSIc.

There are many serious modem composers who write for pure harmonicmusic. These include American composers Ben Johnston, Lou Harrison, WendyCarlos, La Monte Young, Terry Riley and others. These composers have beenfrustrated by the lack of adequate instruments. "My own music has had to solvethe problem of how to get the intonations I need from instruments ... not inclinedto play them," says composer Larry Polansky.P

Emmy nominee film composer Stephen James Taylor has used pureharmonic pieces inhis work including The Lion King's Timon and Pumbaa on theCBS Saturday show. He comments that "what has kept many of my colleaguesfrom exploring the microtonal area has been the lack of user-friendly interface.?"He adds that "I have been waiting for quite some time for someone to come alongand do what you are doing." The introduction of these new instruments will notonly satisfy these artists who have already experimented with pure harmonics, butwill introduce this musical option to the broader community of musicians. What isnow a flourishing niche technique in music among musicians with the enthusiasmand patience to meet the challenge on their own, will become, with the introductionof the appropriate user-friendly tool, a standard technique.

San Francisco Symphony Director Michael Tilson Thomas has featuredLou Harrison and other pure harmonic composers inhis programs, earning areputation for making new music accessible to modern audiences. Thomas is a


keyboard musician dedicated to classical as well as modem symphonic music."Who knows," Thomas asks, "maybe now we can start to develop keyboardinstruments that will make acoustically correct adjustments on themselves as youare playing them. ,,7S This dream is exactly what Tuning Inc. has achieved with itsPitch Palette software.

The pure harmonic keyboard thus opens all music to a valuable option, notjust a new gimmick or digital toy for synthesizers; but a priceless and historicmusical technique based on the very foundations of sound. Will musicians workingthe popular genres - rock, country, jazz - want this option? Every professionalpopular musician works to achieve a unique sound. Having this new option willcertainly appeal to some musicians. No single style or technique is going to claim100 percent of the artists, nor should it. However, any serious musician will, at thevery least, want to experiment with pure tuning, and many will adopt it in theirwork. "The effect of music is heightened by being in tune," says composer TerryRiley, "What happens when a note is correctly tuned is that it bas detail and alandscape that is very vibrant. ,,76 Musicians know the problem with equaltemperament, and they know what true harmony sounds like.

One final advantage to pure harmonic music is this: most western music isderived from one single chromatic scale. That tbis scale is out of tune with naturalharmonics is one drawback, but a further limitation is the exclusion of all otherperfectly useful historic and international scales. Computers not only give us theopportunity to solve the harmonic problem, but once this problem is solved thecomputer opens the door to a whole world of harmonic scales, Javanese scales,Arabic scales, East Indian scales, historic scales, Greek, Persian, Romanesque,Gothic ... the list is literally endless. These musical options will also appeal tocertain musicians and will fit with the growing internationalism of music. Forexample, Egyptian musician Fathi Saleh, Egypt's Cultural Attache to France, isthe creator of the "Musiclabe," an instrument for the generation of the Arabicmusical modes or Maqamat. He wrote to Justonic Tuning Inc. that "I am delightedthat you are working on an idea I have dreamt of for years ... one day somebody


making a machine that can tone to any scale and play in real time.?" A greaterflexibility in scale choice will lead to a greater awareness among musicians of therichness of international, historical, and cultural scale options. This can only begood for the advancement of the musical arts.

Musicians know a purely tuned chord from a tempered chord. Pureharmony is the essence of music. Do musicians want it? Well, some do and somemay not. This is what freedom of artistic choice is all about. In 1947 Americancomposer Harry Parch said "Perhaps the most hallowed of traditions amongartists of creative vigor is this: traditions in the creative arts are per se suspect."He adds that "The extent to which an individual can resist being blindly led bytradition is a good measure of his vitality." On the subject of keyboard tuningflexibility, he predicted "It is quite conceivable that an instrument could be builtthat would be capable of an automatic change of pitch throughout its entire range... the problem of transposition may be considered minor? one for which a solutionwill inevitably be found," and added that "Few persons will know or care what themeans is, what its nature is, or how it came into being, but the rewards ... will begreat.''" It is both this "creative vitality" among musicians, and need for a betterinstrument, that inspired the Justonic system.

It ought to be the job of the innovative music instrument designer to providechoices for artists, not dictate how music should be played. The purpose of newmusical instrument design is not to replace one limited paradigm with another, butrather to expand the musical resources. Artists need tools, and a changing artneeds changing tools. By expanding the palette of sounds available, by givingmusicians new options, the inventors (Gannon & Weyler) of the Justonic systemhave attempted to add tuning precision and flexibility to the musician's tool kit.The future of music is destined to be vastly influenced by digital technologies.This change is already in full swing. The piano gave us simplicity butcompromised pure harmony. Digital technology allows us to retain the firstadvantage and restore the purity that was lost in the original compromise.


It is ironic that computer power and digital signal processing return tomusic some of its most fundamental qualities, but the long story of harmony isrich with such irony. Harmonic proportions are the essence of musical sound, andthis has been known for thousands of years. Musical traditions, musicalinstruments come and go, but the fundamental nature of sound remains. From thebone flute of the Neanderthal, to the bamboo pipes of Ling Lun, to the kithara ofancient Greece, to the keyboard of Bach, and to our present day the nature ofsound has not changed. What does change are the creative instincts of artists andthe tools we build to manifest those artistic visions.

Shakuhachi master Masayuki Koga reminds us in his 1989 The JapaneseBamboo Flute:79

"The wonderousness of the human mind is too great to be transferredinto music only by 7 or 12 elements of tone steps in one octave.

There are millions of steps of microtones, and none is to be thrown away,just like nothing is to be wasted in this world."


Appendix AThe Harmonic Overtone Series

With comparative chromatic scale degrees or approximations

Harmonic Western chromatic scale degreeseries (Of approximation)

lst octaveIII III 1.00000 0.00 528.00 C Tonic

~d octave2 211 III 1.00000 0.00 528.00 C Octave3 311 312 1.50000 701.96 792.00 G Fifth

lrd octave4 4/1 111 1.00000 0.00 528.00 C Octave5 511 514 1.2.5000 386.31 660,00 E Major third6 611 312 1.50000 701.96 792.00 0 Fifth7 7/1 7/4 1.75000 968.83 924.00 Bb B!~~minor seventh

~Ihoctaveg 811 III 1.00000 0.00 528.00 C Octave9 9/1 9/8 1.12500 203.91 594.00 D Major second10 1011 5/4 1.25000 386.31 660.00 E Major third11 WI 1.118 1.37500 551.32 726.00 FUb 114-tone flat diminished fifth12 12/1 312 1.50000 701.96 792.00 G Fifth13 13/1 13/8 1.62500 840.53 858.00 AbU 1I4-tone flat si)(th14 14/1 714 1.75000 968.83 924.00 Bb Blue minor seventh15 1511 15/8 1.87500 1088.27 990.00 B Major seventh

;th octave16 1611 III 1.00000 0.00 528.00 C Octave17 1711 17116 1.06250 104.96 561.00 (C#-) Minor second (7 cents flat of 16/15)

18 18/1 918 1.12500 203.91 594.00 D Major second19 19/1 19/16 1.18750 297.51 627.00 (Eb-) Small minor third (between 615 and 7/6)

20 20/1 514 1.25000 386.31 660.00 E Major third21 211l 21116 1.31250 470.78 693.00 (F-) liS-lone flat fourth22 22/1 1118 1.37500 551.32 726.00 FUb 114-tone flat diminished Fifth23 23/i 23/16 1.4375.0 628.27 759.00 (FU+) liS-tone sharp diminished fifth24 2411 312 1.50000 701.96 792.00 G Fifth25 2511 25/16 1.56250 772.63 825.00 (Ab-) lIS-tone flat minor sixth26 26/1 13/8 1.62500 840.53 858.00 Ablf 1I4-tone flat sixth27 2711 27/16 1.68750 905.87 891.00 (A+) 1/8-fone sharp sixth (21.5 cents sharp of 513)28 28/1 714 1.75000 968.83 924.00 Bb Blue minor seventh29 2911 29/16 1.81250 1029.58 957.00 (Bb+) Sharp minor seventh (l/4-tone sharp of7/4)30 30/1 15/8 1.87500 1088.27 990.00 B Major seventh31 31/1 31116 1.93750 1145.04 1023.00 Bit 1I4~tone sharp major seventh

i!h octave32 32/1 III 1.00000 0.00 528.00 C Octave33 33/1 33/32 1.03125 53.27 544.50 C#b Quartertone, flat minor second

etc ...

