a historical and fractal perspective on the life and saxophone solos of john coltrane

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A Historical and Fractal Perspective on

the Life and Saxophone Solos of John

ColtraneChristine Charyton , John G. Holden , Richard J. Jagacinski &

John O. ElliottPublished online: 30 Jul 2013.

To cite this article:Christine Charyton , John G. Holden , Richard J. Jagacinski & John O. Elliott(2012) A Historical and Fractal Perspective on the Life and Saxophone Solos of John Coltrane, Jazz

Perspectives, 6:3, 311-335, DOI: 10.1080/17494060.2013.806031

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A Historical and Fractal Perspective on

the Life and Saxophone Solos of JohnColtrane

Christine Charyton, John G. Holden, Richard J. Jagacinskiand John O. Elliott

John Coltrane was a dedicated student of many disciplines beyond music religion,

astrology, astronomy, and other sciences. The books he left behind more than suggestit: he was denitely into mathematics and an esoteric application of numbers tomusic. So I thought about those cells as pure numbers and saw how they deneratios known as the Golden Mean, also called the divine proportion. These ratiosare found in proportions of the human body and in nature: in seashells, whentrees begin branching. Its also an established theory of aesthetic perfection: howbuildings or portraits are arranged, or when events occur in Mozart sonatas RaviColtrane, John Coltranes son, in liner notes to A Love Supreme, Deluxe Edition , 2002.1

John Coltranes approach to style remains elusive, even though the saxophonist is the

subject of thousands of books and articles, and his solos have been rendered in over 700

transcriptions by one disciple alone, Andrew White.2

Yet, the comment by Coltranesson Ravi may provide a richer understanding of Coltranes technique. The established

theory of aesthetic perfection Ravi Coltrane points to in the liner note comment is

related to the mathematics of fractal geometry. Fractal objects (Benoit Mandelbrot

coined the term fractal in 19753) are comprised of smaller, nested copies of the

whole object. These kinds of structures turn up in a surprising range of natural

objects and processes such as fern leaves, the silhouettes of mountainous landscapes,

and the boundaries of cumulus clouds. Relationships between nature, mathematics,

and beauty have fascinated theorists since antiquity; Aristotle, Euclid, Sir Isaac

Newton, and Edwin Abbott, for example, were interested in the philosophy of math-ematics, geometry, science, nature and aesthetics.4 Beyond understanding spatial

1Ravi Coltrane (liner notes) in John Coltrane, A Love Supreme: Deluxe Edition, Universal Distribution/Impulse!

Records 5899452 (2002 [1964]): 2324.2One catalog (Libraries Worldwide) lists 2610 books and articles on Coltrane, whether historical/biographical or

analytical. SeeThe Andrew White Comprehensive Catalogue of over 2000 Self-Produced Products,September 23, 2005

2006 Edition(Washington, DC: Andrews Music, 2006).3Benoit Mandlebrot, Stochastic Models for the Earths Relief, the Shape and the Fractal Dimension of the Coast-

lines, and the Number-Area Rule for Islands,Proceedings of the National Academy of the Sciences, vol. 72, no. 10

(1975), 38253828,4Edwin Abbott,Flatland: A Romance of Many Dimensions(New York: Dover Publications, 1992 [1884]); Aristotle,

Aristotles Psychology, trans. by Edwin Wallace (London: Cambridge University Press, 1882); Euclid (trans. and ed.

Jazz Perspectives, 2012Vol. 6, No. 3, 311335, http://dx.doi.org/10.1080/17494060.2013.806031

# 2012 Taylor & Francis


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structures (the commonplace application of geometry) fractal geometry can be applied

to patterns that unfold in time. Application of fractal geometry to temporal sequences

has, itself, a long intellectual pedigree. Jean Baptiste Fourier, Robert Brown and Benoit

Mandelbrot all shared an interest in identifying patterns of motion.5 Whether describ-

ing the fractal structure of a solid object or a pattern developing over time, the pattern

most commonly referred to is 1/f scaling (pronounced one-over-ef) or more pre-

cisely 1/f , where is a scaling exponent. Similarly, temporal fractal patterns can be

perceived as patterns of occurrence over time that can be revealed by applying the ana-

lytic techniques developed by Fourier.

Turning to a selection of John Coltranes solos, the fractal structure we describe is a

characteristic of the melody that unfolds in each piece. Our study concerns relation-

ships across the successive notes or pitches in the pieces. The patterns we identify in

the sequences successive pitches illustrate fractal structure across time. We demon-

strate that self-similar fractal patterns are apparent in the temporal pattern of pitches

in Coltranes solo pieces.

Fractal Analysis of Coltranes Solo Pieces

Several quantitative and descriptive studies of Coltranes music have been reported.6

However, the present analysis is the rst to apply methods rooted in fractal geometry

to Coltranes solo saxophone performances in order to better understand the complex-

ity of his technique. There were two main goals of our fractal analysis. First, we wanted

to examine how fractal geometry could provide a more accurate and quantitative analy-

sis of Coltranes style. We wanted to potentially offer another analysis tool for musi-cians since our mathematical analysis has strong potential for being less culturally

biased by Western traditions. Second, we also sought to evaluate Birkhoffs Theory

of Aesthetic Value (1933) in relation to quantitative fractal analyses of other classical

and jazz performances and compare these analyses to our investigation of the impro-

vised solos of one specic performer, namely, John Coltrane. Previously, 1/f analyses

by Robert Simson, M. D.),The Elements of Euclid, the First Six Books Together with the 11th and 12th, the Errors, by

Which Heon, or Others, Have Long Ago Vitiated These Books, Are Corrected, and Some of Euclids Demonstrations Are

Restored. Also, the Book of Euclids Data in the Like Manner Corrected, (Philadelphia: Homas and George Palmer for

Conrad and Co., 1803); Sir Isaac Newton, (trans. by Mr. Raphson, and rev. and corrected by Mr. Cunn), Universal

Arithmetick: Or, a Treatise of Arithmetical Composition and Resolution. To Which Is Added, Dr. Halleys Method ofFinding the Roots of Equations Arithmetically(London: J. Senex; W. Taylor, T. Warner, and J. Osborn, 1720).5Jean Baptiste Joseph Fourier, The Analytical Theory of Heat(Cambridge: Cambridge University Press, 1878);

Robert Brown, A Brief Account of Microscopical Observations Made in the Months of June, July and August, 1827,

on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and

Inorganic Bodies (The Philosophical Magazine and Annals of Philosophy [New Series], Sept., 1828. [reprint;

orig. pamphlet self-published, 1827]), Taylor & Francis online, http://www.tandfonline.com/doi/abs/10.1080/

14786442808674769, accessed 25 April 2013; Benoit B. Mandelbrot, Multifractals and 1/f noise (New York:

Springer-Verlag, 1998).6For examples of quantitative analysis of Coltranes music, see Paul Aitken, Unity and Form in Miles Davis Blue

in Green, McMaster [University] Music Analysis Colloquium, vol. 4 (2005): 111; Jeff Bair, Cyclic Patterns in

John Coltranes Melodic Vocabulary as Inuenced by Nicolas SlonimskysThesaurus of Scales and Melodic Patterns:

An Analysis of Selected Improvisations,

DMA diss. (University of North Texas, 2003); Walt Weiskopf and RamonRicker,Coltrane: A Players Guide To His Harmony(New Albany, IN: J. Aebersold, 1991).

