Structural Scaling in Bachs-libre

Fractals, Vol. 15, No. 1 (2007) 89–95c© World Scientific Publishing Company

STRUCTURAL SCALING IN BACH’S

CELLO SUITE NO. 3

HARLAN J. BROTHERSDirector of Technology, The Country School

341 Opening Hill Road

Madison, CT 06443, USA

[email protected]

[email protected]

Received April 30, 2006Accepted May 24, 2006

Abstract

The Bourree Part I from Johann Sebastian Bach’s Cello Suite No. 3 provides a clear exampleof structural scaling. The recursive form of this structure can be visualized in the manner of awell known fractal construction — the Cantor set.

Keywords : Fractal; Music; Scaling; Structure; Power Law; Recursion; Bach; Cantor.

1. INTRODUCTION

Music is a profoundly rich and often complicatedform of artistic expression. In contrast to dancing,writing, and the visual arts, music is far less tan-gible. Its ephemeral form is fundamentally psycho-logical in nature.

In a purely physical sense, music is a serial phe-nomenon; a longitudinal sound wave that unfoldsone moment at a time. Psychologically, however,

it possesses what we perceive as depth. That isto say, polyphonic music is heard as the simul-

taneous propagation of a collection of individualsounds that are layered or intertwined. The precisemechanism for how we process, understand, and

appreciate music is not well-understood. Nonethe-less, it seems that the truth of its mysterious powerto emotionally move and inspire us is universally

accepted.

89

90 H. J. Brothers

The fact that musical structure takes shape inthe mind presents unique challenges with regard toassessing the existence, manner, and extent of frac-tal scaling in a given composition. However, the ideathat we might naturally compose in some inherentlyfractal fashion should not be surprising; we are, in afundamental sense, fractal beings living in a fractaluniverse.1,2 Not only are we physiologically fractalwith regard to our nervous, vascular, and respira-tory systems, but expanding research shows grow-ing evidence of underlying fractal structure in ourcreative and artistic endeavors.1

The earliest attempt to measure fractal-relatedscaling in music was carried out by Richard Vossand John Clarke while Voss was a graduate studentat the University of California.3 They found strongindications of 1/f scaling in the distribution ofboth program loudness and the set of zero-crossingsfor the audio signals of several different genres ofmusic.

The idea that music possessed this type of inher-ent power law relation led Voss, and others, toexplore the algorithmic generation of compositionsusing 1/f scaling as a guideline.4,5 It turns outthat music composed in this fashion sounds rela-tively pleasing to most listeners. By contrast, musicgenerated using 1/f0 scaling (associated with white

noise) sounds too random, while music derived from1/f2 scaling (associated with brown noise) soundstoo monotonous. Based on these findings, Gardnerlater wrote a column in Scientific American on frac-tals, scaling noise, and algorithmic composition.6

In discussing music’s requisite mixture of surpriseand expectation, he characterized the significanceof Voss’ and Clarke’s work by saying “...what Vosshas done is to suggest a mathematical measure ofthe mixture.”

Due to the circumstances of their discovery, Vossand Clarke used audio signals as the basis for theiranalyses. However, given the highly complicatednature of musical sound, this method has limita-tions; it is currently problematic to electronicallydiscern and quantify specific musical characteristicsdirectly from audio data.

An alternative approach to ascertaining andassessing the fractal properties of music is to use

a pre-audio representation of a composition. Thiscan be thought of as a way of measuring, in apure sense, what the composer conceived, inde-pendently of a particular orchestration, the per-formance of the musicians, or the acoustic quali-ties of the instruments. The representation can takethe form of either a written score or its electronicanalog, a MIDI data file.a Accurately implementedMIDI representations are ideal for examining dis-tributions with respect to pitch, interval, and noteduration.

As with all scientific inquiry, the appropriatemethodology for measurement and analysis is bestdetermined by a well-informed understanding of thephenomenon being studied.

2. FRACTAL SCALING IN MUSIC

There are many ways in which fractal structure canmanifest itself in music. The four most fundamentalof these can be categorized as follows:

(1) Motivic scaling has been practiced since atleast the late 14th century with the develop-ment of the mensuration or prolation canon(Ref. 7, pp. 197–198).b This type of canon ischaracterized by a melody or rhythmic motifthat is repeated in different voices simultane-ously at different tempos. Modern composerssuch as Martin Bresnick, Gyorgy Ligeti, Con-lon Nancarrow, and Arvo Part have developedsophisticated compositional styles that oftenincorporate forms of motivic scaling. It is easyto explore this approach to composition usingcommercially available MIDI sequencing soft-ware or algorithmic music programs.8

(2) Duration scaling requires that, given a hetero-geneous distribution of values, the overall sta-tistical distribution of note durations satisfies apower law relation. A composition that exhibitsmotivic scaling will also exhibit duration scal-ing if its primary voice consists of isochronalpitches.