JUl 1 ne story OJ narmony


AppendixB

The Derivation and Use of Just Musical Scales

The phenomenon of harmony - or resonance, or sympathetic vibration - is not ahuman construct, but is rather a natural relationship between vibrating bodies

in the real world. We make music by arranging audible vibrations into patterns,and traditionally we have been guided by natural harmonics, The relationshipsamong vibrational frequencies are what we call musical intervals. We measurethese intervals in vibratory cycles per unit of time.

A stretched string or piece of wood or metal may vibrate back and forthonce every millisecond. Another string or piece of wood or metal may vibratetwice in a millisecond. The relationship between these two vibrations is "two-to-one," written as the ratio 2: lor as the fraction 2/1, or inversely as the fraction112. Frequencies that have whole number relationships are called "harmonic."

A pure harmonic musical scale, or "just" scale, is defined as a series ofaudible frequencies determined by applying a series of whole number ratios to anyfundamental base frequency. Just ratios are made up of whole numbers (1,2,3,4,etc.). These numbers represent whole, complete cycles of vibration. Examples ofwhole number ratios, or just ratios, are as follows:

2:1,3:2,5:4

These ratios may also be written as fractions:


2/1, 3/2, 5/4

Harmonic relationships were discovered by human ear at least 4,500 yearsago, and probably much earlier. What we call an "octave" is a precise harmonicratio of2: 1. This was probably the first harmonic relationship discovered, and it isthe most universal of all musical relationships. Most scales in most cultures repeata pattern of tones after every 2:1 interval. However, this is a convention, not ademand of nature. It would be perfectly reasonable, as an artistic exercise, torepeat a scale after every 3: 1 or 5: 1 interval, or at any other point, or not at all.However, the musical ears of most cultures have settled on the 2:1 interval as thebasic unit of musical scales. An "octave" may be defined as any doubling orhalving of any frequency (multiplying by any power of 2).

The western word "octave" refers, of course, to the convention of the eighthdiatonic tone being the double of the tonic. This is a convention, not a law ofnature. However, the convention is not arbitrary, as Gerald Eskelin points out inLies My Music Teacher Told Me. As Eskelin, director of the L.A. Jazz Choir,points out, the major/minor diatonic scale degrees are derived from the mostnatural and useful harmonic relationships. Nevertheless, a just musical scale mayhave any number of tones within an "octave."

The eight notes of the diatonic major scale include both the tonic (l:1) andthe octave (2:1). When we speak: ofa "lz-tone" scale, these 12 tones do notinclude the octave tone.

It is best, when analyzing just scales, to remember the simple harmonicrelationships behind the conventions. In this discussion we will refer to scales ashaving a certain number of tones, and in every case the number of tones does notinclude the 2: 1 "octave" tone. Also, please be aware that "tone" in this contextrefers to any just ratio, and not to a western "wholetone step. If

Serious problems do arise from our musical conventions, namely thepredominant use in western music of a single, mistuned scale, and the exclusion ofpure harmonics and countless other scales.


The equal tempered scale is a mathematical construct created byseventeenth century mathematicians to simplify the construction of keyboardinstruments. The equal tempered major thirds and sixths are sharp of theirharmonic counterparts, minor thirds and sixths are flat. The fourths are sharp, thefifths flat. These deviations can be seen in the table.

Equal Tempered Deviations from Just Intonation

)('),4("': A AI D C CI I) 0' E F PI 0 01 ~JII5I(tb,1 .... 0000 .....= 4".0000 .20.0000 5S0.oooo ....... r 111.0000 • 6$0,0000 700&-.0000 1~ 11.. 0000 ' .... 0000 830.0000

TIttlO .. tcI(l1l) -CIQOO "5i:1W 4i3.8a24 11il.2S14 SS<.J64.\ 061.3lV6 Ilt2SZ' "'.2564 6iM.4S6Q 1l$.ll16 l~W24 IlO.&lOC uo.ooooo....t.u.oe: 0.0000 ~1Uce ·'.1\7' .....1416 ...... ...... I.2SZ4 ~,.,. ........ ...... ,,..,.. t.'100 9.-

o_,-;;;c

-0>

Deviations in cycles-per-second (Hz), Key of A~40 Hz. The temperedflat-7 is 14 Hz sharp of the natural harmonic, the minor third is 5 Hz flat,

the major third is 4 Hz sharp, and so forth.

To use a pure harmonic scale for music, one must alter the frequency ofeach note depending on key modulation and harmonic structure. This dynamicalteration of pitch is the challenge that choral singers or string ensembles facewhen they attempt to execute a piece of music in pure harmony. A well-trained

121


hom or string player knows that the "0" in the key ofE is slightly sharp of the "0"in the key of A. 80 Furthermore, melodic considerations may suggest additionalchanges in pitch. We know, from analysis of recordings, that string players likeHeifetz, Casals, and others have used up to 40 tones per octave in performance. 81

To play western chromatic music in any and all keys, while maintaining justintonation, requires hundreds of such microtonal, or enharmonic, adjustments.

The authors have devised a system that overcomes this problem byanalyzing the modulations and harmonic structures of music in real time, and bymaking the necessary changes in the intonation of each tone. There have beenprevious attempts to solve this problem with both mechanical and electronicmeans.82 Mechanical solutions involving extended keyboards have proved toocumbersome to build and play, limiting their popular appeal. Previous electronicsolutions have been limited in their ability to respond in real-time, as the musicianplays. The purpose of the authors' Justonic software retuning system is to allowthe musician to play the traditional western keyboard (or fretted instrument) in thetraditional manner, without having to step through computer menu selections, andyet also achieve pure harmonic intonation for all harmonic structures in all keymodulations.

The history of musical scale development has been encumbered withattempts to create just scales that eliminate the problem. Such attempts have failedfor quite natural and logical reasons. It is simply not possible to build a just scale,based on harmonic ratios, that can freely modulate to any scale degree and playthe same scale from that point without retuning any of the tones. There is no wayto avoid the fact that in true harmonic music the frequency of each scale degreemust be flexible. The equal tempered scale does not solve the problem, but ratherchooses free modulation over harmonic purity by abandoning harmonic intervalsaltogether.

Harmonic ratios are derived from the harmonic series. These harmonics arealso referred to as overtones or as partials. The authors use the commonconvention which defines these terms in the following manner: An harmonic is a


pure harmonic, a whole number multiple (2x, 3x, 4x, etc.) of any base frequency.An overtone is not necessarily harmonic. For example, bell, drum, and pianostring overtones are not purely harmonic. Flute and hom harmonics tend to bepure harmonics. The series of partials are the same as the series ofhannonics inthat they include the fundamental tone (1: 1) itself.

Long before human theorists knew anything about harmonics, combinationtones, or any other acoustic qualities of sound, human ears used these qualities todiscover the natural harmonies. When ancient vocalists heard themselves sing aperfect unison (1; 1)~or a perfect octave (2: 1), they new it sounded pleasing. Theancient Chinese discovered another pleasing relationship, what we now call theperfect fifth, and they also discovered that the ratio between these two tones was3:2. What they didn't know was that the fifth was the third partial of a tone oneoctave below the tonic. They did not need to know this. Their ears put all thisinformation together and decided that this interval sounded alluring and radiant.The Greek Harmonist Archytas is credited with first documenting what wenowadays call the pure harmonic major third, a 5:4 relationship of tones.

Musicians naturally began to string these pleasant sounding tones togetherinto formal scales. The purpose of this paper is to introduce certain scale choices,and to show how one might organize the process of selecting tones for harmonic,or just, scales. We will look at the harmonic logic behind the creation of a justversion of the western chromatic scale, and we will then investigate how we mightbegin to organize optional scale choices into a system of scales that may beexploited by composers and musicians.

Deriving a just western chromatic scale

A pure harmonic, or just, version of the western 12-tone chromatic scalecan be generated through an analysis of the natural harmonic series, thecombination tones that arise from these harmonics, and inversions of these tones.


It took human musicians several thousand years to come up with this knowledge,but it all derives from quite simple and natural acoustic phenomena.

The harmonic series for a fundamental frequency of 110 cycles per second(Hz) is:

110,220,330,440,550,660,770, ... e~.

The second partial is the octave of our fundamental. The third partial has arelationship ofllthree-to-one,1I (330 Hz to 110 Hz) or 3:1. We reduce this tone tothe lower octave by dividing by two, making it ''three-to-two,'' written as 3 :2.This,we already know is the just perfect fifth in the diatonic scale. Continuing with thisprocess, we see that the fourth partial is another octave, the fifth partial can bereduced to a just third (5:4), and so on.