312 Fractal Analysis of Coltranes Solos
http://www.tandfonline.com/doi/abs/10.1080/14786442808674769http://www.tandfonline.com/doi/abs/10.1080/14786442808674769http://www.tandfonline.com/doi/abs/10.1080/14786442808674769http://www.tandfonline.com/doi/abs/10.1080/14786442808674769


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can be generated by successive tosses of 1 die with 120 sides8. The resulting series is

random and irregular.

A related pattern is referred to as Brownian motion. It is a stochastic process, or a time

series which has independent increments and demonstrates self-similarity.9 Brownian

motion was named after Scottish botanist Robert Brown. In 1827, Brown described

the structure of plants from various locations around the world and sketched their aes-

thetic beauty.10 The term, Brownian motion was later coined in honor of his discovery.

During June, July and August of that year, Brown observed that small particles of pollen

from plants decomposed in an aqueous solution, visible by microscope, had an irregular

motion.11 Browns examination of the Molecules from the grains of pollen adhering to

the stigma particularly inAntirrhinum majus,led to his discovery (p. 479).

Their motion consisting not only of a change of place in the uid, manifested byalterations in their relative positions, but also not unfrequently of a change ofform in the particle itself; a contraction or curvature taking place repeatedly about

the middle of one side, accompanied by a corresponding swelling or convexity onthe opposite side of the particle. In a few instances, the particle was seen to turnon its longer axis. These motions were such as to satisfy me, after frequently repeatedobservation, that they arose neither from currents in the uid, but belonged to theparticle itself.12

This observation led to quantitative data13 and further scientic understanding in

applied mathematics, physics, engineering and other sciences. Brownian motion or

Brown noise can be generated by a series from 1 die with 3 sides (+1, 0, 1), in

which each new toss of the die is added to the sum of the previous tosses14. The incre-

ments are thus independent and random. However, the overall series is more predict-

able than white noise, with less opportunity for randomization and large sudden

changes. Brown noise has a spectral slope of2 (scaling exponent = 2, power pro-

portional to 1/f2), and corresponds to a higher degree of consistency of structure

from pitch to pitch. Successive pitch values are only small random increments or decre-

ments away from previously played notes. The reason such a sequence is referred to as a

motion is because previous pitches serve as the origin for new pitches. The music may

sound like a clear pattern of order but with unpredictability demonstrated byuctu-

ations between pitch increments. Boon and Delcroly have used the term red noise

to refer to the 1/f2 spectrum that others describe as Brown noise15.

8Jeong, Joung, and Kim, Quantication of Emotion.9Yuh-Dauh Lyuu,Financial Engineering and Computation: Principles, Mathematics, Algorithms, (New York: Cam-

bridge University Press, 2002).10Robert Brown, A Brief Account of Microscopical Observations(1827).11Joseph Honerkamp (trans. by Katja Linderberg), Stochastic Dynamical Systems: Concepts, Numerical Methods,

Data Analysis(New York: VCH Publishers, 1994).12Brown,A Brief Account of Microscopical Observations, pp. 466467.13Albert Einstein and Leopold Infeld, The Evolution of Physics: From Early Concepts to Relativity and Quanta

(New York: NY A Touchstone Book, 1938).14Jeong, Joung, and Kim, Quantication of Emotion.15On red noise,see Boon and Delcroly, Dynamical Systems Theory(1995). On Brown noise, see Jeong, Joung,

and Kim, Quantication of Emotion

(1998) and Manfred R. Schroeder, Fractals, Chaos, Power Laws: Minutesfrom an Innite Universe(New York: W. H. Freeman and Company, 1990).



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Poised in between white noise and Brown noise is pink noise, which has a

spectral slope that is approximately1 (scaling exponent = 1, power proportional

to 1/f1 = 1/f). Pink noise has been described as a key feature of the Theory of Aesthetic

Value advanced by George D. Birkhoff because pink noise power spectra have been

considered more pleasing and interesting than either white or Brown noise.16 Pink

noise is complex in the sense that it is self-similar and entails some predictability.

However, pink noise is not as predictable as Brown noise nor is it excessively and

strictly random. Pink noise is structured with persistent, long-term uctuations.

Theseuctuations can be across extensive series, such as unfolding across runs of hun-

dreds of notes, within which are nested even smaller coherent patterns ofuctuations

that are correlated. Pink noise is statistically self-similar and has a fractal structure in

time. Pink noise can be generated from a series of tosses of a randomly chosen

subset of 7 dice from a larger set of 20 dice, each with 6 sides. The sum of the 20

dice determines the next value in the series. Music that resembles pink noise may

sound more complex with greater randomness than music that resembles Brownnoise. According to Jeong, Joung and Kim, the variability of a pink series is neither

as unpredictable as white noise nor as predictable as Brown noise.17 White noise,

Brown noise, and pink noise are all examples of self-similar fractal structures with

power law relationships having different slopes or scaling exponents.

Previously, Richard Voss and John Clarke examined the spectral density of audio

power uctuations in musical pieces such as Scott Joplin piano rags, classical music,

rock music and news and talk radio broadcasts.18 In each case, they identied a

power-law scaling relation consistent with pink noise. Voss and Clarke dene 1/f

noise with a range from 0.5 < < 1.5 as pink noise,19 a denition that is consistent

with other literature describing pink noise.20 Subsequently, Voss and Clarke (1978)

demonstrated that additional pieces such as Mario Davidovskys Synchronism I, II

and IIIand Karlheinz Stockenhausens Momentedemonstrated approximately 1/f 1

spectral densities, while Milton Babbitts String Quartet No. 3, Betsy Jolas Quatuor

III and Elliott Carters Piano Concerto in Two Movementsshowed decreasing corre-

lations at times longer than several seconds, yet were still demonstrating pink

noise.21 George Birkhoff, writing in 1933, emphasized the importance of sequential

16George D. Birkhoff,Aesthetic Measure, (Cambridge, MA: Harvard University Press, 1933). See also Schroeder,

Fractals, Chaos, Power Laws, as well as Voss and Clarke, 1975 and 1978. Schroeder explains: Birkhoffs theory,in a nutshell, says that for a work to be pleasing and interesting it should be neither too predictable nor pack

too many surprises. Translated to mathematical functions, this might be interpreted as meaning that the power

spectrum of the function should function neither as a boring Brown noise, with a frequency dependence of

f-2, nor like an unpredictable white noise, with a frequency dependence off0 (Schroeder 1990, 109).17Jeong, Joung, and Kim, Quantication of Emotion.18Richard F. Voss and John Clarke, 1/fNoise in Music: Music from 1/fNoise,Journal of the Acoustical Society of

America,vol. 63, no. 1 (1978): 258263.19Richard F. Voss and John Clarke, 1/f Noise in Music and Speech, Nature, vol. 258 (27 November 1975):

317318.20See, for example Eric-Jan Wagenmakers, Simon Farrell, and Roger Ratcliff, Estimation and Interpretation of 1/f

Noise in Human Cognition,Psychonomic Bulletin & Review, vol. 11, no. 4 (2004): 579615.21

Richard F. Voss and John Clarke,

1/fNoise in Music: Music from 1/fNoise,

Journal of the Acoustical Society ofAmerica,vol. 63 no. 1 (1978): 258263.

