Music Information Loops:

Described and Analyzed with Sequential Information Analysis

by

Ryan I. Dorin

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Music

New York University

May, 2005

______________________

Elizabeth Hoffman

© Ryan Dorin

All Rights Reserved, 2005

“Humpty Dumpty sat on a wall:

Humpty Dumpty had a great fall.

All the King’s horses and all the King’s men

Couldn’t put Humpty Dumpty in his place again.”

“That last line is much too long for the poetry,” she added, almost out loud,

forgetting that Humpty Dumpty would hear her.

-Lewis Carroll, Through the Looking Glass

DEDICATION

For my family.

iv

ACKNOWLEDGMENTS

My biggest thanks goes to my family, friends, professors, and colleagues for

their support, inspiration, instruction, and patience throughout these graduate school

years.

This project would not have been possible without my dedicated advisor,

Elizabeth Hoffman. Her patient reading and rereading of countless drafts, plentiful

rainbow-colored comments, and her constant support and encouragement is deeply

appreciated.

I would also like to thank the other members of my committee. Stanley

Boorman contributed greatly in the development of this project, and Michael

Beckerman’s endless enthusiasm and support helped keep me enthusiastic about this

dissertation. A special thanks also to my other faculty readers Suzanne Cusick and

Edward Roesner. Thanks also to Louis Karchin.

So many of my friends have contributed to this work, but a few deserve

special mention. I would like to thank Juliana Trivers for helping me prepare for my

defense. Thanks to Wynn Yamami for his assistance in designing the second and third

stage sequential information diagrams. Thanks to Christopher Ariza for his technical

assistance over the years, and to Michael Birenbaum Quintero for helping me develop

the meta loop concept.

Rena Mueller deserves a special thanks for helping untangle me from all sorts

of bureaucratic snafus over the years. Thanks also to Pauline Lum for keeping the

department running smoothly and for always being ready to assist when something has

to get done.

v

PREFACE

The Gospel of the Music Information Loop (Gospel According to Loop)

In the beginning was the word.

And the word was so good, it could only be sung by one melody at a time.

But there were more words than melodies,

so to compensate,

the melodies were often looped.

Melodies composed for the Church were looped throughout the year,

and looped again in yearly cycles.

Everyone sang the same melody line at the same time for quite a while.

But they eventually discovered that it was holy to double a melody line,

but only at the perfect intervals.

Once the cat was out of the bag, it was a slippery slope

from organum into polyphony.

In the beginning of polyphony,

It was most common to add a second voice to a preexisting melody.

In this way, borrowed melodies became looped throughout entire repertoires,

as composers appropriated them into their new work.

Some melodies were looped for hundreds of years, some perhaps thousands.

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As music notation developed, more sophisticated nomenclatures

for describing music information came to be used.

Not long after the advent of rhythmic notation,

composers began manipulating rhythmic loops,

in combination with melodic loops.

Early rhythmic notation employed a collection of information units

called the rhythmic modes.

The rhythmic modes were looped throughout an entire repertoire,

until composers tired of them and rhythmic notation modernized.

The Ars Nova that followed led to an unprecedented development

in the use of loops as an organizational procedure.

A wide array of looping activities quickly spread into compositional practice.

Isorhythm, cantus firmus, parody, canon, contrafacta,

strophic form, fugue, passacaglia, chaconne,

mottos, subjects, motives, theme and variations,

da capo, ritornello, minuet and trio, sonata, rondo,

these all depend upon carefully organized repetitions of particular information units.

The advent of tonality affirmed a notorious loop:

the musical feature of Tonic-Dominant-Tonic.

Schenker exposed this loop, and atonality rejected it altogether.

But the loop of tonality was replaced with the loop of serialism.

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In the beginning of serialism,

12-Tone Row information units and their variants were looped.

Serialization in music was not entirely new,

but serializing a row of all 12 chromatic pitches was.

And the serial loop behind the tone-row spread into other information fields,

and works were created where every bit of music information fit into a serial array.

Around the same time, John Cage threw dice, coins, and sticks

to generate his information organization.

Such precompositional looping procedures became nearly as complicated

as precompositional serial arrays.

Minimalists rejected serialists – such brainy looping was difficult to listen to –

they sought to create music that was an audible expression of

a quasi-mechanical looping procedure.

Once a procedural alteration operation is set up and 'turned on',

it will continue to run until the loop cycles are complete.

Music about loops was not new,

but making the form of a piece an aural articulation of a looping procedure was.

Such innovations relied upon new technology

and the influence of non-Western musics.

Another cat was let out of the bag when music information loops went digital.

The conditions in the information age facilitate

the proliferation of music information loops,

if only because they are so easy to replicate and spread around.

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Loops hit the mainstream in the digital audio realm,

and now every Apple computer comes with a package of public domain music loops,

for use with its music sequencing program.

Perhaps the contemporary ascension of the loop as a music-making tool

opened the door

for the recontextualization of the loop in traditional music composition.

Techniques of music information organization in digital media

give a new perspective on techniques of information organization

in music of the past.

Sequential information analysis is in part a response to this new perspective.

It provides a method for describing the activity

of music information loops in a piece of music.