The first fifteen partials of the natural harmonic series give us the followingset of unique harmonic ratios (in order of appearance, and reduced to a singleoctave);

1:1, 3:2, 5:4, 7:4, 9:8. 11:8, 13:8, 15:8

These ratios are a subset of the theoretically infinite Harmonic Set. Thissubset, above, has a prime limit of 13. We call this set of ratios the FundamentalChromatic Harmonic Set, because these ratios are the basis of the westernchromatic scale of 12 scale degrees in semitone steps. Six of these eight tones fallat scale degrees of the western 12-tone scale. The 11:8 and 13:8 fallapproximately at quarter-tone points between the semi-tone scale degrees. We willreturn to these two harmonic intervals when we discuss extended scales.Withholding these quartertone ratios, we are left with six harmonic ratios:

1:1, 3:2, 5:4, 7:4, 9:8, 15:8


This is the Chromatic Harmonic Set. Re-arranging the ratios in ascendingorder, we have:

1:1, 9:8, 5:4, 3:2, 7:4, 15:8

Or, in familiar notation in the key ofC:

C, D, E, G, Bb, B

This hexatonic scale is the most natural harmonic scale within the primelimit of7, generated directly from the harmonic series. In our modern diatonicnomenclature, these ratios represent the tonic (1:1), second (9:8), third (5:4), fifth(3:2), minor-seventh (7:4), and the major-seventh (15:8). It may not beimmediately clear to musicians unaccustomed to seeing intervals written as ratiosthat a 5:4 is a wholetone larger than 9:8. In any case, musicians ought to be awareof these ratio relationships since they are the very foundation of the musical arts.

It is important to hear these intervals and to understand how they differfrom tempered intervals. The reader can use the Justonic Pitch Palette software toaccess these intervals and hear them. String players can find these tones by ear,simply by playing the harmonics of a base tone. Notice that the positions of theseharmonic tones do not match the tempered positions. Playa triad with these justintervals, and witness the heatless, smooth quality. Add the minor or majorseventh. These chords will sound different from the tempered variety. Bothsevenths sound flat to the tempered-trained ear, but notice that they create verysweet harmonies.

Inversions of any interval are also quite natural. For example, when theancient Chinese or Greek musicians discovered the perfect fifth, 3:2, they alsodiscovered its inversion, the fourth, 4:3. Why? Because there is nothing in thenature of these tones to say which is the tonic and which is the interval. The octave


of the original tonic is the fourth of the fifth. That is, the ratio series, tonic-fifth-octave:

1:1, 3:2, 2:1

(the notes C,G,C) can simply be transposed to make the G the tonic:

2:3, 1:1, 4:3

These are the same tones defined differently, as a tonic in the middle with afourth above and a fifth below. We say, therefore, that the fourth is the inversionof the fifth. Inversions of the six tones of the Chromatic Harmonic Set give us sixnew ratios. Excluding the 2:1 octave, we have five new ratios to consider:

16:9, 8:5, 4:3, 8:7, 16:J5

Bb, Ab, F, D, C#

The 8:7 is slightly sharp (approximately an eighth-tone) of our morecommon second, 9:8. We will save this harmonic ratio for scales extended beyond12 steps, or with sub-semitone steps. The 16:9 falls between the minor and majorsevenths. We will use this ratio as an alternative minor-seventh in certainvariations. In the meantime, withholding these two sub-semi tone steps, we havethree new ratios which fall at western semitone steps, and which we add to ourscale:

8:5,4:3, 16:15

Placing these ratios in their proper ascending order, we now have anharmonic nonatonic (9-tone) scale, the Justonic Inversion Harmonic Scale. In


diatonic tenus, we have added to our scale the minor-second (16: 15), the fourth(4:3) and the minor-sixth (8:5). Our scale now looks like this:

1:1, 16:15, 9:8, 5:4, 4:3, 3:2, 8:5, 7:4, 15:8

C, C#, D, E, F, G, Ab, Bb, B

This scale has two semitone steps, followed by alternating tone andsemitone steps, and a final semitone step to the octave.

In addition to natural harmonics, tones played together also createcombination tones, the addition and subtraction of two tonal values. For example,if two tones are sounded together, one at 440 cycles per second, and another at660 cycles per second, these "Will create two combination tones, one at 220 cyclesper second, and another at 1100 cycles per second. That is, a pure A and E createa sub-octave A and a higher Cfl. These combination tones add significantly to ourperception of pure harmony."

Likewise, if we simultaneously sound the tonic and the minor-sixth (8:5) ofour harmonic scale, above, a combination tone is created that has the precise ratioof 6:5, the minor third.84 This tone is created by the difference between thefrequency of the tonic and the frequency of the minor sixth. Its inversion is themajor sixth, 5:3. Adding these two harmonic tones, we now have an l l-tone scale:

1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 3:2, 8:5, 5:3, 7:4, 15:8

C, Cfl, D, Eb, E, F, G, Ab, A, Bb, B

All the steps of this natural scale. the Differential Harmonic Scale, are ofsemitone magnitude except for the wholetone step between the fourth and the :fifth(4:3 to 3:2). The "missing" semitone is the only tone in our 12-tone western scalethat is not directly generated by the first 15 partials plus inversions and


combination tones of the harmonic series; it is also the theoretical center of theoctave, and in these characteristics it is unique. The most harmonic choice for thistone, the simplest ratio, is 7:5. We may derive this ratio by taking a minor sixth ofa minor seventh and reducing it to our octave. (7/4 x 815 x 1/2 = 7/5). There areother choices, such as 45/32, which we may use as variations, and we will discussthese later. Using 7:5, we now have a natural harmonic 12-tone scale in semitonesteps, and with a prime limit of 7:

1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8

C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B

This classic just scale is the default western chromatic scale in the Justonicsystem, the Justonic Classic Harmonic Chromatic Scale. The equal temperedversion of this scale has equal semitone steps. This just scale has semitone steps,but they are not all equal in size. There are five different semitone steps here,ranging in size from the small step between the thirds and between the six...rths(25:24) to the large semitone between the tritone and the fifth and between themajor and minor sevenths (15:14). Likewise, the wholetone steps vary in size. Thisvariety gives each harmonic scale a distinct feel.

As we have already seen, the chromatic scale has several just variations.The most significant variant is in the eleventh chromatic position, the "minor-seventh." The choices for this scale degree are:

1. 7:4, the natural harmonic, or seventh partial, of the tonic: Thisharmonic ratio is the natural choice as a minor seventh. It is fIat of the equaltempered variety, and is sometimes called the "blues seventh." This minor-seventhis quite expressive and sounds excellent in major chords or with the septimal (7;6)minor third (see scale g. below). This just tone is the most foreign-sounding to


musicians who are used to the much sharper tempered minor seventh, but there isno denying its exquisite harmonic feel.

2. 16:9, the inversion of the 9:8 second: Closest to the tempered minor-seventh.

3. 9:5, the perfect fifth of the 6:5 minor third: The most sharp of thethree choices, this tone is the correct interval when used with a 6:5 minor third.However, since the same result can be achieved in the Justonic system by"rooting" the minor chord on it's own minor-third or fifth, using this ratio as thescale degree minor-seventh is not necessary, and it is quite rough when heard inamajor chord. It is, however, a viable artistic choice.

By substituting these three options for the minor-seventh we have threealternative chromatic scales, differing only in their eleventh scale degree, andnamed as follows in the Justonic system:

(a) The Justonic Classic Harmonic Chromatic Scale: Using the naturalharmonic minor-seventh:

1:1,16:15,9:8,6:5,5:4,4:3, 7:5, 3:2, 8:5, 5:3, 7:4,15:8

(b) The Justonic Inversion Chromatic Scale: Using the inversion of thesecond for the minor-seventh:

1:1, 16:15,9:8,6:5,5:4,4:3, 7:5, 3:2, 8:5, 5:3, 16:9, 15:8

(c) The Justonic Acute Minor-Seventh Chromatic Scale: Using thenatural fifth of the minor-third for the minor-seventh:

1:1, 16:15,9:8,6:5,5:4,4:3, 7:5, 3:2, 8:5, 5:3, 9:5, 15:8


The next most significant scale variant is in the seventh chromatic position,the tritone, or diminished fifth of the diatonic scale. As we have alreadydetermined, there are no direct, natural harmonic choices for this unique degree ofthe chromatic scale which theoretically falls. in the middle of the octave. Thetempered square root of 2 is the mathematical center of the octave, but since thistempered tone makes no harmonic relationships with any other tones, it is notappropriate in a just intonation context. The two most obvious harmonic choicesfor this scale degree are:

1. 7:5,the simplest ratio, is the most harmonic choice, and appears inscales a-c above, and g below.

2. 45:32the ratio derived by taking a third of the second; this tone isslightly sharp of the 7:5 tone.