Jazz Perspectives 315


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order and complexity for aesthetically pleasing music. Citing Voss and Clark, Manfred

Schroeder suggested that Birkhoffs aesthetic value corresponded to 1/f spectra with

scaling exponent between 0 and 2 and falling right near the middle of this range,

which can be interpreted as being close to a scaling exponent of 1.22

Since then, others have investigated spectral densities from analyses of computer

generated music,23 simulated music,24 pitch shift,25 classical music,26 musical output

from a jazz ensemble,27 and classical ensembles.28 Boon and Delcroly analyzed the

pitch changes from a musician who played a synthesizer connected to a computer.29

They found that Mozarts String Trio KV 266and String Trio KV 563 (Divertimento)

along with BachsSecond Suite for Cello,Fugue BWV 870andMusical Offeringhad spec-

tral slopes very close to 2 (scaling exponent = 2, power proportional to 1/f2),

demonstrating Brownian motion. Additionally, Thelonious Monks Epistrophy,

Bill Evans Two Lonely People and Billy Strayhorns Lush Life also had spectral

slopes very close to2, demonstrating Brownian motion.30 These values are not con-

sistent with the range of aesthetically pleasing values suggested by Voss and Clark orSchroeder. They call into question whether or not pink noise is a common and required

characteristic of aesthetically pleasing music.

The present examination of Coltranes transcribed solos is directed at identifying

scaling relations between frequency and power in successive pitches through a power

spectral analysis. Of interest is the range of values of the scaling exponent for Col-

tranes music and its relation to the Theory of Aesthetic Value. Namely, we are inter-

ested in comparing the scaling exponents from Coltranes earlier improvisations that

may be more aesthetically pleasing to many people with Coltranes later work that

was more controversial and considered less aesthetically pleasing. We would like to sys-

tematically and numerically compare Coltranes scaling exponents with Birkhoffs

Theory of Aesthetic Value and its various interpretations (.5 < < 1.5). Do the

22Schroeder,Fractals, Chaos, Power Laws, 109.23Greg Aloupis, Thomas Fevens, Stefan Langerman, Tomomi Matsui, Antonio Mesa, Yurai Nuez, David Rappa-

port and Godfried Toussaint, Algorithms for Computing Geometric Measures of Melodic Similarity,Computer

Music Journal, 30(2006): 6776; Jeong, Joung & Kim, Quantication of Emotion (1998); and Jeff Pressing,

Novelty, Progress and Research Method in Computer Music Composition,Proceedings of the 1994 International

Computer Music Conference-Aarhus, (San Francisco: ICMA, 1994): 2730.24Ricardo Chacn, Yolanda Batres, and Francisco Cuadros, Teaching Deterministic Chaos through Music,

Physics Education,vol. 27, no. 3 (1992): 151154.25Julyan H. E. Cartwright, Diego L. Gonzlez, and Oreste Piro, Nonlinear Dynamics of the Perceived Pitch of

Complex Sounds,Physical Review Letters, vol. 82, (1999) : 53895392.26Kenneth J. Hsu, with Andrew Hsu, Fractal Geometry of Music: Physics of Melody,Proceedings of the National

Academy of the Sciences,vol. 87 (February 1990): 938941; Kenneth J. Hsu and Andrew Hsu, Self Similarity of the

1/fNoiseCalled Music,Proceedings of the National Academy of the Sciences, vol. 88 (April 1991): 35073509; John

R. Hughes, Yaman Daaboul, John J. Fino, and Gordon L. Shaw, The Mozart Effecton Epileptiform Activity,

Clinical Electroencephalography, vol. 29 (July 1998): 109119.27Gilles Peterschmitt, Emilia Gomez, and Perfecto Herra, Pitch-based Solo Location, Proceedings of MOSART

Workshop on Current Research Directions in Computer Music(Barcelona, 2001): 239243.28Yu Shi, Scalings of Pitches in Music, (1995) Eprint arXiv:adap-org/9509001. Retrieved 30 July 2012 from

http://arxiv.org/PS_cache/adap-org/pdf/9509/9509001v1.pdf.29

Boon and Delcroly, Dynamical Systems Theory

.30Ibid.

http://arxiv.org/PS_cache/adap-org/pdf/9509/9509001v1.pdfhttp://arxiv.org/PS_cache/adap-org/pdf/9509/9509001v1.pdf


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scaling exponents of Coltranes solos suggest other relations between fractal structure

and aesthetics? Namely, if is outside of the range .5 < < 1.5, that would indicate a

shortcoming of existing interpretations of Birkhoffs theory.

Method

Selection and Transcription

Two pieces from each year between 1959 and 1967 were selected based on communi-

cation with Lewis Porter, an expert on the life and work of John Coltrane. According to

Andrew White, who has transcribed some 700 of Coltranes recorded performances,31

Coltranes stylistic periods during this time of his solo career include the second period

(from June 1957 through August 1960) when he was practicing technique, the third

period (from September 1960 through August 1965) when Coltrane was primarily

playing without prearranged chord structure, and the fourth period (September1965 through July 1967) when Coltrane was inuenced by the avant-garde.32 Coltranes

saxophone solos have been transcribed by Andrew White and Carl Coan,33 and we

manually inputted notes of the solos directly from the sheet music and solo transcrip-

tions into Humdrum, a freeware music analysis program available on line at: http://

music-cog.ohio-state.edu/Humdrum.34 We then converted the resulting text le

from kern (a numeric representation of time signature, key, pitch, rhythm, duration,

and phrasing) to semits (a numeric representation of absolute pitch in semitones,

e.g. middle C is 0, C# is 1, etc.). The Humdrum program was modied to output

the data in the timebase and format required to conduct a Fourier analysis. The

result was a temporal series of pitches that represented the successive notes in eachpiece. Rests and grace notes were omitted from each series.35 Coltrane also used mul-

tiple notes and ghost notessparingly at times to accent his solo. When more than one

note was performed at the same time, the mean of the pitches in semitones was calcu-

lated and substituted for the original multiple notes, because Fourier analysis requires a

single pitch value at each point in time, to establish a well-dened pitch trajectory. We

needed to perform the analysis in this manner in order to gain a clearer picture of the

31Lewis Porter, personal communication, May 25, 2006. See also Andrew White, The Works of John Coltrane, Vols.