Sequential information analysis considers information loops

in any and all information fields.

Though different repertoires may display different hierarchical arrangements

of the information fields,

sequential information analysis gives equal attention to all of them.

In this way,

sequential information analysis

may be employed wherever

music information loops are found.

So ends the Gospel According to Loop.

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ABSTRACT

This dissertation develops the definition of an information loop, and uses its

various types to describe the constructional logic of certain repetitive musical devices.

An information loop is a musical feature that happens over and over again, either

within a work or amongst numerous works sharing the same musical feature. A

musical feature is understood here to mean a sound, or a method of organizing sound.

The viewing of music, sonic or notational, as “music information” is part of a

methodology that seeks to transcend engrained stylistic, historical, and ideological

boundaries that have traditionally compartmentalized Western music repertoires.

A theoretical framework, called sequential information analysis, is proposed

for analyzing the operation of repetition as it organizes music information. By dealing

with patterns of music information rather than with more style-specific musical

nomenclatures, sequential information analysis allows comparisons between widely

divergent musical types from the last 1000 years of Western music. This broad

analytic tool is demonstrated in applications drawn from a diverse sampling of

western music. Similarities between the creative and logical premises underlying these

examples are discussed.

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TABLE OF CONTENTS

DEDICATION iv

ACKNOWLEDGMENTS v

PREFACE vi

ABSTRACT x

LIST OF EXAMPLES xiii

LIST OF APPENDICES xvi

CHAPTER 1: SEQUENTIAL INFORMATION ANALYSIS

1-1. Introduction 1

1-2. Music Information 3

1-3. Information Units 4

1-4. Information Fields 5

1-5. The Sequential Information Diagram 6

1-6. Information Loops in Repetitive Sequence 7

1-7. Alteration Operations 10

1-8. Information and Sequence 12

1-9. Structural Hierarchies of Sequences 12

1-10. Case Study No. 1: Beethoven Symphony No. 6 mvt. 1, 14 development section

CHAPTER 2: THE INFORMATION LOOP

2-1. Introduction 25

2-2. The Artifice of Reference 31

2-3. Imitation as Repetition 32

2-4. The Mechanical Loop 36

2-5. Loop in the Rhythmic Information Field 40

2-6. Loop in the Textural Information Field 45

2-7. Loop in the Harmonic Information Field 50

2-8. The Sectional Information Loop 65

xi

2-9. The Melodic Information Loop 68 and Case Study No. 2: J.S. Bach Invention in B-flat Major

CHAPTER 3: STRUCTURAL, ORNAMENTAL, AND META INFORMATION LOOPS

3-1. Introduction 87

3-2. The Structural Information Loop 87

3-3. The Ornamental Information Loop 94

3-4. Case Study No. 3: 98 Bars 44-79 from the “Liebestod” in Wagner’s Tristan

3-5. The Meta Information Loop 123

3-6. Conclusion 129

APPENDICES 131

BIBLIOGRAPHY 176

xii

LIST OF EXAMPLES

Ex. 1 Beethoven Symphony No. 6 mvt. 1, mm. 1-4 15

Ex. 2 Beethoven Symphony No. 6 mvt. 1, mm. 139-242 17 melodic and textural information units

Ex. 3 Beethoven Symphony No. 6 mvt. 1, mm. 139-242 18 first sequential information reduction

Ex. 4 Beethoven Symphony No. 6 mvt. 1, mm. 149-236 21 harmonic rhythm

Ex. 5 Beethoven Symphony No. 6 mvt. 1, mm. 139-242 23 sequential information diagram

Ex. 6 Common Schenkerian Ursatz form 29

Ex. 7 Josquin Desprez, Ave Maria, opening phrase 33

Ex. 8 Josquin Desprez, Ave Maria, first information unit 33

Ex. 9 Josquin Desprez, Ave Maria mm. 64-73 34

Ex. 10 Josquin Desprez, Ave Maria 35 information unit with transposition

Ex. 11 Steve Reich, Clapping Music, information unit 37

Ex. 12 Steve Reich, Clapping Music, m. 2 37

Ex. 13 Hanon, Piano Exercise No. 1, mm. 1-4 38

Ex. 14 “Row, row, row your boat” 40

Ex. 15 “Twinkle twinkle little star” 41

Ex. 16 “Twinkle twinkle,” rhythmic information unit 41

Ex. 17 J.S. Bach, Cto. in F Major BWV 978, mvt. II, mm.1-12 42(Bach's transcription of Vivaldi’s Cto.. in G Major, Op. 3, No. 3)

Ex. 18 Bach-Vivaldi, Cto.. in F Major BWV 978, mvt. II, mm.1-12 43 rhythmic information unit

