The Mathematics of Music

James D Emery

Latest Edit: 7/25/2015

Contents

1 Introduction 2

2 Sound Waves 3

3 Logarithms 3

3.1 The Exponential Function and the Natural Logarithm . . . . . 43.2 The Logarithm to the Base b . . . . . . . . . . . . . . . . . . 8

4 Equal Temperament 9

5 Notation 10

6 Scales 11

7 Intervals 11

8 Trigonometry 11

9 Fourier Series 12

10 Sound 12

11 Harmonics 12

12 Pitch 12

13 Chords 12

1

14 Attack and Duration 12

15 Rhythm and Time 13

16 Vibrating Strings 13

17 Pipes: Vibrating Air Columns 13

18 Musical Instruments 13

19 The Piano Keyboard 13

20 Bessel Functions 13

21 Piezoelectricity 14

22 The Oscilloscope 14

23 The Spectrum Analyzer 14

24 Digital Signal Processing 14

25 The Fast Fourier Transform 15

26 Jokes 16

27 Bibliography 17

28 Appendix A: Piano Keys and Frequencies 18

29 Appendix B: Program freqint.ftn: Musical Notes and Fre-

quencies 21

30 Appendix C: Musical Terms 24

1 Introduction

Mathematics and Music have much similarity because they are both inher-ently very beautiful, and further the graphical notation for Music, and Mathe-matics, are both aesthetically pleasing. Writing musical notation and writing

2

Mathematical notation is often found by many to be quite artistically sen-sual in itself, independent of explicit meaning. It is said that mathematiciansoften have great interest and appreciation of Music. This may be true, butI am not absolutely sure of this.

We shall explore some of the mathematics of Music here. The bibliogra-phy contains several references for the Mathematics and Physics of Music,should the reader choose to explore the subject further.

2 Sound Waves

Sound waves are longitudinal waves, with the pressure oscillating in the prop-agation direction.

3 Logarithms

Music involves logarithms in a couple of different ways. The relationshipbetween notes is logarithmic, and human hearing has a logarithmic response.So let us review logarithms.

The logarithm of a product is the sum of the logarithms

log(ab) = log(a) + log(b).

The logarithm of a/b is the difference of the logarithms.

log(a/b) = log(a) − log(b).

In music we are interested in the ratio of frequencies. The octave doublesa base frequency. Thus if A = 440 cycles per second, that is 440 Hertz(abbreviated as Hz), then the octave is 880 Hz. The ratio is

880/440 = 2.

Taking the logarithm we have

log(880/440) = log(880) − log(440).

We have converted a ratio to a difference.

3

Let us take the logarithm to the base 2, written as log2. By definition ify = log2(x) then in words the logarithm of x to the base 2, is the power that2 must be raised to produce x. That is

x = 2y.

For example, if x = 512 then y = log2(x) = 9, because

29 = 512.

Suppose we have a frequency f and an octave frequency 2f , then

log2(2f/f) = log2(2) = 1,

because 21 = 2. So the difference between the octave frequency and the basefrequency is 1, when expressed using log2. A change in frequency is calledan interval. But for a smaller change in frequency we will get a number lessthan 1. So we shall multiply by 100 to get larger numbers. Thus we definethe interval difference between two frequencies f1 and f2 in cents using thedefinition

100 log2(f2/f1).

It follows that the interval between a frequency and its octave is 100 cents.In the next section we show that

log2(x) =ln(x)

ln(2)=

ln(x)

.69314718,

where ln(x) is the natural logarithm of x.

3.1 The Exponential Function and the Natural Loga-

rithm

We define the exponential function as the power series

exp(x) = 1 + x +x2

2!+

x3

3!+ ....

Differentiating the series term by term we have

d(exp(x)

dx= 0 + 1 + x +

x2

2!+

x3

3!+ ... = exp(x).

4

So a fundamental property of the exponential function is that it equals itsown derivative. This is a very natural property, and the reason that theinverse function ln(x) is called the natural logarithm.

The exponential function has the property

exp(a + b) = exp(a) exp(b).

This follows by finding the product of power series.

exp(a) exp(b) = (1 + a +a2

2!+ ..)(1 + b +

b2

2!+ ..) =

=∞∑

k=0

ck,

where

ck =a0bk

k!+

a1bk−1

1!(k − 1)!+

a2bk−2

2!(k − 2)!+ .. +

akb0

k!

=1

k!(c(k, 0)a0bk + c(k, 1)a1bk−1 + .. + c(k, k)akb0),

=(a + b)k

k!.