There are other choices, such as the Pythagorean tritone (729:512), thenearly-equal tritone (140:99), and the inversion of 45:32 (64:45). The avidexperimenter might want to listen to these tritone alternatives. For our purposeswe will substitute 45:32 for 7:5 in the above scales. We have three new scales,named as follows in the Justonic system:

(d) The Justonic Derived Harmonic Chromatic Scale: Using theharmonic minor-seventh and the derived diminished fifth:

1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 7:4, 15:8

(e) The Justonic Derived Inversion Chromatic Scale: Using the inversionof the second for the minor-seventh with the derived diminished fifth:


1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 16:9, 15:8

(f) The Justonic Derived Acute Minor Seventh Chromatic Scale: Usingthe sharp, 9:5 minor seventh and the derived diminished fifth:

1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 9:5, 15:8

We will look at one more variation. The minor third can be moved loweralmost a quartertone, from 6:5 to 7:6. This more flat minor third is sometimescalled the "blues minor third.n85 It has a unique and interesting feel, and the 7:4minor seventh forms a perfect fifth with this minor third. By substituting thisinterval into scale "a", above, we get the following chromatic scale, featuring theseventh harmonic, the harmonic most obscured by equal temperament.

(g) The Justonic Septimal Chromatic Scale: Using the seventh harmonicas a basis for the minor third, tritone, and minor seventh:

1:1, 16:15, 9:&, 7:6, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8

Above we have set out seven versions of the western chromatic scale, eachslightly unique, but each performing essentially the same role of a chromatic 12-tone scale in semitone steps. Western musicians and composers will find that thescalar resources outlined here have creative applications, and that the harmonicintegrity speaks for itself. Chords created with these scales have an harmonicpurity unheard in tempered music.

Subsets, or modes, of these chromatic scales - such as our diatonic majorscale, pentatonic, Gypsy, whole-tone, etc. - can of course be selected at thediscretion of the artist. In the Justonic system we call these subsets "modes II of thechromatic scale. This use of the term mode is different than the Greek orecclesiastical mode which indicates a new, nominal tonic for a given set of


intervals. For our purposes, a mode is any subset of a scale in any configuration.For example, the Pentatonic scale 1:1,9:8,5:4,3:2,5:3 is a selection, or mode, ofthe fundamental chromatic harmonic scale, as is our diatonic major scale of Do,Re, Mi ... etc. A just version of our diatonic major scale is:

1:1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8

All of the classical diatonic and chromatic Greek and ecclesiastical modesin western music are subsets of the chromatic scale. Every scale is a mode of alllarger scales containing its tones. Theoretically, there exists an infinite number ofscales, harmonic or otherwise, and therefore every scale is a mode of some otherscale. Likewise any mode can be considered to be a scale. However, for thepurposes of common usage within the Justonic system, once a fundamentalharmonic scale is selected and identified as such, all subsets of that scale arecalled modes.

When modulating key, or changing harmonic structure within a key, wewant to keep the ratios pure, and this necessitates the retuning of the actualfrequencies. This extra requirement is the price we pay for playing in tune. Thediscerning artist may find this price to be a bargain. The Justonic system helps bymaking the tuning adjustments automatically.

The seven scales described above only scratch the surface of scale buildingusing harmonic ratios. What we will call extended scales are harmonic scalesusing sub-semi tone steps, or more than 12 steps per octave.

Extended Scales

The chromatic scale of semitone steps is only one way to organize tonesinto scales. Thoughout history, in many cultures today, and even for somemodern western composers, scales take on a variety of patterns, each with its own


limitations and merits. For the musical artist, this rich variety of scales representsa bounty of tonal resources available for the creation of music.

The enharmonic scales, scales using sub-semi tone steps - Greek, Arabic,East Indian and other - are not subsets of the chromatic scale. Nevertheless, anysuch scale that can be described by a set of rational ratios can be considered a justscale. For example, one such scale would be a version of the Greek EnharmonicDorian, based on an enharmonic tetrachord" of Didymus, inwhich the just ratiosare:

1:1,32:31, 16:1514:3, 3:2~48:31, 8:5

Here we have two quarter-tone steps followed by a two-tone step,wholetone step, two more quarter-tone steps, and a final two-tone step to theoctave (2: 1).

Arabic Rast-Rast scales also uses two quarter-tone steps. These scalesemploy a "neutral" third, and a "neutral" seventh. A typical Rast scale or mode hasseven or eight tones per octave. As with the chromatic western scales above, theJustonic system uses several varieties of 12-tone "chromatic Rast" fundamentalscales, each of which allows for a number of Arabic Rast modes of seven or eighttones. One example, using traditional Arabic intervals for all the steps, is thefollowing:

1:1, 18:17,9:8,32:27,27:22,4:3,24:17,3:2, 128:81,27:16, 16:9,81:44

The fourth and eleventh steps of this 12-tone scale are quartertone steps; thefifth and twelfth steps (the twelfth step is from 81144 to the octave 2: 1), are tbree-quarter-tone steps. The scale has a prime limit of 17, and uses several ratiosderived from early attempts to mitigate the modulation problem, with an emphasison modulations to the dominant and subdominant. Arabic musicians haveremained loyal to traditional just ratios. However, the lack of instruments that


could freely modulate and make microtonal adjustments has limited Arabic musicto primarily melodic, modal lines with very little harmonic embellishment. Arabiccomposers find this frustrating, but are also hesitant to relinquish their pureharmonic scales. Whereas European music sacrificed harmonic purity for ease ofmodulation, Arabic music has sacrificed polyphonic development for harmonicpurity.

The Justonic system addresses the needs of both traditions. For the Arabicscales, the system retains traditional harmonic ratios, but adds free modulation andextended harmony by making the necessary microtonal shifts.

There are many historic just scales using other than semitone steps. In thecontext of a just scale, "semitone" and "quartertone" are variable, not preciseintervals. We loosely refer to a "semitone" as 1/12 of an octave, but in justintonation scales a semitone may be an interval of 16:15, the slightly more flat18:17as in the Arabic scale above, and in other cases may be 21:20, 135:128,orsome other ratio of similar size as in the just western chromatic scales above. It isonly a convention, after all, to call 116of the octave a "wholetone," and it is onlyin the equal tempered system that all tones and all semitones are of equal size.

We may now return to the harmonic intervals which we gleaned from theharmonic series but did not use in our just versions of the western chromatic scale.Some of these we left out because they formed quartertone intervals, and somewere alternatives to certain scale degrees. These harmonic scale ratios are:

11:8, 13:8, 8:7, 16:9, 9:5

Adding these tones to our chromatic scale. we have a 17rtone scale with aprime limit of 13. and with nine sub-semitone steps created by the inclusion of thenew ratios:

1:1, 16:15, 9:8, 8:7, 6:5. 5:4, 4:3, 11:8, 7:5, 3:2,8:5. 13:8, 5:3, 7:4, 16:9, 9:5, 15:8


We may continue this process by adding ratios such as 10:9, 7:6, and 11:6,giving us a 20-tone scale:

1:1, 16:15, 10:9, 9:8, 8:7, 7:6, 6:5, 5:4, 4:3, 11:8, 7:5, 3:2,8:5, 13:8, 5:3, 7:4, 16:9, 9:5, 11:6, 15:8

The process of scale-building is theoretically infinite, and indeed there havebeen hundreds of scales used throughout history by human cultures. Most of thesescales are derived from (in order of historic discovery and significance) the octave(2:1), the fifth (3:2), the fourth (4:3), and the major third (5:4). These harmonicintervals are all heard in the first five partials of an audible frequency. They trulyrepresent the harmonic heart of world music.

Cataloging the parameters of scales

The authors predict that since computer and signal processing power nowmakes free harmonic modulation possible, and since the Justonic system makes itsimple, western music will naturally move toward a greater variety of scalechoices. The 300-year era of most music being written and played in a single out-of-tune chromatic scale is over. Composers and musicians will naturally expandtheir tonal vocabularies as these scales are made accessible.

And as this phenomenon unfolds, we might be wise to consider how best toorganize and catalog the infinite variety ofhannonic scale choices. The history ofwestern music has left us with a tangle of contradictory conventions, as discussedabove. It is an irony of western music that we have so limited ourselves to onesimple scale, and yet so confounded our young students with a notation system


that begs its own twisted internal logic. It is quite possible that we could solveboth problems, expanding our scale choices, while rationalizing and simplifying(and internationalizing) our notation system.

Inany case, to begin, we must identify the parameters of scales, and thenattempt to organize those parameters into a system that simplifies selection of thescales. The authors suggest the scale parameters in table 3, and invite composersand musicians to suggest other parameters, or changes to those in table 3, thatmight suit the needs of modem music making.