Ex. 19 J.S. Bach, Invention in B-flat major, mm. 14-15 44

Ex. 20. Robert Schumann, Arabesque Op. 18, mm. 1-16, melody 44

Ex. 21 J.S. Bach, C Major Prelude, WTC I, mm. 1-4 46

Ex. 22 W.A. Mozart, Piano Sonata K332, mvt. 2, mm. 1-2 48

Ex. 23 Frédéric Chopin, Nocturne in D-flat Major, Op. 27, No. 2 48

Ex. 24 Franz Schubert, Die Forelle, mm. 1-4 49

Ex. 25 Vivaldi, The Four Seasons, “Winter” mvt. 1, mm. 22-26 51

xiii

Ex. 26 Vivaldi, The Four Seasons, “Winter” mvt. 1, mm. 22-26 52 bass line

Ex. 27 Gimbel and Fox, “Killing Me Softly,” mm. 5-11 53

Ex. 28 Gimbel and Fox, “Killing Me Softly,” mm. 5-11 54 actual bass line and hypothetical extended loop

Ex. 29 Mozart, Piano Sonata K533 mvt. 1, mm. 125-147 56 melodic information units, alteration operations, and hypothetical bassline loop

Ex. 30. Beethoven Symphony No. 6 mvt. 1, mm. 247-278 60 bassline with hypothetical continuation

Ex. 31 Beethoven Symphony No. 6 mvt. 1, mm. 9-12 60

Ex. 32 Beethoven Sym. #6, mvt. I, mm. 247-278 61 information units (IUs)

Ex. 33 Beethoven Sym. #6, mvt. I, mm. 247-282 63 information unit markup score

Ex. 34 Beethoven Sym. #6, mvt. I, mm. 247-282 65 sequential information diagram (SID)

Ex. 35 Chopin, Ballade No. 2 in F major Op. 38, mm. 82-88 67

Ex. 36 Ave mater salvatoris 68

Ex. 37 Rachmaninoff, Sym. #2, mvt. III, 1-4 69

Ex. 38 Britten, “Dirge” from Serenade for Tenor, Horn, and Strings 71 mm. 1-12, melodic information unit; first two iterations

Ex. 39-46 refer to: J.S. Bach Invention No. 14 in B-flat Major, BWV 785

Ex. 39 composite rhythmic information reduction 72

Ex. 40. Primary melodic information unit (MIUA) with variants 73

Ex. 41 Secondary melodic information unit (MIUB) with variants 74

Ex. 42 Cadential melodic information unit (CIU) with variants. 75

Ex. 43 Melodic Information Unit Score Markup mm. 1-20 76

Ex. 44 First-stage sequential information diagram 79

Ex. 45 SID overlay on score 82

Ex. 46 Second-stage sequential information diagram 84

Ex. 47 J.S. Bach Goldberg Variations bass line information unit 89

Ex. 48 Guillaume de Machaut, “Bone pastor Guillerme” (1324) 91 Sequential information diagram of structural information unit

xiv

Ex. 49 Machaut, “Bone pastor Guillerme” 92 Sectional melodic information unit with repetitions

Ex. 50 Machaut, “Bone pastor Guillerme” 93 Sectional rhythmic information unit, mm. 1-97

Ex. 51 Machaut, “Bone pastor Guillerme” 94 rhythmic information, mm. 97-142

Ex. 52-64 refer to: Wagner, Tristan und Isolde, “Liebestod”, mm. 44-76

Ex. 52 MIUA and variants 99

Ex. 53 MIUB and variant 99

Ex. 54 MIUC and variant 100

Ex. 55 RIUA and variants with optional alteration operation 100

Ex. 56 RIUB and variants 100

Ex. 57 RIUC 101

Ex. 58 BIUA1 and variant 101

Ex. 59 BIUA1 and variant Harmonic Information Units 102

Ex. 60. Sequential Information Diagram, first-stage reduction 104

Ex. 61 Sequential Information Diagram, second stage reduction 108

Ex. 62 Sequential information diagram, third-stage reduction 113

Ex. 63 Information unit markup score 116

Ex. 64 First-stage sequential information diagram overlaid on score 120

Ex. 65 12-bar blues harmonic progression 123

xv

LIST OF APPENDICES

APPENDIX A: Introduction to Compositions 131

APPENDIX B: be el kakku 134

APPENDIX C: Rowing and Fibbing 149

APPENDIX D: The Brave Beetle 158

xvi

Chapter 1: Sequential Information Analysis

1-1. Introduction

This dissertation is an introduction to the music information loop. The music

information loop is a phenomenon created by patterned repetitions of information

units. Information loops may arise from repeated musical objects, or from repetitive

procedures for manipulating those objects. “Information” here describes the

abstraction of sound, or patterns of sound, into a representational idea of those

sounds. The various components of musical information are categorized according to

function into information fields. In characterizing information loops, we enlarge the

notion of musical repetition beyond its typical denotation of a sound, or pattern of

sounds, played over again. With the abstraction of musical information, we can better

describe methods of organizing sounds that are repeated over again. In this way, I hope

to provide a valuable recontextualization of how we think about and define

repetition in the organization and construction of music.

The resulting analytical survey categorizes various types of information in

repertoire from the Middle Ages to the present by assessing the role of the

information loop in the creation of musical form and content. Partly to demonstrate

the utility and value of employing the information loop concept in music analysis, this

dissertation investigates music on a deliberately far-reaching historical time scale. We

will discover that information loops have always been, and continue to be, a

fundamental tool of musical construction by composers of Western music. The

fundamental properties of the loop allow for comparative analyses that eliminate

stylistic, historical, cultural, and ideological boundaries. Such boundaries have

traditionally compartmentalized Western music repertoires into like-minded

categories.