We have used here the binomial theorem, and the binomial coefficients

c(k, j) =k!

j!(k − j)!.

We have shown that

exp(a + b) = exp(a) exp(b).

We define a number e by

e = exp(1).

One can prove that e is an irrational and transcendental number.We have

em = exp(1)m =m∏

i=1

exp(1) = exp(m).

Alsoexp(1/n)n = exp(1) = e.

5

Thuse1/n = exp(1/n).

Thus for any rational number r

er = exp(r).

If x is irrational, then ex is not yet defined. However, if ex is to be a continuousfunction we must define, for all real x

ex = exp(x).

From the power series definition of exp(x), we see that it is a monotoneincreasing function, so it has an inverse. Define the natural logarithm as theinverse of the exponential function

ln(x) = exp−1(x).

Since exp(0) = 1, we have ln(1) = 0, and since exp(1) = e, we have ln(e) = 1.If ln(x1) = y1 and ln(x2) = y2, then x1 = ey1 and x2 = ey2 Then

x1x2 = ey1+y2,

henceln(x1x2) = y1 + y2 = ln(x1) + ln(x2).

Similarly, if r is a rational number, then

ln(xr) = r ln(x).

Thenxr = exp(r ln(x)).

If a is an irrational number, we define

xa = exp(a ln(x)).

Notice that iff(x) = ax = exp(x ln(a)),

thenf ′(x) = exp(x ln(a)) ln(a) = f(x) ln(a).

6

Hence for any real number a,

ln(xa) = a ln(x).

Letting y = ln(x), We have

d(ln(x))

dx= 1/

d(exp(y))

dy= 1/ exp(ln(x)) = 1/x.

Now given a real number a, we define the power function

ax = exp(ln(ax)) = exp(x ln(a)).

The logarithm to base a is the inverse of the power function. We write

loga(x) = y,

wheny = ax.

Example We shall show that

limn→∞

(1 + 1/n)n = e.

By taking the logarithm we have

limn→∞

ln((1 + 1/n)n)) = limn→∞

ln(1 + 1/n)

1/n

Let x = 1/n . We shall use the mean value theorem. The numerator andthe denominator each go to zero, so we may replace the numerator and thedenominator by their derivatives (L’Hospital’s Rule).

limx→0

ln(1 + x)

x

= limx→0

1/(1 + x))

1= 1.

Solim

n→∞

ln((1 + 1/n)n) = 1.

Thereforelim

n→∞

(1 + 1/n)n = e.

7

3.2 The Logarithm to the Base b

Given a positive real number b, we define the logarithm y = logb x of x tothe base b, by

x = by.

So we have

ln(x) = y ln(b) = logbx ln(b).

Therefore the logarithm to the base b may be defined in terms of the naturallogarithm as

logb x =ln(x)

ln(b).

So logb(x) has the same properties as ln(x), such as

logb(x1x2) = logb(x1) + logb(x2)

andlogb(x

a) = a logb(x).

Common logarithms use base b = 10. We have approximately

ln(10) = 2.30258509.

So we have approximately

log10(x) =ln(x)

2.30258509,

and less approximately,

log10(x) =ln(x)

2.30.

Similarly

log2 x =ln(x)

ln(2)≈

ln(x)

.69314718.

Now consider the relation between logb x and logc x, for two different basesb and c. Notice that logb x ln(b) and logb x ln(b) are equal, because they areboth equal to ln(x). Therefore

logb(x) =logc(x) ln(c)

ln(b).

8

=logc(x)

ln(b)/ ln(c)

=logc(x)

logc(b).

4 Equal Temperament

The octave is divided into 12 notes and intervals, with successive frequencyratios all equal, say to a number α. The notes in the octave are the twelvenotes with frequencies f0, f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, and f11. Now f12 =2f0. And for each k,fk+1/fk = α. Thus

f12/f0 = (f1/f0)(f2/f1)...(f12/f11) = α12

2f0/f0 = 2 = α12,

soα = 21/12.

So the frequencies are related to a fundamental frequency such as A4 withfrequency 440 Hz, by a formula

f(n) = αn−49× 440 = 2(n−49)/12

× 440,

where n is the key number on the piano. A is the 49th key on the piano. Wecan solve this for n, given the frequency f .

n = 12 log2(f/440) + 49

Thus middle C with frequency 261.63 gives

n = 12 ln(261.63/440)/ln(2) + 49 = 40,

and middle C is the 40th key.The frequency ratio 3/2 is supposed to be especially pleasing. Because

the frequency of a vibrating string is inversely proportional to its length, astring that has been shortened to 2/3 of its length, will have frequency equalto 3/2 of the frequency of the original string length. In the C major scalewith notes C,D,E,F,G,A,B of the equal tempered scale we have the ratio ofthe frequency of the fifth note G to the first note C is

332/261.63 = 1.498,

9

which is very close to 3/2. This is seven semitones. Indeed we have

α7 = 1.498307076876682 ≈ 3/2.