(3) Pitch-related scaling can take several differentforms. While a detailed account is beyond the

aMIDI is an acronym for Musical Instrument Digital Interface. It is a communications protocol that encodes information aboutcharacteristics like timing, pitch, volume, and instrument selection. This data is then interpreted by a dedicated synthesizeror computer sound card and played live in much the same fashion as a scroll-driven player piano performs.bLe Ray Au Soleyl, composed by Johannes Ciconia (ca. 1335–1411) is one of the earliest known examples of a three partmensuration canon.

Structural Scaling in Bach’s Cello Suite No. 3 91

scope of this article, the first clearly success-ful pitch-related analysis of music was devel-oped by the father and son team, Kenneth andAndrew Hsu, who found evidence of intervalscaling in the works of Bach, Mozart, and acollection of Swiss folk songs.9 As in the caseof duration scaling, it is the statistical distri-bution of elements that satisfies a power lawrelation.

(4) Structural scaling refers to large scale patternsin characteristics such as song form or dynam-ics. These patterns display a nested or recur-sive form.

It is important to note that, regardless of thetype of scaling under consideration, in order to ful-fill a power law relation, any inherent pattern ina group of musical elements requires the presenceof a minimum of three distinct levels of scaling.10

This requirement reflects the fact that the log-logplot of a power law relation appears linear; at leastthree data points are needed to assert a linearrelationship. For example, a mensural canon withonly two unique voices cannot be said to possess apower law relation and hence cannot be consideredfractal.

3. BACH’S CELLO SUITE NO. 3

Johann Sebastian Bach is widely acknowledged forhis exceptional skill in combining aesthetic sensibil-ity with mathematically derived musical transforms(Ref. 7, pp. 446–449).11 Although his works werenot, in all likelihood, “algorithmic” in the modern,recursive sense of the word, the structural scalingfound in the first Bourree from Cello Suite No. 3suggests an innately hierarchical construction. Thepattern of this scaling satisfies an easily measuredpower law relation and can be viewed from a fractalperspective.

Based on the written score, this paper will exam-ine the structure of first section of the first Bourreewith respect to phrasing. Musical phrasing is analo-gous to linguistic phrasing in that it deals with howcertain sequences of notes are “naturally” associ-ated with each other.

This particular piece of music can be decomposedinto phrases that occur in a pattern of the form

AAB, wherein the duration of each A section is“short” relative to each B section, which lasts twiceas long. Figure 1 shows the first 16 measures of thispiece (eight measures repeated) along with horizon-tal braces that mark phrase groupings.c

The first beat of music is a phrase, m1, thatbegins with a double eighth note pickup followedby a quarter note (see Appendix for a list of termsalong with relevant note durations and their sym-bols). Phrase m1 forms the pattern:

Short (1 quarter beat) Short (1 quarter beat)Long (1 half beat).

The same can be said for the next three-notephrase, m2. Figure 2 shows a graphical represen-tation of these “Level 1” phrases. In the followinggraphs, blue notes constitute short elements, rednotes constitute long elements, and gray notes areof no concern. The x-axis is time and the y-axis ispitch.

Referring back to Fig. 1, the phrases m1 and m2,are each one-beat tonic phrases leading to the two-beat dominant phrase, m3. Together, m1, m2, andm3 form the antecedent, s1, which consists of thepattern.

Short (1 beat) Short (1 beat) Long (2 beats).

The last half beat of measure 2 begins with a dia-tonic transposition of m1, m1↓, which is followedby a diatonic transposition of m2, m2↓.d Thesetwo one-beat dominant phrases (the second pair ofthree-note phrases in Fig. 2) are followed by thetwo-beat tonic variation of m3, m3v↓, which com-pletes the consequent, s2. Figure 3 shows a graphi-cal representation of these “Level 2” phrases.

The first eight beats of music can therefore bedivided into two four-beat subjects, s1 and s2.These directly precede a long, eight-beat continua-tion, s3. Taking s1, s2, and s3 together, once again,we find the pattern:

Short (4 beats) Short (4 beats) Long (8 beats).

Figure 4 shows a graphical representation of these“Level 3” phrases.

What we find then is a nested series of scaledphrases of the same form, AAB. This pattern, infact, continues to the extent that the first eight mea-sures repeat, giving us two “short” sections thatare then followed by an extended and harmonically

cAn excerpt from a recording by Pablo Casals on the Naxos Historical label (CD 8110915-16) can be heard at http://www.brotherstechnology.com/math/fractals-supp.html.dIn music analysis, the symbol “↓” and “v” respectively denote “download transposition” and “variation.”

92 H. J. Brothers

Fig. 1 Analysis of the first 16 measures of the Bourree from Suite No. 3.