1

On the other hand, employing such a wide repertoire for this discussion by no

means implies that information loops are employed in the organization of every piece

of music. We might be surprised, however, to discover information loops operating in

many places where overt repetition is not immediately apparent. And in some cases,

information loops are employed with such great abstraction that it is not possible to

hear the loops at all. Since we can find information loops in music throughout the

ages, a broad survey reveals historical trends in the types of music information

composers employed while generating loops. Information loops may operate within

different structural hierarchies, and their behavior may vary from structural ornament

to structure itself.

The structural mechanism of the loop both generates and depends upon

sequences of repeated information. Information units may be organized into

sequenced, or sequential, patterns. Our analytical method concerns itself with the

management of these repetitive patterns. Our survey will demonstrate the

pervasiveness of repetitive sequences of different types of music information in all

sorts of music compositions. It will become evident how repetitive sequences of

information units allow a composer to generate substantial amounts of music out of a

small amount of music information. Such loops also tend to provide a sense of unity

in the organization of the information content, giving a work its form.

In order to represent the organization of sequential information loops I have

developed a musical score “mark up language” with notational shorthand. What

results is a series of sequential information diagrams which represent the

underlying patterns of information loops in a musical work or passage. “Sequence”

here is used in the broadest sense as a unit of information that is reiterated.

Sequential information analysis is used to illuminate the music information that

repeats over and over again, and to organize this information into patterns of

2

information loops. Understanding how this information is organized sets into stark

relief any non-repeating information operating across a work’s form.

There are no stylistic limitations or ideological barriers in sequential information

analysis, as traditional questions of musical meaning are often secondary here.

Sequential information analysis is concerned primarily with representing the

organization in the structure of a musical work. There is no theoretical limit to the

different sorts of musical information that can be sequenced. By treating music as

organized patterns of information units, instead of employing more style-specific

musical nomenclatures, sequential information analysis provides a consistent

methodology for engaging in comparative analyses between widely divergent musical

types from the last 1000 years of Western music. It is my hope that this venture will

prove valuable in furthering our understanding of the creative and logical premises

behind the organization of sound in Western music.

1-2. Music Information

In a real sense, a discourse on music information was opened when Pythagoras

first explained the phenomenon of musical pitch as ratios of whole numbers. It then

became possible to discuss certain properties of sound according to mathematical

formulas.1 But music information itself can be manifested in different ways. Recorded

sound, for example, may be mathematically quantified, transmitted, stored,

reproduced, and synthesized –– with the proper technology. The physical sound is

converted to a stream of digital information which can then be decoded back into

sound. That is to say, it is not the actual sound which is stored, but rather it is the

1. See Lucent Technologies 1998. The article celebrates the fiftieth anniversary of the publishing of Claude Shannon’s “Mathematical Theory of Communications.” Shannon developed information theory, where information refers to a communication being transmitted over a communication system. Sequential information analysis is not related to information theory.

3

information which describes to a mechanical interface how to reproduce the sound.

Likewise, a musical score is a set of instructions detailing how to reproduce the

sound, inscribed on paper for interpretation by the interface (eyes) of a machine

(musician). The transmission of music, in other words, depends upon the transmission

of music information describing how to make the music.

Much like a language, music communicates by connecting independent pieces of

sonic information into larger streams of information through time. These temporal

streams gather more pieces of information and order them, thereby continuing to

gather meaning through contextual association until their connected pattern has made

a complete statement, thought, idea, or feeling. As with language, there can be many

different ideas which, when put together, create through their association, opposition,

and integration a deeper meaning than they could as separate entities. All forms of

communication, including music, operate on this notion. The operation of

information loops in the language of music information often appears analogous to

grammar in natural language. In order to investigate information loops, we will first

determine the way music information is organized, and then turn to the actual

content of the communication itself.

1-3. Information Units

The information content of a piece of music can be arranged and sequenced into

information units which may be viewed as both the building blocks and dynamic

generators of musical form and content. Once abstracted from their formal context,

the information units may be viewed as instructions, or templates, each describing a

systematized organization of a pattern, or patterns, of sound.

Music analysis has traditionally found specific ways to describe certain

information units. Motive, theme, melody, set class, tone row and rhythmic cell are all

4

identifiers of information units in specific repertoires. But it is the contextual

specificity of these terms that make their use counterproductive for comparative

analysis between the organization of information in widely divergent repertoires.

Recontextualizing the building blocks of a composition as information units gives us a

flexible concept that is applicable in virtually any musical context.

1-4. Information Fields

In order to better conceptualize the interplay of the music information in a

composition, let us distinguish various elements of music as specific information

fields. Music information fields may include, but are not limited to, pitch, rhythm,

meter, harmony, texture, melody, timbre, register, and time. Music information fields must

be distinguished from information fields in general, and they simply describe the

information directly related to the nature of sound. Other types of information fields

describe structural systems of information organization, not the actual sound itself.

Information may be abstracted from an individual field, or from combinations of

fields with weighted intensities. Sequential information analysis is used to

contextualize the interactions between these different information fields. It allows us

to conceptualize numerous permutations of various information fields, and to

measure their relative interactions with a particular information unit.