This is called a perfect fifth in the C major scale, C up to G, being five keys.On a stringed instrument reducing a string length by 1/3 producs a fifth,because the reciprocal of 3/2 is 2/3.

Similarly,α5 = 1.334839854170034 ≈ 4/3

gives a perfect fourth, C to F is four keys in the C major scale, which isproduced by shortening a string by 1/4.

Powers of α = 21/12

α1 = 1.059463094359295

α2 = 1.122462048309373

α3 = 1.259921049894873

α4 = 1.259921049894873

α5 = 1.334839854170034

α6 = 1.414213562373095

α7 = 1.498307076876682

α8 = 1.587401051968200

α9 = 1.681792830507429

5 Notation

See a beginning music theory book, such as the book by Henry in the bibliog-raphy, for information on the staff (grid of five lines), the treble clef, the bassclef, and the notation representing notes, such as whole notes, half notes,quarter notes etc.

I may construct a computer program to display the staff and the clefs. Imight draw the clefs with a program producing a Postscript graphic usingeither Bezier or B-spline curves.

10

6 Scales

Each octave is divided into 12 semitones. A scale such as the C major scaleconsists of a subset of these frequencies. The C major scale consists of thewhite keys on the piano, with C the tonic (first key) The major scale has thepattern WWHWWWH, the W standing for a whole step and the H standingfor a half step (semitone).

The A minor scale consists of the white keys on the piano starting withA as the tonic. The minor scale has the pattern WHWWHWW.

If we start anywhere on the piano, that is at any piano key X, we canstill form the pattern WWHWWWH, and produce a major scale. However,in the general case, the keys needed will not necessarily be just the whitekeys, some of the piano keys may be the black keys. So in this case we wouldbe playing in the key of X major, for some key we call X. Similarly, if wehave the pattern WHWWHWW, and again we would use both white andblack keys. We would be playing in the key of X minor. Equal temperamenttuning allows this to be done, since the frequency ratio between successivekeys will always be equal. This will not happen in the JUST temperamentsystem of tuning.

7 Intervals

An interval is a distance between frequencies, or a ratio of frequencies.

8 Trigonometry

The fundamental functions in trigonometry are the sine function and thecosine function. Fourier series is an infinite series of sine and cosine terms ofthe form

an cos(nω)

andbn sin(nω),

where ω is the angular fundamental frequency, and nω is the nth harmonicfrequency.

11

9 Fourier Series

If a musical instrument plays a note in fundamental frequency ω, then thequality of the sound or timbre, is characteristic of the particular instrument.That is a flute sounds differently than a piano. This depends on the quantityof each harmonic. A tone consisting say of just the first harmonic is a verypure tone, perhaps coming from an electronic keyboard. And of course itdoes not sound very great, because the absence of the higher harmonics,allows it to have little character or quality.

See the bibliography and the documents fourier.pdf and fouran.pdf.

10 Sound

See a beginning university Physics book.

11 Harmonics

Harmonics concerns the intervals between notes, such as the octave, thefourth, the fifth and so on. Harmonic analysis concerns Fourier Series, andtransforms and related material such as Haar Measure.

12 Pitch

Pitch is frequency.

13 Chords

Chords are sets of three or more keys played together to obtain harmonicsounds.

14 Attack and Duration

As a note is played on a musical instrument it has a time duration and astarting form known as the attack.

12

15 Rhythm and Time

dot diddy dot diddy dot diddy .......

16 Vibrating Strings

See books on vibration theory and partial differential equations, and mydocuments Vibration, Damped Harmonic Vibration, and Harmonic

Mechanical Vibration

17 Pipes: Vibrating Air Columns

See an elementary Physics book, or for an advanced treatment a book onmathematical acoustics.

18 Musical Instruments

19 The Piano Keyboard

The standard piano has 88 keys, this includes seven complete octaves, theseare called the 1st, 2nd, 3rd, 4th, 5th, 6th, and 7th, octaves. The startingnotes of these octaves are C1, C2, C3, C4, C5, C6, C7. Middle C is C4, withfrequency 261.63 Hz. There is a partial octave at the beginning of the key-board, consisting of the 3 keys A0, A0#, and B0, and there is a partial octaveat the end of the keyboard consisting of the single key C8. Thus there are3 + 7 × 12 + 1 = 88 keys. Key A4 = 440Hz is the 49th key of the piano.