Fig. 2 A graphical representation of the Level 1 phrases for the first eight measures (with pickup). Blue notes are membersof “short” groups and red notes are members of “long” groups.

Fig. 3 Graph of the Level 2 phrases.

Fig. 4 Graph of the Level 3 phrases.


different 20 measure, “long” section. Interestingly,although Bach wrote the piece with a repeat symbolat the end of this 20-measure section, anecdotal evi-dence suggests that some cellists choose to performit without this second repeat.12 Performed in thisfashion, the Bourree Part I exhibits a full four levelsof structural scaling symmetry.

4. VISUALIZING THE

STRUCTURE OF THE

COMPOSITION

The Cantor set is a classic fractal that can beconstructed by beginning with a line segment andsuccessively removing the middle third of eachremaining segment.13 In the limit, it has zerolength. Because the substance of the line disappearsrapidly, its structure is often illustrated by reverse-stacking successive iterations of the set, beginningwith the full line segment. The combined iterationsthen take the form of what is sometimes referred toas a “Cantor comb” (see Fig. 5).e

The Cantor set can be modeled using L-systems,a formal set of rules and symbols that can be usedto generate a wide variety of fractal constructions.14

Specifically, it is produced by the substitution mapA→ABA, B→BBB. In this light, it is easy tosee that the A and B sections do not necessar-ily have to correspond to the classic “line, gap”sequence. Indeed, depending on the field of inter-est, one could assign an unlimited number of related

characteristic pairs such as “red, blue,” “loud, soft,”or most broadly, “same, different.”f

In this composition, the hierarchical nesting ofthe AAB pattern (short, short, long), can be rep-resented in the form of a Cantor map wherein eachA section is, itself, composed of the pattern AAB.Figure 6 shows a map of the first 16 measures of thepiece.

Here, the red regions each represent four beats(two measures); the blue regions, 1 beat; and theyellow regions, quarter beats (eighth notes). Thewhite regions represent the everything that is notpart of a short phrase at each scale of measurement.

Visualizing the composition in this fashion helpsillustrate not only the overall structure of the piece,but also the relative frequencies of its constituentelements. In this case, it is easy to see that, acrosslevels, as the length of the short phrase increasesby a factor of 4, its frequency decreases by a factorof 1/2.

Figure 7 shows a log-log plot of the count, N ,against the inverse of the size, s (as measured inbeats), for the short phrases at a given level of scal-ing. The straight line that results confirms the exis-tence of a power law relation. The slope, m, of aline generated in this fashion can be interpreted asthe dimension of the object being measured. In thiscase m = 1/2. It is important to emphasize, how-ever, that this particular value has relevance only inrelation to the characteristic of phrase structure; itshould not be construed as an overall fractal mea-sure for the composition.

Fig. 5 “Cantor comb.”

Fig. 6 This Cantor map of the first 16 measures illustrates the nested structure of the phrasing.

eThe term “Cantor comb” was originally used to describe the product of the Cantor set with an interval. Its usage hassubsequently broadened to include comb-like objects generated by a Cantor process.fExamples of musical compositions based on substitution maps can be heard at http://www.brotherstechnology.com/math/fractals-supp.html.

94 H. J. Brothers

Fig. 7 Log-log plot of the “short phrase” count against theinverse of their respective sizes as measured in beats.

5. CONCLUSION

Benoit Mandelbrot’s seminal term fractal becamepopular with his publication in 1982 of The Frac-

tal Geometry of Nature.g It did not take longbefore composers such as Gyorgy Ligeti and CharlesWuorinen began to describe their work in relationto fractal structure.15−17

The fact that Bach was born almost three cen-turies before the formal concept of fractals cameinto existence may well indicate that an intuitiveaffinity for fractal structure is, at least for somecomposers, an inherent motivational element in thecompositional process. Six centuries of explorationand development of the mensural canon argue infavor of this possibility. Further support is sug-gested by the numerous examples fractal structurein other areas of artistic endeavor; scaling symmetryarises in art, architecture, and literature.1

Fractal analysis of music is in its infancy. It is cer-tainly possible, given Bach’s superlative skill at pat-tern manipulation and his prodigious body of work,that the clear, if simple example presented here rep-resents a rare or even unique case of phrase-relatedscaling in the pre-Mandelbrot era. Nonetheless, it ishoped that the simple fact of its existence will helppromote research that will reveal further examplesof scaling symmetry in the rich and vast body ofmusical expression.

ACKNOWLEDGMENTS

The author would like to thank Silas Meredithof the Hopkins School for identifying the musical

subject of this article. Thanks also to Eric Nathanfor his help with formal analysis and to MatthewPickett for pointing the author to the work of Mar-tin Bresnick. Professor Bresnick was, in turn, verygenerous with his time and expertise. The authorwould like to express deep appreciation to BenoitMandelbrot of Yale University for his fundamentalsupport of this research.