The metaphor of the information field is accurate for conceptualizing the

multidimensional relationships between each of the above musical elements. Each

field is limited to a range of very specific types of information, but the range of

possible information for each type is practically infinite. Information fields are

isolated from each other into special niches of information organization, magnifying

our awareness of their participation in the organization of a work. Fields may

encompass complex interactions of information, and, in a given work, we may find

5

many levels of information organization within a single field. Interplay between the

different information fields is characterized by either a static or dynamic

interrelationship.

When different information fields operate in synchronized fashion upon an

information unit, they have a static interrelationship, and we say these fields are

conflated. Conflated fields often function as units of information which have one

corresponding sound. For example, a motive in a work by Beethoven will typically

synchronize information from the fields of rhythm and melody to complete its design.

The motive is characterized by a melodic shape expressed with a particular rhythmic

shape. The melodic and rhythmic information fields are conflated in the expression of

the information unit. In practice, overt musical repetition is typically characterized by

the conflation of several information fields.

We define our own information fields in each sequential information analysis we

create. The fields become our organizational categories for the information content.

Going through the exercise of defining and conceptualizing information fields allows

us to plot their interaction in the organization of a work. Sequential information

analysis is largely interested in describing these interactions. In addition, we will see

that different musics favor different information fields. It is interesting to discover

composers from different eras applying the same repetitive systems, but to

information in different fields.

1-5. The Sequential Information Diagram

Placing aesthetic value on the balance between form and content in a work has

long been of primary concern in Western music. In sequential information analysis,

the music information may be described as the content, and its organization as the

form. Because the meaning of the content changes through the nature of its

6

association, opposition, and integration (in a given context), form and content are

forever intertwined.

For enhanced analytic comprehensibility and comparison, sequential information

analysis makes use of a visual aid called the sequential information diagram. The

representation of information in the diagram is loosely based on the graphical

interface of a generic computer sequencer. Sequential information diagrams are useful

for illustrating the layout of the patterns of repetitive sequences in musical works. In

effect, they provide a graphical model of key features of the form and content of the

music information in a work.

The sequential information diagram serves fundamentally to articulate the

repetitive features of a musical work. This is in contrast to many types of traditional

music analysis which often focus on the non-repetitive features of a work. For this

reason, sequential information analysis is not meant to replace or contradict any

existing analytical methods. It is meant, rather, to complement other analytical tools.

Graphically representing the organization of musical information generates

analytical data, and these data may be freely interpreted. For ease of reference, the

diagrams are formatted to match the layout of the musical score upon which they are

based. The style of the diagram can vary, but its purpose is to articulate the iterations

of information units, thus revealing repetitive patterns operating in various

information fields.

1-6. Information Loops in Repetitive Sequence

A repetitive sequence is a device that presents repetition-with-alteration of units

of music information. Typically in a repetitive sequence, some basic information unit

is repeated consecutively a number of times, and with each repetition, a particular

operation is performed on the information unit to alter it in some way. Despite this

7

alteration operation, some specific feature of the information unit is left unchanged in

each subsequent iteration. There are numerous style-specific operations that may be

performed to alter some aspect of the information unit while leaving crucial

distinguishing aspects unchanged. In a repetitive sequence, an information loop is

generated by the music information which is repeated without alteration.

Traditionally, the term “sequence” has applied to two specific types of sequences

– the melodic sequence and the harmonic sequence. Indeed, both of these are

common in sequential information analysis, but as sequential information analysis

treats all information fields as having equal potential for sequential repetition, the

melodic and harmonic sequences are only two out of many possible types of

sequences. In addition, loops with conflated information fields are common in

repetitive sequences.

The relationship between repetitive sequences and information loops has always

been evident in music composition, but the resonance between these two phenomena

is particularly evident in the realm of computer music sequencing. For today’s

computer-assisted composer, music sequencing is a common activity in which

multiple tracks are layered against each other in a graphical interface on a computer.

The tracks commonly contain digital sound samples.2 Each sample is a distinct music

information unit, and can contain varied types of music information. Tracks are laid

out on an xy grid, where time is measured along the x axis. This process of sequencing

facilitates the practice of allowing the tracks or samples to loop themselves in a

predetermined pattern. Or the samples may be cut and pasted into different tracks at

different times. Our information units, the sound samples, are represented as movable

rectangular boxes within the grid. Alteration operations may be performed on each

iteration of each information unit. Available operations depends upon the given

2. “Sample” has come to commonly mean a fragment of digital audio.

8

software used for manipulating the samples. Operations on the information unit

typically include adding reverb, changing pitch, reversing, changing durations,

performing spectral filtering, etc.

Looping is an efficient production method as it may generate a sizable portion of

sound from a small collection of material. Sequencing in this instance creates a series

of sound events from the repeated application of predetermined sound events. There

are few restrictions on the kinds of information that may be found in these

predetermined sound events. Sound events are arranged into a sequence, or patterns

of sequences, and the final product is generally called a piece of music.