1 through 14, (Washington, DC: Andrews Music, 19732006).32

For discussions of Coltranes stylistic periods, Andrew White,Tranen Me (A Semi-Autobiography): A Treatise onthe Music of John Coltrane(Washington, D.C.: Andrew Musical Enterprises, Inc., 1981).33White,Tranen Me(1995); Carl Coan,John Coltranes Solos[transcriptions, with commentary by Ronnie Schiff

and Ravi Coltrane] (Milwaukee, WI: Hal Leonard Publications, 1995).34David Huron, Music Information Processing Using the Humdrum Toolkit: Concepts, Examples, and Lessons,

Computer Music Journal, vol. 26 no. 2 (July 2002): 1126. According to the Humdrum web site, the kern(after

the German word for core) representation scheme can be used to represent basic or core information for period-

of-common-practice Western music. The kern scheme allows the encoding of pitch and duration, as well as acciden-

tals, articulation, ornamentation, ties, slurs, phrasing, glissandi, barlines, stem-direction and beamingas syntactical,

rather than an orthographic, information, and so is appropriate for our fractal analysis. David Huron,The Humdrum

Toolkit: Software for Music Research(Ohio State University School of Music, 1999):http://music-cog.ohio-state.edu/

Humdrum/index.html, accessed 1 August 2012.35

For a rationale for omitting grace notes and rests, see Yu Shi,

Correlations of Pitches in Music,

Fractals,vol. 4,no. 4 (1996): 547553.

http://music-cog.ohio-state.edu/Humdrumhttp://music-cog.ohio-state.edu/Humdrumhttp://music-cog.ohio-state.edu/Humdrum/index.htmlhttp://music-cog.ohio-state.edu/Humdrum/index.htmlhttp://music-cog.ohio-state.edu/Humdrum/index.htmlhttp://music-cog.ohio-state.edu/Humdrum/index.htmlhttp://music-cog.ohio-state.edu/Humdrumhttp://music-cog.ohio-state.edu/Humdrum


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more pervasive fractal patterns in Coltranes solos. For the spectral analysis, each note

was represented as an equivalent sequence of sixty-fourth notes, the smallest unit of

sound that Coltrane played in his solos. For example, a quarter note was represented

as a sequence of 16 sixty-fourth notes. Subtracting the average pitch from each series

and dividing by its standard deviation yielded 18 normalized relative pitchseries suit-

able for a power spectral analysis.

Results

Except for one unusually short piece, Lush Life, the series ranged in length from

306 to 1,162 successive pitches from Coltranes solos, which corresponded to equiv-

alent series of sixty-fourth notes ranging in length from 2,640 to 9,232 (M = 4,842,

SD = 2,513). All series were padded with zeros to bring the length to the nearest

higher integer power of 2 for the Fast Fourier algorithm used to calculate the

spectra. To obtain reliable estimates, the spectral analysis used the Welch methodof decomposing the overall sequence for each musical piece into subsets of length

1,024, with 50% overlap between successive subsets of sixty-fourth notes. The

power spectra for these subsets were averaged for each musical piece. This technique

of spectral analyses conducted on overlapping subsets of the data produces more

reliable estimates than conducting a single spectral analysis over all of the data at

once. The advantage of this method is statistical. This method averages separate

spectra derived from successive blocks of the entire note sequence and better separ-

ates the low spectrum frequencies.36 For Lush Life, the overall sequence of 238

pitches was equivalent to 1,006 sixty-fourth notes. The subsets used in this spectralanalysis were 256 notes in length with a 50% overlap between successive subsets of

sixty-fourth notes. The resulting power spectra, for all pieces except Lush Life,con-

tained 511 frequencies ranging from approximately 2 sixty-fourth notes per cycle (or

0.5 cycles/sixty-fourth note) to a single cycle of 1,024 sixty-fourth notes (or .00098

cycles/sixty-fourth note). On a log10 scale, these frequencies correspond to 0.3

and 3.0, respectively (horizontal axes in Figures 16). The power spectrum for

Lush Life contained 127 frequencies ranging from approximately 0.5 cycles/sixty-

fourth note to .0039 cycles/sixty-fourth note. On a log10 scale, these frequencies

correspond to0.3 and2.4.

Spectral Scaling Analyses. The spectral plot for each musical piece is shown on the

right side of Figures 1 6 for each of John Coltranes saxophone solos from 1959

1967 in successive order. Each point represents power, which can be interpreted as a

measure of relative change from the average pitch. Log power is proportional to the

log frequency, i.e., log(). Thus, as in all fractals, the ratio that captures this relation

stays constant across changes of power amplitude and frequency, which is illustrated

by the constant slope of the lines in Figures 16.

36

For a rationale of the Welsh method, see Harold Ynedstad,

Stationary Temperature Cycles in the Barents Sea:Cause of Causes,ICES Annual Science Conference Island (1996): 15.



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Figure 1Mutual information and spectral plot, respectively, for songs from 19591960.



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These graphs clearly demonstrate a linear trend on double logarithmic axes. The

spectral slopes ranged from 1.64 to 1.77 (scaling exponent between 1.64 and

1.77), which is characteristic of an anti-persistent fractional Brownian motion.37

Namely, the successive notes are less predictable than prototypical Brown noise

(= 2), but more predictable than prototypical pink noise ( = 1). Unlike prototypical

Brownian motion ( = 2), the successive, generally small changes in pitch are negativelycorrelated, oranti-persistent. The note progression tends to reverse relatively often.

Table 1 lists the absolute values of the slopes, the scaling exponents, and the corre-

lation coefcients for these linear patterns. The correlation coefcient rranged from

.88 to .98, which indicates strong linear relationships. These linear trends indicate a

fractal or self-similar structure in the succession of pitches. However, for many of

the pieces there is also evidence for a succession of distinct peaks or resonances at

Table 1. The scaling exponent, , for each piece.

TimePeriod Year

NoteCount Song Album

ScalingExponent

r

2nd

1959 777 Giant Steps Giant Steps 1.76 0.962nd 556 Some Other Blues Coltrane Jazz 1.77 0.953rd 1960 867 My Favorite

ThingsMy Favorite Things 1.77 0.98

3rd 664 Equinox Coltranes Sound 1.67 0.963rd 1961 946 Impressions Impressions 1.74 0.933rd 1,162 Spiritual Live at the Village

Vanguard1.68 0.97

3rd 1962 532 Tunji Coltrane 1.69 0.973rd 420 Nancy Ballads 1.70 0.943rd 1963 238 Lush Life John Coltrane, Johnny

Hartman1.75 0.98

3rd 396 Alabama Live at Birdland 1.76 0.963rd 1964 995 Acknowledgement A Love Supreme 1.72 0.963rd 602 Crescent Crescent 1.64 0.953rd 1965 649 Ascent Sun Ship 1.74 0.954th 306 Welcome Kulu Se Mama 1.75 0.944th 1966 529 Naima Live at the Village

Vanguard Again!1.71 0.93

4th 716 Crescent Live in Japan 1.66 0.954th 1967 782 Jupiter Interstellar Space 1.65 0.884th 621 Expression Expression 1.65 0.94

Mean 623.29 1.71 0.95SD 212.98 0.05 0.02

Notes: Sincef = 1/f , the scaling exponent is the absolute value of the spectral slope given bytting a regression

line on a double logarithmic plot of the power spectrum;ris the correlation coefcient between the frequency and

power in logarithmic units. The pieces in boldhave a series of regularly spaced peaks in their mutual information

graphs in Figures 16.