Table 3: Scale parameters

I. Intonation Family;A. Harmonic: based on whole number harmonic seriesB. Overtone: based on enharmonic overtone series

1. Drums2. Bells3. Stretched piano4. etc.

C. Tempered: non harmonic1. meantone2. equal3. etc.

D. Free Continuum: No restriction, using the entire range of audible tonesat the discretion of the composer.

II. Intonation Genus:A. Harmonic: All tones directly taken from the harmonic series.B. Inversion: Harmonics plus inversions


C. Derived: Harmonics, inversions, plus tones derived by taking intervals ofexisting tones.

D. Free: No restriction on tone selection

m. Prime limit: All harmonic scales have a prime limit, the largest prime numberharmonic necessary to create the scale. Prime numbers 1 and 2 areassumed, so this category would start with 3, then 5, 7, 11, 13, 17, etc.

IV. Scale step construction (Tetrachord family, after Greek tradition, E.M.Wilson, 1986-7, and 1. Chahners, 1993)81:A. Diatonic: wholetones and single semitones

1. soft2. intense3. equable4.ctc.

B. Chromatic: consecutive scmitones and 3/2-tones (minor third), withwholetone disjunction between tetrachords.

C. Enharmonic: with sub-semitone steps.D. Hyperenharmonic. with steps larger than a major third and smaller than

a quartertone (smaller than 36/35, and in the range of the 81:80 syntonic comma,the "eighth-tone").

E. Reduplicated: with two exactly repeating intervals in each tetrachord,Arabic and others.

F. Mean: Calculated mean divisions.1. Arithmetic2. Geometric3, Harmonic4. Logarithmic5. etc.

G. Tempered: divisions of a tempered fourth


L Equal2. Non-equal

H. Non-tetrachordal: with scale divisions based on other than tetrachordaldesign.

1. Pentachordal2, Hexachordal3. etc.

I.Mixed; mixed tetrachord construction in a single octave scale.J. Miscellaneous: Chalmers' catch-all family which includes unusual

historic [Archytas, AI Farabi, et. al.) and modem (I. Xenakis, H. Partch, J.Carrillo, et. al.) tetrachords.

K. Free: no tetrachordal rule limiting scale construction.

V. Characteristic Interval (per Chalmers): A sub-classification of the above, bylargest interval in the tetrachord, or in the scale.

VI. Repetition pattern:A. 2x: traditional harmonic octaveB. 2x enharmonic: stretched piano octave, etc.C.Non-2x:

1. 3x, 5x, etc.2. Variable3. Non-repeating4. Mixed5. Free

VII. Number of Tones: Because of the prevalence of 12-tone systems (in westernmusic, and in world musical instrument construction), 12-tone scalesmight best receive their own category, with extended and subtendedscales as the other two categories, with appropriate subcategories.


A.12-toneB. Extended (more than 12-tones per repetition unit)C. Subtended (less than 12-tones per repetition unit)

vrn. Cultural origin: It may be useful to composers and musicians to be able tocatalog and retrieve scales by musical tradition or cultural origin,with subcategories identifying individual originators or documentersof scales, such as Ling Lun, Archytas, Ptolemy, Al-Farabi ... HarryPartch, Kathleen Schlesinger, Murray Barbour, John Chalmers, etc.

IX. There may well be scalar parameters overlooked by the authors that areimportant to composers and musicians. As this cataloging of scales isa work in progress, we shall let the categories remain open.

There are other important characteristics of scale use, such as notationsystem, modal system, note change parameters, pitch bend, tuning resolution, andmodulation system (spiral. closed, etc.) that also require a close look. and a logicalcataloging. These qualities may be considered independently of scale choice, andmay best be included in a set of operational parameters.

From an artistic point of view, there is nothing in nature to preclude anytone or interval from being used to make music. The historic rules governing scalechoices reflect cultural taste and aesthetic habit, but do not constitute laws ofnature. Nevertheless, human musical scales are predominantly influenced by thenatural ratios of the harmonic series, the basis of harmony. Much of the historictinkering with the harmonic intervals has been an attempt to mitigate themodulation problem, and avoid the necessary retuning with each change of key ortonal center.

There are modem composers who have experimented with scale choice(Harry Partch, Julian Carrillo, Wendy Carlos, La Monte Young, Lou Harrison,Robert Rich, Ben Johnston, Erv Wilson and many others). However, for 700 years


most western music has restricted itself to one chromatic scale in semitone steps.This limitation is a serious artistic compromise. The trade-off is fOT simplicity, butmusicians might well ask themselves if such a trade-off serves their art.

Would painters accept a restriction to a small, fixed set of colors because itwas easier to build paint sets that way? Would poets accept the limitation ofrhyming couplets because printers found it easier to set type that way? No?Consider that this is precisely what we have accepted in our choice of a singlescale for the last 300 years!

Musicians have been somewhat misled. by keyboard manufacturers andtheorists since the days of Bach. Keyboard manufacturers found that themechanical challenge of creating a just intonation, free modulation, fixed-keyinstrument was beyond the limit of their technological resources. In short, theygave up. Theorists too often ventured down the road of searching for an acceptablecompromise rather than precisely defining and solving the problem. Teachers wereleft to tell their students that the equal tempered intervals were the bestcompromise available, and that we all had to suffer the rough, beating chords.

Young string players and horn players have been taught to hear and playapure major third when playing with string and horn ensembles, then taught to playa tempered third when performing with a piano. This has caused untold confusionamong young players who never entirely understood the problem and yct could notresolve the conflict so plain to their ears. Choirs learned to sing those beautiful,resonant harmonic triads that send chills down the spine and suggest the presenceof angels above. Then, when singing with pianos and guitars they are forced intotempered triads that wobble and beat.

It is true that the keyboard, and the system of equal temperament, havecontributed to the development of our music. From the dramatic modulations andpower chords of Beethoven to the chromatic fluidity of modem jazz, westernmusic has taken full advantage of equal temperament. However, the loss of pureharmonics and scale variety was the price paid to gain this simplicity. Now, withthe aid of computer power, we can solve the problem that 17th century keyboard


manufacturers could not solve. We can also enrich our music with the greatspectrum of scale choices that are available. The scales outlined above are littlemore than a starting place in this investigation. Please see table 4 for someadditional scale choices.

It is the belief of the authors that it is the responsibility of the music theoristand music instrument maker to serve music first. It is our hope that we mightcontribute in some small way to a new flowering of western and world music bymaking these scale choices available, and by making dynamic just intonation withthese scales not only possible, but simple. For the modem composer and musicianthis expanded palette of tones can only contribute to greater variety, subtlety, andpower of expression. May the muse be with you.

======


-1-

IZth-root-of-Z, 13

-2-

24-tone equal temperament, 95

-3-


-4-


-7-

72·tone multiple division equal

temperernent, 106

143

Index

-9-


-A-

A cappella, 4, 7,44,117,120,121

Aaron, Pietro, 8, 63

Acoustic Guitar magazine, 94

Acoustics, 5, 99

Agam, 53

Aleatory, 90

Al-Farabi, 52 .