In traditional classical music, instead of samples there are motives, themes,

rhythms, textural patterns, etc. But composers have still tended to fashion their music

out of small pieces of information which can be manipulated with prescribed

alterations, with value often placed on an economy of means and efficiency of

information distribution.3 From a practical standpoint, this efficiency allows a

composer to fill more time with a minimum of information. Repetition is also helpful

because it facilitates listener comprehension of the structural organization of a work.

When information is repeated, less effort need be spent on grasping the content and

more contemplation may be given to grasping its organization. In addition, the

repetitive sequence, and the logic of repetitive patterns in general, has cognitive roots

3. For example, economy of means is a hallmark of Beethoven’s compositional craft, and his music is admired for the powerful statements it makes with the manipulation of a small number of information units. It might be said that it was a sophisticated distillation of these values which led to the notion of ‘organicism’ in a musical work. Both the notion of organicism and the notion of efficiency are dependent on generating musical form through information repetition.

9

in human perception.4

1-7. Alteration Operations

Alteration operations occupy their own information field, and relate to the music

information fields on a structural and procedural level. Different repertoires often

employ different types of alteration operations. Nevertheless, it is surprising how few

in number are the fundamental operations we tend to find in music. Common

alteration operations are often subsets of transposition, retrograde, and inversion in

the pitch information field, and augmentation and diminution in the rhythmic

information field. In addition, an alteration operation may be an empty set, in which

case the information unit is repeated without alteration. These basic operations are

used in the mechanism of information manipulation. These operations are found in all

repertoires, as well as in sound manipulation with digital technology. Put simply,

these alteration operations allow us to take an information unit and make it higher or

lower, play it forwards or backwards, turn it upside down, and play it faster or

slower.5

Consider this comment made by C.P.E. Bach about his father, J.S. Bach, listening

to a particular fugue for the first time:

4. Rahn 1993, 49. Rahn goes so far as to state that “All musical structure derives from repetition” and that humans depend upon repetition in order to perceive structure. See also Drake and Bertrand 2001. Drake and Bertrand’s first three conclusions on universals in temporal processing are these: 1) we tend to group into perceptual units events that have similar physical characteristics or that occur close in time; 2) processing is better for regular than irregular sequences. We tend to hear as regular sequences that are not really regular; 3) we spontaneously search for temporal regularities and organize events around these perceived regularities.

5. These are the basic alteration operations, but are by no means the only ones. Changes of orchestration or instrumentation of an information unit, reharmonizations of a melody, and repeating a melodic information with different lyrics, are other common alteration operations.

10

When he (J.S.) listened to a rich and many-voiced fugue, he could soon say, after the first entries of the subjects, what contrapuntal devices it would be possible to apply, and which of them the composer by rights ought to apply, and on such occasion, when I was standing next to him, and he had voiced his surmises to me, he would joyfully nudge me when his expectations were fulfilled.6

Fugal technique in fact involves a highly repetitive process whereby units of

information – subjects and countersubjects – are manipulated with various alteration

operations – transposition, diminution, augmentation, inversion – and placed in

counterpoint amongst several voices. Extrapolating from the above quotation, it may

be said that, in a given context, J.S. Bach could fathom immediately the combination

of alteration operations possible to use on the information units, and how these

combinations could be used in counterpoint. Indeed, Bach’s mastery of tonal

counterpoint cannot be distinguished from his mastery of alteration operations.

Though sequential information analysis focuses on identifying repetitive

information sequences, its application is not limited to analyzing overtly repetitive

music. At times the structural primacy of the alteration operations obscures the

existence of covert sequences,7 but sequential information analysis is particularly

useful in exposing covert sequences. The analysis may then serve to help the ear

discover information patterns which were not immediately evident with a purely

aural investigation.

6. Wolff 1998, 397.

7. Hanninen 2003. Her discussion of recontextualization depends in large measure upon the subjective response of the listener. Her article is an attempt to explain how and why we can hear repetitive musical things differently, even if they are repeated continuously in the context of the work. In phenomenology, the analysis of the music cannot be separated from the experience of listening to it. Sequential information analysis, however, is concerned with the the patterns of organization of musical information, not the experience of listening to it.

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1-8. Information and Sequence

Nearly any kind of music information can be sequenced. My discussion will

reveal repetitive sequencing of rhythmic cells, motivic shapes, harmonic motion, tonal

designs, rhythmic structures, textures, and ordered pitch collections. It also leaves

room for further exploration.

Different musics favor sequencing of different types of information, and present

different contextual interrelationships between the units of information. In addition,

different musics place the operation of these sequences on different hierarchical levels

within the form and content of the work. When musics tend to feature similar types

of sequential repetition of information at similar hierarchical levels, they can be

related and put into categories. For example, J.S. Bach’s Goldberg Variations,

Passacaglia in C minor, and Chaconne in D minor all fit into the category of pieces that

are formed through repetitive sequencing of a tonal design. There is a rich historical

precedent for works in this category. This category of music construction became one

of the fundamental twentieth-century forms with the advent of the twelve-bar blues.