37Andras Eke, Peter Herman, James B. Bassingthwaighte, Gary M. Raymond, Don B. Percival, Michael Cannon,

Istvan Balla, and Cornelia Ikrnyi,

Physiological Time Series: Distinguishing Fractal Noises from Motions,

Euro-pean Journal of Physiology,vol. 439, no. 4 (2000): 403415, (DOI) 10.1007/s004249900135.



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the highest frequencies. In order to address these peaks we conducted another type of

analysis using a mutual information measure.

Average Mutual Information Analyses. The left side of Figures 16 depicts the logar-

ithm of average mutual information between the pitches at different delays ranging

from 1 sixty-fourth note to 128 sixty-fourth notes. Average mutual information can

be considered a nominal measure of correlation.38 Thus, the higher the mutual infor-

mation, the more predictable one pitch is from a preceding pitch. One difference

between a mutual information analysis and an auto-correlation analysis or spectral

analysis is that the mutual information analysis can detect both linear and nonlinear

patterns of association.

There is an overall declining trend in the plot of mutual information versus delay,

indicating that the prediction of more distant pitches is more uncertain. This pattern

is a direct consequence of the fractal nature of the pitch sequences. If depicted on

double-logarithmic axes, the relation between mutual information and delay is linearand a statistic akin to the spectral slope can be derived from the slope of the mutual

information plot.

What is notable about these plots is that several musical pieces have a regularly spaced

series of relative peaks in the mutual information measure, which can be interpreted as

dominant rhythmic intervals. If eighth notes constitute the dominant rhythmic interval,

this would be consistent with relative peaks in mutual information separated by delays of

8 sixty-fourth notes. For example, in Figure 1Giant Steps (1959) showed consistent

peaks at delays which are multiples of 8 sixty-fourth notes, illustrating that the rhythmic

pattern of the eighth-notes was emphasized as a dominant theme.

Note that the presence of a regular dominant rhythm is contrary to the notion of

self-similarity, in that a particular time scale is characteristic of the music. The

fractal trend in the power spectra and the dominant rhythm revealed by the mutual

information measure in some pieces may therefore create a particular kind of complex-

ity in the music. For some pieces, the peaks in the mutual information measure

occurred at irregular intervals and in some cases there were no apparent peaks

(Figures 16). Both of these tendencies indicate that the sequence of pitches is less pre-

dictable. In contrast, some pieces, such as Giant Steps, have relatively pronounced

dominant rhythms with larger (more negative) spectral slopes. In these cases, the

mutual information measure, in conjunction with the spectral slope, indicates thatthe music is more predictable. Other pieces from Coltranes later third and fourth

periods, Jupiter (1967) and Crescent (1964), have dominant rhythms, but lower

(less predictable) power spectral slopes.39

For Giant Steps, the average mutual information measure shows a consistent

peaked pattern at eighth-note intervals that are indicative of the structural chord

38Henry D. I. Abarbanel, Analysis of Observed Chaotic Data (New York: Springer, 1996).39John Coltrane, Giant Steps,Giant Steps, Rhino/Atlantic A2 1311 (2007 [1959]). John Coltrane, Jupiter,Inter-

stellar Space, Impulse! ASD 9277 (2000 [1967]). John Coltrane, Crescent,

Crescent, Impulse! 1764902 (2008[1964]).



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changes within the piece and the upbeat tempo. In Giant Steps,the rapid harmonic

changes are more predictable, and these changes represent the second period of Col-

tranes style (from June, 1957 through August, 1960), a period, Coltrane acknowledged,

when Thelonious Monk exerted a major inuence on his playing.40 During this period,

Coltrane worked to develop his creative technique. Giant Steps, his most famous

piece of this period, remains a jazz standard that demonstrates virtuosity and technical

expertise. For Coltrane, the pattern may be more predictable than his fourth period.

Impressions (1961), and less predominately Ascent (1965), also have a similar

eighth-note rhythmic pattern with stylist chord changes. Impressions is stylistic of

Coltranes third period with the classic quartet (from September, 1960 through

August, 1965), which was inuenced by music without prearranged chord pro-

gressions.41 At this time, Coltrane collaborated with Eric Dolphy and was inuenced

by Ornette Coleman, both of whom were embracing non-progressive harmonies.

Another early piece with a steep spectral slope (scaling exponent = 1.77) is Some

Other Blues.42

The mutual information also shows a consistent peaked pattern ateighth-note intervals that is indicative of the structural chord variations within the

piece. Its upbeat tempo is less robust and pronounced than Giant Steps and

Impressions. Some Other Blues, My Favorite Things, Acknowledgement and

Ascent also sound like the eighth-note rhythmic pattern is accented, yet within

them, there is also a smooth melodic line that is present. This eighth-note pattern is

not accentuated as robustly as Giant Steps and Impressions. Instead, these tem-

poral sequences are more subtle and smooth. Furthermore, Acknowledgementand

My Favorite Things have average mutual information that is less robustly peaked

than Some Other Blues.43

Other pieces have shallower, less predictable spectral slopes (scaling exponent

closer to 1.6) and an absence of regularly spaced peaks in the mutual information

measure. Such pieces include Tunji, Spiritual, Equinox, Crescent (Live in

Japan version), and Expression.44 The average mutual information for Spiritual

and Equinox is smoother than the other pieces mentioned above, despite these

pieces being indicative of Coltranes early third stylistic period. These two pieces

display relatively smoother average mutual information patterns, indicating less of

an emphasis on the rhythmic pattern. These pieces are reective of Coltranes third

period where many songs were modal in style, without the constraints of prearranged

chord patterns.45

These two pieces have a slower tempo with lyrical melodies.

40Lewis Porter,John Coltrane and His Music(1983); Andrew White, Trane n Me(1981).41Lewis Porter,John Coltrane and His Music(1983); Andrew White, Trane n Me(1981).42John Coltrane, Some Other Blues,Coltrane Jazz, Rhino/Atlantic 1354 (2008 [1959]).

John Coltrane, My Favorite Things,My Favorite Things, Atlantic 1361 (1990 [1960]).

John Coltrane, Acknowledgment, A Love Supreme, Impulse! GRD 155 (2003 [1964]).43Coltrane, My Favorite Things (1990 [1960]);

Coltrane, Acknowledgment(2003 [1964]).44John Coltrane, Tunji,Coltrane, Impulse! 215 (2002 [1962]).