Alves, William, 106

AntheiI, George, 97

Arabic music, 51, 114

Archicernbalo, 63

144 The story oj Harmony

Archytas, 11, 12, 11

Aristotle, 25, 34, 37

Aristoxenus, 10, 37, 38, 40

Aritlunetic proportion, 33,53,88

Armstrong, Louis, 83

Atonal,82

Aulos, 29, 88

Automatic luning, 104

Blues third,S I, 67

Bosanquet, R.H.M., 80

Bowed instruments, S6

Branca, Glenn, 96

Brown, Colin, 80

Busoni, Ferruccio, 88

Buxtehude, Dietrich, 6, 71

-c--B- Cage, John, 83

Cahill, Thaddeus, 89

Carillo, Julian, 87

Carlos, Wendy, 95, 112

Chalmers, John, 32, 97

Chanting, 17

China, 17,50,56

Chord inversions, 106

Chromatic, 31, 88

Chromatic scale, 82

Chrornclodeon, 92

Coker, Jerry. 84

Colctrane, John, 83

Combination tones, 69

Babylonians, 22

Bach, 1.S., 8, 14,65,68,71,72

Bach, Nicholas, 3,4,5, 12, 13,68

Bamboo flutes, 17

Bamboo pipes, 20

Barstow, Dr. Robert, III

Beach Boys, 89

Beethoven, Lugwig Van, 73

Benade, Arthur H., 96

Bernoulli, Daniel, 71

Blue minor third, 84

Blue notes, 84

Blues, 83, 84, 86


Comma of Didymus, 21, 27, 44,85

Commas, 5, 27

Common keys, 8

Computer, 94,98

Computer Music Journal, 95

Consonance, 54,56, 79, 91

Counterpoint, 56

Cowell, Henry, 82, 88, 89

Cristofori, Bartolommeo, 14,68

Curwen, John. 76

Cycle of 53 fifths, 67

Cycle of fifths, 26. 37

-D-

d'Arezzo, Guido, 54

Dame, Steve, 101

Davis, Miles, 83

Debussy, Claude, 82, 93

Descartes, Rene, 10,64,68

Diatonic, 31, 45,54

Didgeridoo, 17

Didymus, 5,44

Differential tones. See combination tones

145

Digital,16

Digitized pitch. 16, 100

Disjunctive tone, 33, 46, 49

Dissonance, 38, 55,79,82, 100

Distant keys, 8, 58

Doni, Giovanni Battista, 67

Dorian, JJ, 46,88

Doty, David, 90, 97

Drumming, 17

Dual identity, 91

Duncan, Dudley, 84

-E-

Ecclesiastical,33

Egyptians, 17, 22

Electronic Musician, 97

Emancipation of the dissonance, 82

Enharmonic, 31. 49. 61, 88

Enharmonic organ, 76

Eno, Brian, 86

Equal semi tones, 63

Equal temperament, 4, 13, 14,32.38, SO,

58,60,65,67,68,70,81,83,87,95

146 The story of Harmony

-F-

Galileo, 64, 76

Gannon, Bill, 99, 114

Gannon, Oliver, 101

Glarean, Henricus, 60

Glover sisters, 75

Glover. Sarah. 75. 76

Guitar interface, 101

Guitars, 87. 112

Equal tempered thirds, 11

Eratosthenes, 43

Erhu,56

Euler, Leonard, 71

Fang, King, 40

Faux bourbon, 55

Fetis, Francois, 74

Fifth harmonic, 21, 34

Fifths, 5, 19,20

Fixed-tone. 27. 35. 41,56. 58, 62, 65

Fixed-lone instruments, 65, 100

Flutes, 17, 18

Fourier, Jean Baptiste, 76

Fourths,5,19,31,34

Franco of Cologne, 55

Frets, IS, 16,50,56,62,81,84,85,98

Fundamental tone, 19,27,91,93

-H-

-G-

Halberstadt keyboard, 56, 57

Halberstadt organ. 6

Haley. Jeffrey, 100

Hammond organ, 89

Handel, Georg Friedrich, 14,65, 70

Handy, W.C., 84

Harmonia, 30

Harmonic division, 33

Harmonic fitlh, 19,67

Harmonic fourth, 67

Harmonic proportion. 38, 53.61.70. 73.