I might create a computer program to display any portion of the pianokeyboard and display chords by the colored keys struck down.

20 Bessel Functions

The vibration of a flat plate, or of a drum involves Bessel functions. Besselfunctions arise from Bessel’s differential equation. They also arise in theX-ray diffraction pattern from crystalized DNA, and played a roll in thediscovery of the structure of DNA.

13

21 Piezoelectricity

Piezoelectricity arises from pressure applied to certain crystals, and inverselymoves when electricity is applied to piezoelectric material. It can both gen-erate and sense sound waves and ultrasound waves. Thus it has much appli-cation and affect in music.

22 The Oscilloscope

The oscilloscope displays wave shapes on a cathode ray tube, a tube like thepicture tube on a television set. An oscilloscope usually employs electrostaticdeflection of the electron beam, using charged plates, so that the voltagesand resulting electric field deflect the beam, and make the image ray trace,produce a curve on the screen.

A television traditionally used magnetic deflection of the beam controlledby coils of wire mounted on the tube.

23 The Spectrum Analyzer

A spectrum analyzer computes the amplitudes of the various harmonics mak-ing up a sampled signal. So for example can determine the character of agiven musical instrument. Once these harmonic amplitudes are known theycan be regenerated electronically with oscillators, to produce synthesizedsounds reproducing more or less the sound of an instrument, or even a hu-man voice.

24 Digital Signal Processing

Looking at the image of a sound produced by an instrument, we see that thesewave shapes are quite complicated and require determining many harmonics.There is much computation needed for this. The signals are sampled atdiscrete intervals, and so are processed by computers. If the sampling issufficiently dense, essentially all of the practical content of the wave shape canbe captured. Although the fourier series has in general an infinite number ofterms, and an infinite number of harmonics, we can only hear a small number

14

of them with our imperfect ears, so only a finite number of the harmonicsneed to be determined.

25 The Fast Fourier Transform

The Fast Fourier Transform is an algorithm the does a very fast calculation ofthe finite Fourier Transform, which makes the calculation practical for manyapplications for very large data sets. Otherwise the calculation would notbe useful for most common applications, such as, for example, Digital SignalProcessing, the determination of molecular structure from X-Ray crystallog-raphy, and the computation of 3d models using CAT scans (Computer AidedTomography), and doing a harmonic analysis of signals such as in analyzingthe timbre of musical instruments.

15

26 Jokes

[1] What do you call twenty violinists playing the Blues? A senseless act ofviolins.

[2] The phone is Baroque, please call Bach later! I wonder when I should callagain? He is probably not himself right now, but I think he will be Bach ina minuet to fix it.

[3] What do you get when you drop a piano down a mine shaft? A flat miner.

[4] What do you call someone who hangs around with musicians? A drummer.

[5] What is the difference between a fiddle and a violin? The fiddle has”strangs.”

[6] Did you hear about the bass player who was so depressed about his poortiming he threw himself behind a train?

[7] Why was Prince Esterazy often unable to find his court musician? Hewas Haydn.

16

27 Bibliography

[1] Culver Charles A., Musical Acoustics, MacGraw-Hill, 1956.

[2] Emery James D., Fourier Analysis, fouran.tex, stem2.org/je/fouran.pdf

[3] Emery James D., Joseph Fourier, stem2.org/je/fourier.pdf

[4] Emery James D., Quick Calculus, stem2.org/je/calcq.pdf

[5] Fischer J. Cree, Piano Tuning, Dover 1975, (Reprint of 1907 Edition).

[6] Halliday David, Resnick Robert, Physics, John Wiley and Sons, 1966.

[7] Helmholtz Hermann Ludwig Ferdinand von, On the Sensation of Tone,Dover, 1954, (republication of the Ellis translation of 1885, of Die Lehre vondem Tonemphfindungen.) New Introduction by Henry Margenau.

[8] Henry Earl, Fundamentals of Music, Prentice Hall, 1988.

[9] Jeans James Sir, Science and Music, Dover, 1968, (Original Publicationby Cambridge University Press, 1937).

[10] Lindsay R. Bruce (editor), Physical Acoustics: Benchmark Papers

in Acoustics, Dowden, Hutchingson and Ross, Inc., 1974.