John Coltrane, Spiritual,Live at the Village Vanguard, Impulse! A-10 (1998 [1961]).

John Coltrane, Equinox,

Coltranes Sound, Atlantic SD 1419 (1999 [1960]).45Andrew White, Trane n Me.



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The average mutual information pattern is inconsistent for Naima, Crescent

(Live in Japan) and Expression; the pattern has some eighth-note changes yet is

inconsistent.46 This version of Naima was recorded during the Live at the Village

Vanguard Again! sessions, some seven years after the original studio version.

Naima, along with this version of Crescent (Live in Japan) and Expression are

a part of Coltranes fourth period (September, 1965 through July, 1967) during

which he played with his second wife, Alice McCloud Coltrane (piano), Rashied Ali

(drums), Pharoah Sanders (saxophone), and Jimmy Garrison (bass).47 Although

these songs still displayed a pattern consistent with 1/f scaling, Crescent (Live in

Japan) and Expression have slopes that indicate less predictability than the earlier

pieces in his second period. The rhythmic patterns and improvisation during

Coltranes fourth period tended to be more free-form, with less structural constraints.

However, the version of Crescentfrom the 1964 studio album, during Coltranes

third period, demonstrated more structure.48 The mutual information for Crescent

(1964) showed peaks at sixteenth note intervals, yet Crescent(Live in Japan) did not.The 1964 version of Crescenthas a very melodic and lyrical introduction and then

builds into a solo that emphasizes the sixteenth-notes, while the Live in Japan

version is more avant-garde and builds up with several runs that are quick and aggres-

sive. However, the dominant theme and variations are slower and passionate, leading to

aggressive shrieking, thus being more irregular or unpredictable.

The mutual information for Welcome is irregular. The scaling exponent is more

consistent with some of the earlier periods, yet Welcomestems from the beginning of

Coltranes stylistic fourth period.49 Jupiter is also part of Coltranes fourth stylistic

period, yet may be more characteristic of the end of that period, in that the spectral

slope is shallower ( is further from 2); the piece is less predictable.50 Furthermore,

the mutual information emphasizes 3 eighth-note sequences (delays of 24 sixty-

fourth notes), which is characteristic of Jupiter. Jupiter has many fast runs. As

we listen to Coltranes music, we think we know where he is going with his style,

yet, as is characteristic in the fourth period, he switches the pattern of his playing

which results in surprising the listener, with less predictability, greater randomization

and greater variation.

Both the third and fourth periods have distinct features that allow experienced lis-

teners to identify Coltranes voice. Nancy (with the Laughing Face) and Tunji

also had slopes in the middle of the range and displayed relatively constant mutualinformation at long delays, which may represent a combination of smooth melodic

playing with some distinct rhythmic patterns that vary within the piece.51 However,

46John Coltrane, Naima,Live at the Village Vanguard Again!, Impulse!/Universal Distribution 001582702 (2011

[1966]); John Coltrane, Crescent, Live in Japan, Impulse! 4102 (1991 [1966]); John Coltrane, Expression,

Expression, Impulse! 131 (1993 [1967]).47Lewis Porter,John Coltrane and His Music, 1983; Andrew White, Trane n Me, 1981.48Hereafter, Crescent(1964). See John Coltrane, Crescent, Impulse! 1764902 (2008 [1964]).49John Coltrane, Welcome, Kulu S Mama, Impulse! A-9106 (1967 [1965]).50John Coltrane, Jupiter,Interstellar Space, Impulse! ASD 9277 (2000 [1967]).51

John Coltrane,

Nancy (with the Laughing Face)

Ballads, Impulse! A32 (2008 [1962]); Coltrane,

Tunji

(2002[1962]).



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the peaks in the average mutual information graph are not very regular, demonstrating

that there is not a dominant rhythmic pattern. These compositions were recorded in

1962 as part of Coltranes stylistic third period. The consistent lack of a dominant

rhythmic pattern may be a stylistic feature of Coltranes lyrically expressive solos,

which is especially true for ballads. Alabama is also a lyrically expressive solo.52

The sound is smooth yet with an impressive message. Alabama is irregular in its

mutual information despite the steeper (more predictable) spectral slope closer to

2, (scaling exponent = 1.76, power proportional to 1/f1.76). The complexity of Col-

tranes music during his third and fourth periods would suggest that novice listeners

would be less likely to appreciate the later work due to greater variation and less pre-

dictability. The irregularity in rhythm may be a feature that Coltrane developed in the

third and fourth period, whereas the earlier second and third period pieces tended to

emphasize the eighth-note metric unit more, which may be more typical of straightfor-

ward jazz. However, as Coltrane developed his technique, he concurrently developed

his style. The fourth period is when Coltrane learned to create his own rules, ratherthan to follow if also to bend conventions of melodic contour, phrasing,

harmony, and rhythm, as in his stylistic second period. Coltrane also did not

conform to a specic dominant rhythmic pattern in much of his later work, but also

uctuated between dominant and non-dominant rhythmic patterns, beginning with

his third stylistic period.

Lush Lifeis a much shorter piece, so the statistical estimate of the power spectral

slope and average mutual information may be limited by greater statistical uncertainty

than the other pieces.53 Lush Life also has a similar slope to Giant Steps, Some

Other Blues,and Impressions; yet the mutual information may suggest a somewhat

more irregular rhythmic pattern for this ballad.54 Perhaps this is more stylistic of Col-

trane, being that he tended to be lyrically expressive and melodic. Coltrane is also

expressive on other ballads including Alabama,not rigidly conforming to a standar-

dized rhythmic pattern and performing with rubato.55 This rhythmic variation from

slowing and speeding in lyrical phrasing could diminish the peaks in the average

mutual information analysis.

Discussion

All eighteen solos from 1959 through 1967 during Coltranes second, third and fourth

period styles display fractal scaling. Our ndings suggest that Coltranes second period

was a time with greater predictability and rhythmic emphasis on the eighth-note, while

later work, especially during the end of Coltranes fourth period, was less predictable

and demonstrated more irregular rhythmic structures. The present examination of

52John Coltrane, Alabama,Live at Birdland, Impulse! MCAD 42001 (2008 [1963]).53John Coltrane, Lush Life, John Coltrane and Johnny Hartman, GRP/Impulse! A40 (1995 [1963]).54John Coltrane, Giant Steps,Giant Steps, Rhino/Atlantic A2 1311 (2007 [1959]). John Coltrane, Some Other

Blues(2008 [1959]). John Coltrane, Impressions,Impressions, Impulse!/Universal Distribution 1764899 (2008

[1961

1963, issued 1963]).55John Coltrane, Alabama,Live at Birdland, Impulse! MCAD 42001 (2008 [1963]).