87,91,93Gafori, Franchino, 59

Galilei, Vincenzo, 61, 63


Harmonic ratios, 46,47,63. See harmonic

proportion

Harmonic series, 19,64,66,82

Harmonic seventh, 105

Harmonic third, 45, 54, 55, 56, 67

Harmonic triad, 66,79

Harmonics, 24,39,97,98, 106

Harmorus~,30,3J,33,37,38,85,87

Harmonium, 80

Hannony, 19,29,39,56,58,70

Harrison, Lou, 89, 90, 112

Helmholtz, Herman, 15,63,69,72,77,85,

107

Herf, Franz, 96

Hervey, Patty, 101

Hexachord,54

Hindemith, Paul, 82

Historical tunings, 73

Homophonic, 39

Hom, Paul, 102

Hydraulos, 41

Hymns, S9

147

-1-

Inversions, 79, 106

Ives, Charles, 90

-J-

James Broadwood piano factory, 74

Jarrett, Keith, 90

Jazz, 83, 84, 86

Jena, Germany, 1,3,4, 12.72

Johnston. Ben. 93. 94. 112

Jorgensen, Owen H., 15,73

Just intonation, 4, 32, 100

Just Intonation Network, 90, 97

Justonic, 101, 103, 104, 105, 107, 114

-K-

Kadir, Abdul, 52

Keislar, Douglas, 97

Kepler, Johannes, 10,64

Key, 104

Keyboard temperament, 7

Keyboards, IS, 16,54,56,67,73,81,83,

88


Kirck, George, 97

Kithara, 29, 30, 39

Koga, Masayuki, 115

Koto,34

Kronos Quartet, 95

Kurzweil 1<2500,102

-M-

-L-

Major, 51, 79

Major thirds, 32,54

Malcolm, Alexander, II

Manual tuning, 104

Maqamat,52

Martin D-35, 85

Marvin, Jameson, III

Meantone, 5, 8,60,61,65,70,95Lagrange, Joseph Louis, 71

Lanfranco, Giovanni Maria, 60

Led Zeppelin, 89

Leibniz, Gottfried, 10, 68

Leipzig, 2

Lemmens, Nicolas Jacques, 75

Lissajous, Jules, 76, 77

Lloyd, Llewelyn, 81

Lil,20

Lun, Ling, 20, 25

Lute, 4, 52,62

Lydian, 46

Lyra, 29, 67

Meistersingers, 55

Melodic scales, 19, 53, 56

Melody, 18, 19, 70

Mercator, Nicolas, 40, 67

Mersennc, Marin, 6, 10

~I, 96, 97, 101, 104

Minnesingers, SS

Minor, 79

Minor semi lone, 51

Minor thirds, 32, 44,54,67,70

Minor wholetone, 51

Minstrels, 55

Mixolydian mode, 23, 88

Modes, 52


Modulation, 79, 106

Monochord, 24

Monophonic, 19,29,39

Monophony, 90

Monteverdi, Claudio, 59

Moog synthesizer, 95

Moog, Robert, 94

Motets, 59

Movable doh, 75

Mozart, W.A., 65, 72

Music theory, 43, 47

musical key, 103, 104

-N-

Nahawand, 51

Nath, Pandit Pran, 96

National Association of Music Merchants

(NAtv1},.i),96

Natural keys, 8

Nature of sound, 91

Neanderthals, 17, 18

Neidhardt, Johann Georg, 3, 9, 12,59

Neutral thirds, 50, 105

149

New World Records, 90

Notation, 19,21,22,39,44,125,135,139

-0-

O'Keefe, Steve, 86

Octave, 14,20,41

Old Testament, 19

Oliveros, Pauline, 96

Organ, 54,57,68, 78

Organs, 74

Organum, 53, 54

Overtone series. See harmonic series

Overtones. See harmonic series

-p-

Parker, Charlies, 83

Partch, Harry, 34, 37,53,80,88,90,91,

93,97, 114

Pentachord, 51

Pentatonic scale, 21, 25

Phrynis, 29

Piano, 14,68,73,74,92,98, 112

Pitch Palette, 103


Plato, 34

Polansky, Larry, 97, 112

Pole, William, IS

Polyphony, 39, 54, S8

Poole, Henry, 78, 80

Prime harmonic, S1

Profit-5, 96

Ptolernaeus, Claudius. See Ptolemy

Ptolemaic Sequence, 61

Ptolemy, 45, 47

Pure chord, 100

Pure fifths, 8, 70

Pure fourths, 70

Pure harmonic, 32

Pure harmony, 70

Pure thirds, 54, 70

Pythagoras, 23, 25, 26

Pythagorean, 54

Pythagorean comma, 26, 37

Pythagorean thirds, 45, 54,55,59

Pythagorean tunings, 5

Pythagorean wholetone, 32

-Q-

Quartertone,32,49,87,91,96

-R-

Radio Cologne studio, 94

Radio Corporation of America, 94

Radiodiffusion studio, 94

Rarneau, Jean-Philippe, 70

Ramis, Bartolomcus, S9

Rast,51

Ratios, 32

Ravel, J. Maurice, 82

Rayna, David, 97

RCA Mark sound synthesizer, 94

Reference frequency, 103, 104

Rhapsodes, 23

Rich, Robert, 97, 102

Riley, Terry, 96, 112, 113

Rock music, 85, 86

Root tone. See fundamental tone

Rosenthal, Henry, 97


-8-

Safiyu-d-Din, 51

Salinas, Franco de, 6 I

Sandard pitch, 12

Sauveur, Joseph,68

Scales, 33, 103, 105, 106

Schlesinger, Kathleen, 88

Schlick, Arnold, 60

Schneider, John, 86

Scholar's lute, 34

Scholz, Carter, 97, 102

Schonberg, Arnold, 82

Scriabin, A.N., 82, 93

Selecting the tuning root. 105

Semite tribes, 9

Semitone, 32, 63

Sequence track, 104

Serial techniques, 82

Series of fifths, 33, 37, 49

Seventh harmonic, 28, 3 J, 34, 74, 84, 110

Shirazi, Mahrnud, 52

Silk road, 43

Simultaneous harmony, 53

151

Singing, 17

Sixths, 33, 55, 56

Smith, Dave, 96

Smith, Dr. Robert, 14, 70

Sol-faists, 76

Sound,15

Sound waves, 71

Split-kcy keyboards, 56, 59, 70

Stockhausen, Karlheinz, 83

Stoltzman, Lucy, 90

Stradivari, Antonio, 68

Stravinsky, Igor, 82

Sumerians, 22

Summation tones. See combination tones

Sweet Honey and the Rock, 107

-T-

TanburofBaghdad,51

Tartini's tones. See combination tones

Tartini, Giuseppe, 68

Taylor, Margaret, 102

Taylor, Stephen James, 112

Tcheng-tien, Ho, 10, 50

iS2

Telhannonium, 89

Temperament,6, 15,57,73,100

Tempered chord, 100

Tempered harmony, 59

Tenny, James, 96

Terpander,23

Tetrachord, 3/,32,45,46,49, 5I

The Nylons, 107

Therernin, Leon, 88

Third. See pure third. See harmonic third

Thirds, 33, 110

Thomas, Micheal Tilson, 112

Thompson, Perronet, 76, 80

Three-quarter tone, 50

Timotheus,29

Tonality, 70, 74, 82, 83

Tonic, 27

Tonic Sol-faists. See Sol-faists

Troubadours, 55

Trouvere poets, 55

Tsai-yu, Chu, 10

Tuning compromise. See

Tuning fork, 12, 14

The story of Harmony

Tuning fork tonometer, 74

Tuning resolution, 100

Tuning root, 103, 104, 105

Tuning root, automatic, 104

Tuning root, manual, 104, 105

Twelve-tone, 82

Two streams of music, 41, 81, 98, III

-v-Varese, Edgard, 89

Vaughan, Sarah, 83

Vincentino, Don Nicola, 60

Violins, 60, 68

Virtual DSP Corp., 101

Vocal music, 65

Voice, 39, 63, 91, 98

-w-Waage, Harold, 97

Wave lengths, 93

Weigel, Erhard, 4

Well temperament,S, 8, 13,65,70

Well Tempered Clavier, 8, 14,65,70


Wcrckmeister, Andreas, 8, II -.x-Weyler, Rex, 99, 114

Xenharmoniko, 97

Wholetone, 27, 32, JS, J8, 41

Wholetone scales, 82 -y-

Wholetones, lOS Yasser, Joseph, 96

Young, La Monte, 96, 112Wiu\inson, Scott, 97

Wolf notes, 7, 8, 13,58

Wosta of Zalzal, SO -Z--

lalzal,50

Zarlino, Gioseffo, 61, 63, 91

Notes

Chapter 1

1 An account of this tuning contest appears in Johann Sebastian Bach, by PhilippSpina, translated by Clara Bell and J.A. Fuller-Maitland, London, 1884; vol. I, 137 f.Also see Equal Temperament, by 1. Murray Barbour, Michigan State College Press,1951, pp.85-87.

2 Mersenne, Marin F.; Harmonie Universelle. Paris; Sebastien Cramoisy, 1636.Translation: Roger E. Chapman, The Hague, Martinus Nijhoff, 1957, pp. 89-90.

3 Aaron, Pietro; Thoscanello De La Musica. Vinegia: Impressa per Bernardino etMattheo de Uitali, 1523. Reprint: Broude Brothers Limited, New York, 1969.


4 Holder, William; A Treatise of the Natural Grounds, and Principles of Harmony.London; 1. Heptinstall, 1694.

5 Werckmeister, Andreas; Musicalische Temperatur. Quedlinburg, Theodori PhilippiCalvisii, 1691~reprint: Utrecht, The Diapason Press, 1983.

6 Jorgensen, Owen R; Tuning. East Lansing; Michigan State University Press, 1991.The definitive text on tuning systems from the 17th to the 20th century, withinstructions for tuning these historical temperaments by ear.

7 Barbour, James Murray; Tuning and Temperament. Michigan State University Press,1951; reprint: Da Capo Press, 1972.

g Jorgensen, Owen H.; Tuning; op. cit.; pp. 63. Jorgensen cites Alexander Malcolm's ATreatise of Musick, Speculative, Practical, and Historical; Edinburgh, 1721.

9 Partch, Harry; op. cit., 259.

10 Smith, Robert; Harmonics, Or The Philosophy of Musical Sounds. Cambridge; W.Thurlbourn and T. Merrill; 1949; reprint: New York; Da Capo Press, 1966.

11 Helmholtz, Hermann; On the Sensations of Tone as a Physiological Basis for theTheory of Music. Fourth German edition, 1877; translated, revised, corrected, withnotes and additional appendix by Alexander J. Ellis. Reprint: New York, DoverPublications, Inc., 1954.

12 Jorgensen, Owen H.; Tuning; op. cit.; pp. 4, 6.

Chapter 2

13 Folger, Tim; and Shanti Menon, Discover; December, 1996.14 Gordon, R.K, trans. Anglo-Szxon Poetry. London, J.M. Dent & Sons Ltd; 1926.

Widsith, possibly the oldest poem in the English language (c. 7th C.) is a fictionalaccount of a wandering minstrel recounting heroic lore. The quoted passage makesclear that instruments and voices sang harmony. The work itself makes clear thatsong was the primary form of historical record.

15 Yasser, Joseph; A Theory of Evolving Tonality. New York; American Library ofMusicology, 1932. Yasser cites Chinese historian Sze Ma-chi'en, a contemporary ofPtolemy, second century A.D.


16 Chalmers, John H.; Divisions of the Tetrachord.Hanover; Frog Peak Music, 1993.Chalmers cites M. Duchesne-Guillemin, Revue de Musicology, 49:3~17, 1963; and55:3-11, 1969.

Chapter 3

17 When we speak: of the Greek modes, we are faced with the poor scholarship ofHenricus Glareanus, who in 1547 published the 12 ecclesiastical modes, gave themGreek names, but got all the Greek names wrong. Glareanus's "Mixolydian," thedominant seventh mode, is the Greek Ionic, and his "Ionic" is the Greek Lydian, etc.The Catholic Church adopted Glareanus's terminology, and this has caused confusionever since. Modem usage usually conforms to Glareanus, not Greek terminology,except when discussing actual Greek modes. This confusion could be settled once andfor all if scholars simply adopted the original Greek names. After all, why should wehonor Glareanus's poor scholarship by adopting his erroneous system. In this historywe will use the Greek names, preceded by the word "Greek" to make this clear. Whenwe are referring specifically to the ecclesiastical modes, we will identity them as such.

18 Hope, Robert C.; Medieval Music: An Historical Sketch. London; Elliot Stock, 1894.

Chapter 4

19 Partch, Harry; op. cit., 238.

20 The nomenclature here can lead to some confusion. The musical "third" is based onthe "fifth harmonic," and the musical "fifth" is based on the "third harmonic," Thenumbering of the harmonics is natural and simple enough. The fundamental is thefirst harmonic, and subsequent harmonics are numbered "second," "third," and soforth, The musical "third" is so called because it is the third note in a diatonic majorscale, and likewise, the "fifth" is the :fifth note. The fact that the third harmoniccreates the musical fifth, and visa versa, is simply ali accident of our diatonicconvention.

21 Chalmers, John H.; Divisions of the Tetrachord. Hanover; Frog Peak Music, 1993.The definitive treatise on scale construction based on tetrachords. For musicians


looking to expand their scale resources, this book will provide you with enough scaleoptions to :fill a lifetime of investigation. Chalmers catalogs over 700 tetrachordsincluding the diatonic, chromatic, and enharmonic genera, a section onhyperenharmonic tetrachords (with sub-quartertone intervals), and other unusualconstructions.

22 For a discussion on the mathematics of these and other proportional means, seeChalmers (Op. Cit.) pp. 29-31, and Partch (Op. Cit.) pp. 104-106.

Chapter 5

23 Smith, Sir William; Dictionary of Greek and Roman Biography and Mythology.Boston, Little & Brown, 1849. The biographies of Pythagoras and Archytas werementioned by later writers, but have never been found.

24 Schlesinger, Kathleen; "Further Notes on Aristoxenus and Musical Intervals," TheClassic Quarterly, 27:88-96; April, 1993. Also see Chalmers (Op. Cit.) pp. 17-23.

25 Aristoxenus; The Harmonics. Edited and translated by Henry S. Macran, Oxford,1902; reprint: Hildesheim, Georg Olms, 1974, pp. 188.

26 This point of view is logical in relation to the use of the monochord, but admittedlyspeculative as far as Greek choral singing goes. For an alternative view see A Historyof 'Consonance' And 'Dissonance' by James Tenny, Section I, "The pre-polyphonicera."