[11] Machlis Joseph, The Enjoyment of Music, W. W. Norton, 1967.

[12] Machlis Joseph, Forney Kristine, The Enjoyment of Music, 8th Edi-tion, 1999, W. W. Norton.

[13] Mason Warren P., (editor), Physical Acoustics: Principles and

Methods, Volume 1-Part A, 1964, Academic Press.

[14] Morgan Joseph, The Physical Basis of Musical Sounds, Robert E.Krieger Publishing Company, 1980.

17

[15] Olson Harry F., Music, Physics, and Engineering, 2nd Edition,Dover, 1967.

[16] Siegmeister Elie, Harmony and Melody, Two Volumes, 1966, Wadsworth.

[17] Suits B. H., Notes: Physics of Music, Frequencies for equal-

tempered scale, www.phy.mtu.edu/suits/notefreqs.html, Michigan Tech-nological University.

[18] Winckel Fritz, (Translated from the German by Thomas Binkley), Mu-

sic, Sound, and Sensation: A Modern Exposition , Dover, 1967

[19] Wikipedia, Piano Key Frequencies,https://en.wikipedia.org/wiki/Piano key frequencies

[20] Wood Alexander, (Revision Bowsher J. M.), The Physics of Music,Sixth Edition, University Paperbacks, Methuen and Company, London, 1961.

28 Appendix A: Piano Keys and Frequencies

The 88 piano keys and their frequencies:

88 piano keys:

1 A0 27.500000000000

2 A#0 29.135235094881

3 B0 30.867706328508

4 C1 32.703195662575

5 C#1 34.647828872109

6 D1 36.708095989676

7 D#1 38.890872965260

8 E1 41.203444614109

9 F1 43.653528929125

10 F#1 46.249302838954

11 G1 48.999429497719

12 G#1 51.913087197493

13 A1 55.000000000000

14 A#1 58.270470189761

15 B1 61.735412657015

16 C2 65.406391325149

17 C#2 69.295657744218

18 D2 73.416191979352

19 D#2 77.781745930520

20 E2 82.406889228217

21 F2 87.307057858251

18

22 F#2 92.498605677908

23 G2 97.998858995437

24 G#2 103.82617439499

25 A2 110.00000000000

26 A#2 116.54094037952

27 B2 123.47082531403

28 C3 130.81278265030

29 C#3 138.59131548844

30 D3 146.83238395870

31 D#3 155.56349186104

32 E3 164.81377845643

33 F3 174.61411571650

34 F#3 184.99721135582

35 G3 195.99771799087

36 G#3 207.65234878997

37 A3 220.00000000000

38 A#3 233.08188075904

39 B3 246.94165062806

40 C4 261.62556530060

41 C#4 277.18263097687

42 D4 293.66476791741

43 D#4 311.12698372208

44 E4 329.62755691287

45 F4 349.22823143300

46 F#4 369.99442271163

47 G4 391.99543598175

48 G#4 415.30469757995

49 A4 440.00000000000

50 A#4 466.16376151809

51 B4 493.88330125612

52 C5 523.25113060120

53 C#5 554.36526195374

54 D5 587.32953583482

55 D#5 622.25396744416

56 E5 659.25511382574

57 F5 698.45646286601

58 F#5 739.98884542327

59 G5 783.99087196350

60 G#5 830.60939515989

61 A5 880.00000000000

62 A#5 932.32752303618

63 B5 987.76660251225

64 C6 1046.5022612024

65 C#6 1108.7305239075

66 D6 1174.6590716696

67 D#6 1244.5079348883

68 E6 1318.5102276515

69 F6 1396.9129257320

70 F#6 1479.9776908465

71 G6 1567.9817439270

72 G#6 1661.2187903198

73 A6 1760.0000000000

74 A#6 1864.6550460724

75 B6 1975.5332050245

76 C7 2093.0045224048

77 C#7 2217.4610478150

78 D7 2349.3181433393

19

79 D#7 2489.0158697766

80 E7 2637.0204553030

81 F7 2793.8258514640

82 F#7 2959.9553816931

83 G7 3135.9634878540

84 G#7 3322.4375806396

85 A7 3520.0000000000

86 A#7 3729.3100921447

87 B7 3951.0664100490

88 C8 4186.0090448096

20

29 Appendix B: Program freqint.ftn: Musi-

cal Notes and Frequencies

c freqint.ftn 7/15/15, musical notes and frequencies

implicit real*8(a-h,o-z)

dimension f(500)

character*4 key(100)