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Coltrane endures as one of the most respected jazz musicians by numerous contem-

porary jazz and rock musicians59; fractal geometry offers additional insight regarding

the dimensionality and complexity of his music. In the short span of Coltranes

career and musical output, he recorded hundreds of compositions and released 35

albums as leader or co-leader; many are considered jazz standards.60 Some albums,

although less accepted at the time, such as Giant Stepsand My Favorite Things, are

now acclaimed as classics. Yet, Coltranes music may be perceived as contradicting

fractal, mathematical interpretations of Birkhoffs Theory of Aesthetic Value through

exhibiting Brown noise rather than pink noise. He also developed novel techniques

such as multiphonics (playing more than one note at the same time). His multi-

tonic changes (rapid harmonic changes) created polypentatonic possibilities (improvi-

sations in the pentatonic scales) in compositions such as Giant Steps,61 now con-

sidered a masterwork, but which was described by critics of the time, Ben Ratliff

reports, as suffering from rhythmic stiffness and melodic tameness.62 Our analysis

of Giant Steps did indicate that the rhythm was more predictable than much ofhis later work. During his second period Coltrane was focused on developing a techni-

cal expertise and his performances were more predictable.

However, Coltranes improvisations did not t with the typical conventions and

were not judged as aesthetically pleasing, but rather were harshly criticized as anti-

jazz as early as 1961.63 Coltrane began emphasizing more rhythmic irregularity

than the eighth-note rhythm characteristic of his second period, yet he uctuated

between eighth-note rhythmic patterns and rhythmic irregularity during his third

period, and this rhythmic irregularity became more characteristic during his fourth

period. It is well known that Stravinskys Rite of Spring, a challenging piece of music

with strong dissonant qualities, caused a riot in its rst performance. Performances

in the later years of Coltranes life were described as angry, blaringly abrasive,

and harsh, at, querulous and at times, vindictive.64 A session at New Yorks

Village Theater (later Bill Grahams Fillmore East) in December of 1966 caused a

number of persons in the audience to walk out, yet others were enthusiastically shout-

ing.65 His later work (19641967), however, continues to be misunderstood and criti-

cized todayeven among jazz acionados.

59See, for example, Frank Kofsky, John Coltrane and the Jazz Revolution of the 1960s(Atlanta, GA: Pathnder,

1998); Eric Nisenson, Ascension: John Coltrane and His Quest(1st Da Capo Press ed.), (New York: Da CapoPress, 1995 [New York: St. Martins Press, 1993]); Lewis Porter, John Coltrane and His Music (1998). Coltrane

has been lionized and even made a canonical gure since his death: see, for example, (no byline), John Coltrane

As a Saint at The Church of John Coltrane, The Hufngton Post(12 March 2011,http://www.hufngtonpost.com/

2011/03/11/john-coltrane-as-a-saint-_n_833744.html, accessed 2 August 2012).60Yasuhiro Fujioka, with Lewis Porter and Yoh-Ichi Hamada (eds.),John Coltrane: A Discography and Musical Bio-

graphy(Newark, N.J.: Scarecrow Press, 1995).61Masaya Yamaguchi (Jay Sweet, ed.), A Creative Approach To Multi-Tonic Changes:Beyond Coltranes Harmo-

nic Formula,Annual Review of Jazz Studies, vol. 12 (2002): 147167.62Ratliff, Coltrane: The Story of a Sound, 130.63Nisenson,Ascension: John Coltrane and His Quest, 119.64Ratliff,Coltrane: The Story of a Sound, 131; Nisenson,Ascension: John Coltrane and His Quest, 203; Fraim,Spirit

Catcher, 64.65Fraim,Spirit Catcher.

http://www.huffingtonpost.com/2011/03/11/john-coltrane-as-a-saint-_n_833744.htmlhttp://www.huffingtonpost.com/2011/03/11/john-coltrane-as-a-saint-_n_833744.htmlhttp://www.huffingtonpost.com/2011/03/11/john-coltrane-as-a-saint-_n_833744.htmlhttp://www.huffingtonpost.com/2011/03/11/john-coltrane-as-a-saint-_n_833744.htmlhttp://www.huffingtonpost.com/2011/03/11/john-coltrane-as-a-saint-_n_833744.htmlhttp://www.huffingtonpost.com/2011/03/11/john-coltrane-as-a-saint-_n_833744.html


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Our fractal analysis supports the notion, advanced by Lewis Porter, that What

seems to be chaotic [in Coltranes later work] is just the opposite.66 Coltranes

work during his fourth period still displays fractal qualities. The fractals of this later

work are often even less predictable than his early work, yet are still robustly fractal.

Before his death at age 40, Coltrane strove to have no boundaries or limitations to con-

strain his improvisations. Coltrane constantly challenged himself to expand upon his

technique throughout his career even when he was already an accomplished musician.

A long-standing speculation is that 1/f scaling in music arises as a consequence of

the inherent aesthetics of music itself. Some patterns are thought to be intrinsically

more pleasing to the human ear than others. Jeong, Joung and Kim, for example,

suggested that most listeners prefer 1/f1 music to music with an uncorrelated white

noise (1/f0) structure or highly correlated Brown noise (1/f2) structure.67 In fact,

the form of 1/f 1 scaling that comprises pink noise represents a compromise between

complete randomness (white noise) and highly constrained Brownian patterns of varia-

bility. Essentially the Theory of Aesthetic Value, described by George Birkhoff, andsimilar hypotheses suggest that interesting pieces of art somehow strike balances

between the expected and the unexpected, between repetition and contrast.68

Various ranges of the scaling exponent, , have been interpreted as being consistent

with Birkhoffs theory. Manfred Schroeder suggested that scaling exponents

between 0 and 2 fall into the Theory of Aesthetic Value, yet he suggested that the

middle, closer to 1, may be more aesthetically pleasing.69 Voss and Clarke suggested

that listeners aesthetically prefer pink noise (0.5 < < 1.5) over Brown and white

noise.70 Both Schroeder and Voss and Clarke may have interpreted Birkhoff differently

regarding the Theory of Aesthetic Value. Based on our analysis of Coltranes solo

improvisations, we propose that the numerical range for the Theory of Aesthetic

Value also include anti-persistent fractional Brownian motion. Coltranes earlier

improvisations were even more Brownian than his later improvisations, which were

closer to the pink range already interpreted as the Theory of Aesthetic Value by Voss

and Clarke. However, it is also important to note that the music that they analyzed

lasted for hours and had more variability (and noise) than the solos from one single

musician.

There is merit to Birkhoffs Theory of Aesthetic Value perspective and to Schroeders

suggestion that this theory may be quantied in terms of spectral slopes or scaling

exponents. However, none of Coltranes solos display pink noise. Instead, we havedemonstrated that they are best characterized as fractional Brownian Motion,71

which was also found to characterize classical and jazz music, but using different

pitch encoding than in the present study. Yet Coltranes solos still display order and

66Lewis Porter,John Coltrane: His Life and Music(1998): 270.67Jeong, Joung, and Kim, Quantication of Emotion.68Birkhoff,Aesthetic Measure; Yu Shi, Correlations of Pitches in Music,547553.69Schroeder,Fractals, Chaos, Power Laws.70Richard Voss and John Clarke, 1/fNoise in Music and Speech,(1975); and 1/fNoise in Music: Music from 1/f

Noise

(1978).71Eke et al., Physiological Time Series.