REFERENCES

1. M. L. Frame and B. B. Mandelbrot, Fractal Geome-

try: A Panorama of Fractals and Their Uses, http://classes.yale.edu/Fractals/Panorama/welcome.html.

2. B. B. Mandelbrot, The Fractal Geometry of Nature

(W. H. Freeman and Co., New York, 1982).3. R. F. Voss and J. Clarke, 1/f noise in music and

speech, Nature 258 (1975) 317–318.4. R. F. Voss and J. Clarke, 1/f noise in music: music

from 1/f noise, J. Acoust. Soc. Am. 63(1) (1978),258–263.

5. R. F. Voss, Random fractals: self-affinity in noise,music, mountains, and clouds, Physica D 38 (1989)362–371.

6. M. Gardner, Mathematical games: white and brownmusic, fractal curves and one-over-f fluctuations,Sci. Am. 4 (1978) 16–32.

7. D. J. Grout and C. V. Palisca, A History of Western

Music, 7th edn. (W. W. Norton & Co, Inc., 2005).8. M. L. Frame, G. Booth and H. J. Brothers, Fractal

Music Composer, http://gingerbooth.com/course-ware/fracmusic.html.

9. K. Hsu and A. Hsu, Fractal geometry of music, Proc.

Natl. Acad. Sci. USA 87 (1990) 938–941.10. M. L. Frame and H. J. Brothers, Fractal Geom-

etry: Labs, Fractal Music, http://classes.yale.edu/fractals/labs/fractalmusiclab/FracMusicBground/Frac.html.

11. D. R. Hofstadter, Godel, Escher, Bach: An Eternal

Golden Braid (Basic Books, 1980).12. T. Janof, Conversation with Anner Bylsma, Tutti

Celli, Newsletter of the Internet Cello Society, 4(6)(1998).

13. E. W. Weisstein et al., Cantor set, from Math-

World — A Wolfram Web Resource, http://math-world.wolfram.com/CantorSet.html.

14. E. W. Weisstein, Lindenmayer system, from Math-

World — A Wolfram Web Resource, http://math-world.wolfram.com/LindenmayerSystem.html.

15. J. Rockwell, Laurels at an auspicious time forGyorgy Ligeti, The New York Times, Cultural Desk

(11 November 1986).

gThe earliest published reference to the word fractal can be found in Mandelbrot’s 1975 book entitled Les Objets Fractals:

Forme, Hasard et Dimension (Flammarion, Paris).


16. D. J. Soria, Gyorgy Ligeti, Musical Am. 107(4)(September 1987) 12, 14, 15 and 27.

17. C. Wuorinen, Biography, http://www.charleswuori-nen.com/documents/bio.cw.doc.

Appendix

A.1. Glossary of Musical Terms

Antecedent: The first of a pair of musical state-ments that complement each other in rhythmicsymmetry and harmonic balance. As with a rhymedcouplet in poetic verse, the dynamics set in motionduring the first section are completed in the secondsection.

Beat: The underlying pulse of a composition. Thisis often indicated by the movement of the handsor baton of the conductor or by the click of ametronome.

Consequent: The second of a pair of musical state-ments that complement each other in rhythmicsymmetry and harmonic balance. (See Antecedent

above.)

Continuation: A melodic development of a subjectthat extends an anticipated resolution of harmony.(See Subject below.)

Diatonic: The notes that occur naturally in a scalewithout being modified are said to be diatonic.

Dominant: The fifth tone of a major or minorscale. A chord built on the fifth tone of a major scale(and certain minor scales) is a dominant chord. Adominant chord tends to engender a sense of antic-ipation that is resolved by returning to the tonic

chord. (See Tonic below.)

Measure: A metrical division of music marked offby vertical lines (bar lines). Each measure containsa fixed number of beats. (See Beat above.)

Pickup: A note or short group of notes that pre-cede the first strong metrical beat of a composition.

Subject: A melody, motif, or theme upon which acomposition is based.

Tonic: The note upon which a scale or key is based;the first note of a scale or key. A chord built on thisnote is the tonic chord.

Transposition: The process of changing the key ofa composition. A diatonic transposition uniformlyshifts a group of notes in a given scale to othernotes in the same scale by a specified number ofscale steps. (See Diatonic above.)

A.2. Duration Values and TheirSymbols

The musical excerpt under consideration consistsof quarter notes and eighth notes. A quarter notelasts twice as long as an eighth note. Each measureof this composition contains the equivalent of fourquarters notes (or 8 eighth notes) worth of music.

1 quarter

2 eighths 4 eighths

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Structural Scaling in Bachs-libre