n=100

ndom=2

key(1)=’C0’

key(2)=’C#0’

key(3)=’D0’

key(4)=’D#0’

key(5)=’E0’

key(6)=’F0’

key(7)=’F#0’

key(8)=’G0’

key(9)=’G#0’

key(10)=’A0’

key(11)=’A#0’

key(12)=’B0’

key(13)=’C1’

key(14)=’C#1’

key(15)=’D1’

key(16)=’D#1’

key(17)=’E1’

key(18)=’F1’

key(19)=’F#1’

key(20)=’G1’

key(21)=’G#1’

key(22)=’A1’

key(23)=’A#1’

key(24)=’B1’

key(25)=’C2’

key(26)=’C#2’

key(27)=’D2’

key(28)=’D#2’

key(29)=’E2’

key(30)=’F2’

key(31)=’F#2’

key(32)=’G2’

key(33)=’G#2’

key(34)=’A2’

key(35)=’A#2’

key(36)=’B2’

key(37)=’C3’

key(38)=’C#3’

key(39)=’D3’

key(40)=’D#3’

key(41)=’E3’

key(42)=’F3’

21

key(43)=’F#3’

key(44)=’G3’

key(45)=’G#3’

key(46)=’A3’

key(47)=’A#3’

key(48)=’B3’

key(49)=’C4’

key(50)=’C#4’

key(51)=’D4’

key(52)=’D#4’

key(53)=’E4’

key(54)=’F4’

key(55)=’F#4’

key(56)=’G4’

key(57)=’G#4’

key(58)=’A4’

key(59)=’A#4’

key(60)=’B4’

key(61)=’C5’

key(62)=’C#5’

key(63)=’D5’

key(64)=’D#5’

key(65)=’E5’

key(66)=’F5’

key(67)=’F#5’

key(68)=’G5’

key(69)=’G#5’

key(70)=’A5’

key(71)=’A#5’

key(72)=’B5’

key(73)=’C6’

key(74)=’C#6’

key(75)=’D6’

key(76)=’D#6’

key(77)=’E6’

key(78)=’F6’

key(79)=’F#6’

key(80)=’G6’

key(81)=’G#6’

key(82)=’A6’

key(83)=’A#6’

key(84)=’B6’

key(85)=’C7’

key(86)=’C#7’

key(87)=’D7’

key(88)=’D#7’

key(89)=’E7’

key(90)=’F7’

key(91)=’F#7’

key(92)=’G7’

key(93)=’G#7’

key(94)=’A7’

22

key(95)=’A#7’

key(96)=’B7’

key(97)=’C8’

c do i=1,60

c lc=lenstr(key(i))

c write(*,’(1x,a)’)key(i)(1:lc)//’0’

c write(*,’(1x,a)’)key(i)

c enddo

c do i=1,13

c write(*,’(1x,i5,1x, 3(1x,g15.8))’)i,f(i),f(i+1)/f(i),f(i+1)-f(i)

c end do

f(58)=440.

a=2.d0**(1./12.d0)

write(*,*)’ a = ’,a

write(*,*)’ a^12 = ’,a**12

write(*,*)’ fifth, alpha^7 = ’, a**7

write(*,*)’ ln=’,log(exp(5.d0))

write(*,*)12*log(261.63/440)/log(2.d0)+49

write(*,*)’ powers of alpha’

do i=1,9

write(*,*) i, a**i

end do

write(*,*)

c do k=1,88

write(*,*)’ 88 piano keys: ’

do k=10,97

f(k)=f(58)*a**(k-58)

j=k-9

write(*,’(1x,i4,1x,a,1x,g21.14)’)j,key(k),f(k)

end do

one=1.

zero=0.

pi=4.*atan(one)

nfile1=1

c open(nfile1,file=’q.txt’,status=’unknown’)

nfile2=2

open(nfile2,file=’q.eg’,status=’unknown’)

end

c

c+ lenstr nonblank length of string

function lenstr(s)

c length of the substring of s obtained by deleting all

c trailing blanks from s. thus the length of a string

c containing only blanks will be 0.

character s*(*)

lenstr=0

n=len(s)

do 10 i=n,1,-1

if(s(i:i) .ne. ’ ’)then

lenstr=i

return

endif

10 continue

return

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30 Appendix C: Musical Terms

[1] Just Temperament. This is a tuning based on Pythagorean-like meth-ods, using frequencies based on rational number multiplication of a funda-mental frequency, and on exact classical chords, produced for example on theclassical stringed instruments of the Greeks.

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