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complexity that is aesthetically pleasing. Although the comparison of audience reac-

tions to music is complicated by the evolution of cultural aesthetics over time, some

of Coltranes pieces have been called dissonant or avant-garde sounding to the

novice listener, while others, such as Giant Steps, are quite pleasing to the listener,

even though earlier on some critics did not accept this music. Yet both display a Brow-

nian trajectory in their pitch sequences. Within these spectra, Giant Stepshas greater

predictability and to the novice listener is more pleasing to the ear. In contrast, the later

work of Jupiter and Crescent (Live in Japan) demonstrate less predictability and

may sound like there is less order to the average ear. The average ear may be frustrated

since the motion in the piece is less predictable. However, these pieces display Brow-

nian motion, which involves both order and unpredictability between successive notes.

What can be learned from the qualitative patterns that unfold in John Coltranes,orany

individuals, life experience? Proposing easily testable hypotheses in this realm is difcult.

Nevertheless, suppose one transfers the metaphorical concept of metastable dynamics to

the qualitative realm of an individuals life experience. It is possible to frame creativity,intelligence, and especially development in a manner that is qualitatively consistent.

Robert Sternberg observed in 2001 that many descriptions of intelligence share an

emphasis on adaptation, the ability to effectively mesh with ones surrounding environ-

ment.72 He described wisdom as balancing the forces of change implied by creativity

with the stability or inertia of an existing adaptive state implied by intelligence. For

Sternberg, the continuous interplay of intelligence, creativity, and wisdom form a dia-

lectic spiral. Clearly, Sternbergs hypothesis emphasized interactive relationships. More

broadly, contemporary developmental scientists describe the outcomes of development

as a probabilistic bi-directional interplay between the constraints supplied by genetic

and neural activity, behavior, and the physical, social and cultural environment.73

Previous analyses that found pink noise in musical performances focused primarily

on the patterns of oscillation in the relative volume (loudness) of the pieces, and did

not assess the patterns of the played notes, as we have done here. This outcome

leads us to speculate that aesthetic value may reside in the quantitative patterns that

emerge in artistic performances. However, aesthetic value is often judged in the

context of the milieu of human and artistic culture, and analyses such as those in

the present study may inuence judgment and acceptance of creative works.

In conclusion, Coltranes improvised solos are comprised of fractal patterns that

appear to contradict those predicted by the pink noise identied in classical, jazzand blues music compositions.74 Instead, our ndings conrm that Coltranes work,

including his later avant-garde improvisations, displays similar numeric complexity

72Robert J. Sternberg, What Is the Common Thread of Creativity: Its Dialectical Relation To Intelligence and

Wisdom,American Psychologist, vol. 56 (2001): 360362.73See, for example, Gilbert Gottlieb, Normally Occurring Environmental and Behavioral Inuences on Gene

Activity: from Central Dogma to Probabilistic Epigenesist, Psychological Review, vol. 105 no. 4 (October 1998):

792802.74See Kenneth J. Hsu and Andrew Hsu, Fractal Geometry of Music: Physics of Melody(1990), and Self Simi-

larity of the 1/fNoiseCalled Music,(1991). See also Hughes et al., The Mozart Effecton Epileptiform Activity

(1998); Richard Voss and John Clarke, 1/fNoise in Music and Speech,

(1975), and 1/fNoise in Music: Musicfrom 1/fNoise(1978).



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and order75with spectral slopes close to 2, or more specically scaling exponents in

the average range between 1.6 < < 1.8. Although Jean Pierre Boon and Olivier

Delcroly analyzed performances by a musician using a synthesizer connected to a com-

puter to encode pitch, which is different from the present study, they also found that

the spectral slopes for pieces by Wolfgang Amadeus Mozart, Johann Sebastian Bach,

Thelonious Monk, Bill Evans and other classical and jazz musicians were even closer

to2 (approximate range of the scaling exponents in between 1.8 < < 2.2).76

In comparison with his earlier work, Coltranes later fourth period illustrates greater

complexity, unpredictability and aesthetic value that may be misunderstood by the

average listener. In essence, Coltrane continually developed his self-expressions on

the saxophone that are evident from his early work during his second period and

especially through his later work during his fourth stylistic period. Through exploring

the spatio-temporal properties and stochasticity of Coltranes music,77 we see evidence

of eminent creativity. Some researchers suggest that specic conditions such as ow

(creating in the moment), intrinsic motivation, tolerance for ambiguity and risktaking lead to high levels of creativity within individuals.78 John Coltrane was able to

take his music to another spatio-temporal level. As a result Coltrane was able to with-

stand the constraints of time by sounding absolutely current in any era for multiple

generations of listeners. Perhaps this is the reason why Coltrane will continue to be

considered one of the most innovative musicians in jazz and music generally.

Acknowledgements

We thank Andrew White, Lewis Porter, Shane Ruland, Sean Ferguson, Ted McDaniel,

David Huron and Glenn Elliott for tools, assistance, andadvice in this analysis of John Col-

tranes music. We also thank Paul Jones for technical consultation and Alex Charyton for

his constructive feedback and support. Last, we thank Steven Pond for his attention to

detail, Robert Weisberg for suggesting the concept of a case study, and John Taggart for

introducing us to the work of John Coltrane. May we inspire others regardingthe complex-

ity of music as well as understanding the complexity of the music by John Coltrane.

Abstract

John Coltranes relevant biographical events are discussed along with a fractal analysis

examining the sequential structure of pitches in his saxophone solos. Eighteen solos

75Birkhoff,Aesthetic Measure.76Boon and Delcroly, Dynamical Systems Theory.77Peter Schuster (ed.),Stochastic Phenomena and Chaotic Behaviour in Complex Systems: Proceedings of the Fourth

Meeting of the UNESCSO Working Group on Systems Analysis, Flarrnitz, Karnten, Australia, June 610, 1983(Berlin,

Germany: Springer-Verlag, 1983).78Milhaly Csikszentmihalyi,Flow: The Psychology of Optimal Experience(New York: Harper Collins, 1990); Dean

K. Simonton, Genius, Creativity, and Leadership: Historiometric Inquiries (Cambridge, MA: Harvard University

Press, 1984); Robert J. Sternberg (ed.), Handbook of Creativity(Cambridge, UK: Cambridge University Press,

1999); John L. Sherry, Flow and Media Enjoyment, Communication Theory, vol. 14 no. 4 (November 2004):

328

347. Howard Gardner, Creating Minds: An anatomy of creativity as seen through the lives of Freud, Einstein,Picasso, Stravinsky, Eliot, Graham and Gandhi (New York: NY: HarperCollins, 1993).



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a historical and fractal perspective on the life and saxophone solos of john coltrane

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