Sequential information analysis in fact shows a link between works built upon the

blues sequence and the aforementioned works by Bach, placing them both in the

category of repetitive sequencing of tonal design.8

1-9. Structural Hierarchies of Sequences

As already discussed, sequence in music involves a specific operation performed

on a reiterated information unit. The sequence generates numerous versions of an

information unit from a single template. Information is organized according to how

8. Much of the fun of sequential information analysis comes from discovering how different composers at different times and in different places all created their music with such similar procedures of information management. Engaging in this pastime serves to create a richer appreciation of music in its historical context.

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the generated information units are related to, and differ from, the original template.

Often the repetitive sequencing of an information unit functions in conjunction with

other music information which is not being sequenced. The repetitive sequence then

becomes a stable vehicle for conveying the other non-repeating information which

more directly advances the musical form. This is common where repetitive sequences

provide rhythmic and textural vitality to a harmonic progression, or where a cyclical

harmonic progression becomes the structure for improvisation.

In order to establish the hierarchical position of the repetitive sequence, we must

locate it within a dynamic foreground/background structural information field.

“Foregound” and “background” here only partially follow their reductive associations.

Following reductive notions of background, the less structural information that a

sequence can account for, the further in the background it is typically placed. For this

reason, the repetitive tonal sequence in a twelve-bar blues or in the Bach works listed

above would be classified as a background structure because the sequence itself

cannot adequately explain much of the surface information like rhythm, melody,

texture, and register. What the repetitive tonal sequence provides in these instances is

an architectural grid upon which other patterns of information may operate.9

Structural hierarchies are used both in the organization of repetitive sequences

and in the organization of information loops. It will be shown that information

repetition often results in an organizational hierarchy of sonic space where repetitive

structures tend to move towards the background, and non-repetitive information

tends to play a more significant role in advancing a work’s form. In these cases,

engagement with the motion and change of non-repeating information is enhanced by

its articulation over the background repetitive sequence.

9. Architectural grid, foreground, and background, will be elaborated upon in the discussion of the artifice of reference in Chapter 3.

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1-10. Case Study No. 1: Beethoven Symphony No. 6 mvt. 1, development

The next example will present a sequential information analysis which engages

numerous information fields. The first 103 measures of the development section from

the first movement of Beethoven’s Symphony No. 6 is excellent for introducing the

sequential information diagram.10 This development section is generated with just a

few information units, and displays overtly repetitive features which are easy to

recognize. As was previously mentioned, a motive in a work by Beethoven will

typically conflate the melodic and rhythmic information fields, and such is the case in

Symphony No. 6. For this reason, I have chosen to label these synchronized

information units as melodic information units, with the understanding that each

melodic information unit has a particular and consistent rhythmic design. When a

melodic information unit alters its characteristic rhythm, it is labeled as a variant of

the melodic information unit.

Creating a sequential information diagram often requires several stages of

information reduction, and decisions must be made as to which information the

diagram will represent, and which it will exclude. This case study will present a two-

stage information reduction process. The first stage will present information in the

melodic (and, by association, rhythmic), textural, and harmonic information fields.

We will also see an example of a sectional information unit. The first-stage reduction

will include other information such as specific pitch and register content. The second

stage of the reduction will remove the harmonic information, which may be gleaned

by comparing the diagram to the first reduction.

Before creating a sequential information diagram, we must recognize and label

the information units that the diagram will feature. As it turns out, the three melodic

information units employed in the first 103 measures of the development section are

10. The remainder of the development section will be discussed in Chapter 2.

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all presented in measures 1-4 of the movement:

Example 1. Beethoven Symphony No. 6 mvt. 1, mm. 1-4

In the context of the development section, these four bars may be subdivided

into three melodic information units, each with a characteristic rhythmic design. They

are labeled accordingly MIUa, MIUb, and MIUc. In the development section,

Beethoven presents a variant of MIUb. This variant deconstructs the five-note

information unit and employs only the last two notes, expressed in the development

section as an interval of the descending fourth.11 The first variant will be labeled

MIUb2, and with it Beethoven generates two additional variants, which will be

labeled MIUb2.1 and MIUb2.2.

It should be emphasized that these information units are labeled according to

their use in the development section of this movement. The phrasing of bars 1-4

could be reinterpreted so that MIUa encompasses the first note of bar 2. In addition,

the last note of bar 2 might be viewed as a pickup to MIUc. Indeed, this phrasing of

MIUa might be valid in bars 143-146. The use of either interpretation does not take

away from the fact that an information unit is repeated the specified number of times.

11. One of the transpositions of MIUb in the development produces a descending fourth in the last two notes of the information unit. In this context, it is clear how the descending fourth derives from MIUb.

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In any case, the fragmentation of bars 1-4 in the development section encourages a

bar-by-bar interpretation of the information units.12

The next example is a chart showing all of the melodic and textural information

units employed in the first 104 measures of the development. The textural

information units (TIU) are notated to synchronize with the melodic field. TIUb, a

tremelo-like figure, is measured to the bar as a convenience. It is possible to define

this unit in smaller, or larger, groupings.

Following the chart is a presentation of the first reduction of measures 139-242.

The information units labeled are those which are to be used in the second reduction.

A number with an ‘x’ (e.g. 2x) over the bar lines indicates how many times the next

section is to be repeated.