27 Ibid, pp. 188-1&9.

Chapter 6

28 Mersenne, Marin F.; Harmonie Universelle, (Op. Cit.), pp. 84.29 Ibid, pp. 84. This and the following are taken from Mersenne's paraphrase in which

he explains all of Ptolemy's musical theories.


Chapter?

30 Barbour, James Murray; Tuning and Temperament. (Op. Cit.) pp. 155. Barbour citesMaurice Courant, Chine et Coree, Encyclopedie de la musique et dictionnaire duconservatoire, Paris, 1913.

31 Partch; Harry; Genesis of a Music; (Op. Cit), pp, 244.32 Saleh, Dr. Fathi; The Music/abe, An Instrument for the Generation of Arabic Music

Maqamat. Paper for the Arabic Music Conference, Cairo University, 1996. Dr. Salehhas designed an ingenious instrument for representing the complex system of Arabicscales and modes, or Maqamat.

Chapter 8

33 Partch, Harry; Genesis of a Music; (Op. Cit.), pp. 245.34 Tenny, James; A History of ,Consonance' and 'Dissonance.' New York, Excelsior

Music Publishing Company, 1988, p. 20, Tenny cites Hucbald, 10th C., as the earliestunmistakable reference to simultaneous «consonance."

,35 Helmholtz, Hermann. Sensations of Tone; Op. cu.. p. 196.36 Odington, Walter; De Speculatione Musicae; edited by Frederick F. Hammond.

Stuttgart; American Institute of Musicology, 1970.37 Woldridge, fiE,; The Oxford History of Music. London, Oxford University Press,

1929; reprint, New York, Cooper Square Publishers, 1973; pp. 294-6.

Chapter 9

38 Barbour, James Murray; Tuning and Temperament. Op. Cit.39 Ibid. p. 45.40 Ellis, Alexander; in the translator's additions of Herman Helmholtz; Sensations of

Tone (Op. Cit.), p. 547.41 Partch, Harry; Op.Cit.; p. 250. Partch cites James Barbour, Equal Temperament from

Ramis (1482) to Rameau (1737).


112 Zarlino, Gioseffo; u Institutioni harmoniche. Venice, 1558. (part Three) The Art ofCounterpoint Translated by Guy Marco and Claude Palisca; New Haven, YaleUniversity Press, 1968.

43 Partch, Harry; Op.Cit.; p. 254. Quoted from James Barbour's, Equal Temperament.44 Helmholtz , Hermann. The Sensations of Tone; Op. Cit., p. 326

Chapter 10

4S Mersenne, Marin F.; Harmonie Universelle, (Op. Cit.), p. 29.

46 Jeans, Sir James; Science and Music. New York, Macmillan, 1937; p. 175.47 Perrett, Wilfrid; Some Questions of Musical Theory. Cambridge, W. Heffer and Sons,

Ltd.; 1926; p. 62.48 Rameau, Jean-Philippe; Treatise on Harmony. Translated, with introduction and

notes by Philip Gosset New York, Dover Publications, Inc., 1971; pp. 141. 142, 152.49 Ellis, Alexander, translator's notes; in Hermann Helmholtz's The Sensations of Tone;

p. 548. Ellis cites Dr. Smith's Harmonics, or the Philosophy of Musical Sounds, 2ndedition, pp. 166-7.

Chapter 11

so Jorgensen, Owen H., Tuning; (Op. Cit.); p.l.51 According to Alexander Ellis in his "Additions by the Translator in Hermann

Helmholtz's The Sensations of Tone (Op. Cit.), pp. 548-9.52 Helmholtz, Hermann. The Sensations of Tone, (Op. Cit.). This story is recounted by

Alexander Ellis, as told to him by Fetis, in footnote, p. 280.53 Helmholtz, Hermann. The Sensations of Tone, (Op. Cit.); p, nc. The following

quotes are taken from the final three chapters, beginning on this page.54 Ibid; p. 319.ss Partch, Harry; Op.Cit.; p. 267.


Chapter 12

56 Parkhurst, W. and de Bekker, L.; The Encyclopedia of Music and Musicians. NewYork; Crown Publishers; 1937; p. 274, p. 588.

57 Lloyd, Llewellen, S.; Intervals, Scales, and Temperaments, 1950. Reprint: St.Martin's Press, 1979.

58 Schonberg, Arnold; "Problems of Hannony," Modem Music, 11: 167-187; May/June,1934,

59 Cowell, Henry; New Musical Resources. New York, Alfred A Knopf, 1930.60 Hindemith, Paul; A Composer's World. Cambridge; Harvard University Press, 1952;

pp.55-6. .

61 Hindemith, Paul; The Craft of Musical Composition, V.1. New York, AssociatedMusic Publishers, Inc., 1942; p. 155.

62 Duncan, Dudley; "Septimal Harmony for the Blues," Ill, The Quarterly Journal ofthe Just Intonation Network; vol. 5, n.2, Spring, 1989.

63 Helmholtz, Hermann. The Sensations a/Tone, (Op. Cit.); p. 323.64 Eno, Brian; Keyboard, March, 1995.

65 O'Keefe, Steve; Piano newsletter, Port Townsend, Washington; 199566 Schneider, John; "Fine Tuning," Acoustic GUitar, May/June, 1994.

Chapter 13

67 Busoni, Ferruccio; Sketch of a New Esthetic of Music. New York; Schirmer Books,1911; p. 24.

68 Harrison, Lou; interview by David Doty, IIJ(Op. Cit.); V.3, N.2; Spring, 1987.69 Partch, Harry; Genesis of a Music; (Op. Cu.v; pp. 225-226.

70 Duckworth, William; Talking Music. New York; Schirmer Books; 1995; p.146.Duckworth interviews Ben Johnston, Lou Harrison, John Cage, La Monte Young,Lauri Anderson, and other "experimental composers."


71 Johnston, Ben; "Rational Structure in Music;" 1/1; (Op. Cit.); V.2, N.3, Summer,1968. This piece was reprinted from the original in the Proceedings of the AmericanSociety of University Composers.

Chapter 15

72 Marvin, Jameson; "Choral Singing in Tune." Choral Journal, 32 (5), 1991.73 Polansky, Larry; 1/1; (Op. Cit.); Winter, 1985

74 Stephen James Taylor, in correspondence with the authors and Justonic Tuning Inc.,October 5,1996.

7S ''Michael Tilson Thomas on Contemporary Music, Notation vs. Interpretation, andthe Keyboard as the Conduit for Musical Thought". Keyboard, July, 1996.

76 Riley, Terry; interview with William Duckworth; Talking Music. (Op. Cit.).77 Fathi Saleh, incorrespondence with Justonic Tuning Inc., October 15, 1995.

?8 Harry Partch, Genesis of a Music, The University of Wisconsin Press, 1949; p. 136-7.

79 Masayuki Koga, The Japanese Bamboo Flute, p. 123.

Appendix B

80 Holland, Jack; "An Introduction to 'In Tune' The Scale of Just Intonation, tI BrassBulletin, n.40, pp. 58-64, 1982. Holland describes a simplified "3 positions" system.

81 Podnos, Thcodor H.; Intonation for Strings, Winds and Singers. (Scarecrow Press,Inc. 1981).

82 See: Keislar, Douglas; "History and Principles of Microtonal keyboards," ComputerMusic Journal, vol. 11, n.l, 1987. And: Robert Rich; "Features of Tuneable MIDISynthesizers," 1/1. The Quarterly Joumal a/the Just Intonation Network, vol.8, n.l,1993.

83 Carlos, Wendy; "Tuning: At the Crossroads", Computer Music Journal, 11 :29-43, nl,1987; this is a good account of how harmonic partials contribute to our sense ofharmony.


resulting differential tone will be 264 Hz, a C below the tonic A. We bring this C intoour A440 octave by multiplying by 2, giving us C528. This C of 528 Hz is exactly 6/5times 440 Hz. For a detailed treatment of combination tones see Arthur H. Benade,Fundamentals of Musical Acoustics (Dover, 1976, 1990); and Herman Helmholtz, Onthe Sensations of Tones, (Op. Cit.)

85 Duncan, Dudley; IISeptima! Harmony for the Blues," 1/1. The Quarterly Journal ofthe Just Intonation Network, vol.S, n.2, 1989.

86 Chalmers, John R., Divisions of the Tetrachord. (Op. Cil.)87 Wilson, Ervin M., "The Marwa permutations," Xenharmonikon 9, 1986; "The Purvi

modulations," Xenharmonikon 10. Also see Chalmers, Op. Cit.

rex weyler and bill gannon - the story of harmony

Documents

q u e s t i o n s

b e i n g d e b

t h e s t o r y o f

i t i n g

appendix b

doug weyler

ibolya weyler

lisa weyler