12. The fact that the start and end points of the melodic information units can be reinterpreted according to their context is not unusual. The identification of the MIUs is crucial for sequential information analysis, but the specifics of their labeling is often a convenience.

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Example 2. Beethoven Symphony No. 6 mvt. 1, mm. 139-242melodic and textural information units

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Example 3. Beethoven Symphony No. 6 mvt. 1, mm. 139-242first sequential information reduction

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Example 3. continued

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Example 3. continued

In this reduction, there are certain informational features which have been

ignored, the most obvious being instrumentation. Nevertheless, we are able to get a

good sense of the information content from this passage. Simply with the

implementation of repeat signs, 104 measures have been reduced to 47 measures,

without a loss of melodic, rhythmic, textural, registral, or harmonic information.

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Taken out of context, this passage is in some respects more reminiscent of Philip

Glass than Ludwig van Beethoven.13 In addition to the excessive looping of the

melodic material, certain sections in this example display considerable harmonic stasis.

A B-flat major triad is sustained for twelve bars (mm. 151-162). This is followed by

twenty-eight bars of a sustained D major triad (mm. 163-190). After an interruption

from a contrasting information unit in m. 191, this pattern is repeated at measure 197

with twelve bars of a G major triad (mm. 197-208) followed by twenty-eight bars of

a sustained E major triad (mm. 209-236).

Example 4. Beethoven Symph. No. 6 mvt. 1, mm. 151-236,harmonic rhythm

This diagram reveals several repetitive information units. An information unit

that is continuously repeated with no alteration is the major triad. This is an

underlying information loop in this passage, and it occurs in the harmonic field. Inside

that loop we notice other repetitive features, such as the durational proportions

between the two sections. We can explain the sustained triadic harmonies as another

variety of harmonic information loop, since harmonic stasis here is achieved by

continuously reiterating a single harmonic information unit without alteration.

13. This could be made even more apparent if, as a fun exercise, the number of times each bar repeats were doubled. An exercise such as this illuminates the relationship of repetitive tendencies between two composers, Beethoven and Glass, who are not typically thought to have much in common.

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Some repetition is more latent. The motion between harmonies 1-2 and 3-4 is

inverted, not repeated. First a major third up, then a minor third down. But this

motion is merely an alteration operation on the diatonic third information unit, which

is itself repeated. Other non-repeated information depends upon tonal convention.

Each harmonic unit has its own special tonal relationship to the tonic harmony, and

to each other. In fact, the harmonic motion – B-flat to D, and G to E – absolutely

thwarts conventional expectations of tonal progression, even though it is joined

together with an authentic cadence. It is a delightful harmonic progression, and it is

the only non-repeating information in this passage. Besides this harmonic progression,

the entire information organization of this passage consists of sequentially repeated

information units.

Using our chart of melodic and textural information units, as well as our first

sequential reduction, we will now create a sequential information diagram. There can

be flexibility in fashioning sequential information diagrams. For this example, I have

chosen to maintain a basic representation of the registral location of the information

units. This registral representation is analogous to the treble and bass staves in the

first-stage information reduction. The diagram itself reads from left to right and top

to bottom, like music notation. Barlines are maintained for structural clarity, and

repeat signs are used as a repetitive shorthand, just as they are in the first-stage

reduction.

The largest information reduction between the first and second stages of the

analysis involves the conflation of measures 151-196 and 197-242 into a single

repeated sectional information unit (labeled SIUa). The harmonic relationships in this

passage have been remarked upon, and these sectional information units are otherwise

congruent in their organization of melodic, textural, and registral information.

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Example 5. Beethoven Symphony No. 6 mvt. 1, mm. 139-242sequential information diagram

It should be reemphasized at this point that the sequential information diagram is

not a representation of the music, but rather is a representation of the organization of

the music information in this passage of the work. In this sequential information

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diagram, we have 104 measures of music represented as 15 measures of organized

music information.

There is one more repetitive feature which has not yet been remarked upon in

this example. One which may be treated as a subset of the sectional information unit.

The dynamic shape of the sectional information unit (mm. 151-196) is indicated in

the first-stage reduction (Example 3). In this section, the dynamics gradually increase

in loudness from piano to fortissimo (mm.151-175). The fortissimo is emphasized with

TIUb, which in the score is presented as an orchestral tutti. In mm. 183-187, there is a

diminuendo back to piano, and the remainder of the SIU is soft. This dynamic process is

repeated in the second statement of the sectional information unit (mm. 197-242),

revealing a synchronization of information fields.

Repetitive sequencing facilitates the creation of sizable portions of music with a

small number of information units, thus allowing us to focus on the non-repeating

information. Leonard Meyer has suggested that “the extensive use of a single motive

in a development section does not . . . serve primarily to actualize the ‘full potential’

of the motive. Rather motivic redundancy allows the listener to attend to the increase

in harmonic information that accompanies the modulations characteristic of

development sections.”14 In this development section, we have seen significant

harmonic redundancy as well, and very little non-repeating information in general.

Sequential information analysis reveals how the music in this example is derived from

looping a small amount of music information. One wonders why we continue to call

this the “development” section of the sonata form. The designation of “repetition”

section might be a more appropriate label.

14. Meyer 2000, 292.

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