September 4, 2008

A Fractal in Bach's Cello Suite

Johann Sebastian Bach surely did not have fractals in mind when he composed six suites for solo cello several centuries ago. Nonetheless, at least one movement has the repeating structure on different scales that is characteristic of a fractal.

Harlan J. Brothers of The Country School in Madison, Conn., contends that the first Bourrée in Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this musical structure can be visualized as a fractal construction called the Cantor set, he says.

Brothers' findings appear in the paper "Structural Scaling in Bach's Cello Suite No. 3," published in the March 2007 issue of the journal Fractals.

Examining only the written score, Brothers focused on the phrasing in the first section of the first Bourée. Musical phrasing refers to the way certain sequences of notes are naturally associated with each other, Brothers says.

Brothers detected repeated use of the pattern AAB on different scales, where each B section lasts twice as long as each A section.

Analysis of the first 16 measures of the Bourrée from Bach's Suite No. 3. Courtesy of Harlan Brothers.

For example, the piece starts off with two eighth notes and a quarter note (m1), repeats that pattern (m2), then continues with a phrase (m3) that is twice as long. The same pattern of short, short, long (s1) is repeated (s2), followed by a longer sequence (s3).

Analogously, the first eight measures are repeated, giving two "short" sections that are followed by a 20-measure "long" section.

"Interestingly, although Bach wrote the piece with a repeat symbol at the end of this 20-measure section, anecdotal evidence suggests that some cellists choose to perform it without the second repeat," Brothers noted in his paper. "Performed in this fashion, the Bourrée Part I exhibits a full four levels of structural scaling symmetry."

The structure of Bach's music resembles that of a classic type of fractal known as a Cantor set. Start with a line segment. Remove the middle third. Then remove the middle third from the remaining pieces, and so on. The result is a "Cantor comb."

Four levels in the creation of a Cantor comb. Courtesy of Harlan Brothers.

The hierarchical nesting of the AAB phrasing in the first Bourrée produces a similar pattern.

"The fact that Bach was born almost three centuries before the formal concept of fractals came into existence may well indicate that an intuitive affinity for fractal structure is, at least for some composers, an inherent motivational element in the compositional process," Brothers concluded.

Brothers has set about establishing a mathematical foundation for the classification of fractal music and correcting widespread misconceptions about fractal music. His efforts have revealed that musicians have been composing a form of fractal music for at least six centuries. One example is a type of canon in which different voices repeat the same melody or rhythmic motif simultaneously at different tempos.

Music can exhibit a wide variety of scaling behavior, Brothers says. He has himself written a number of compositions illustrating such properties. And he is keen to have others find further examples of scaling symmetry in what he describes as "the rich and vast body of musical expression."


Anonymous said...

The only misconceptions regarding fractal music that Brothers is revealing are misconceptions of his own regarding the claims and methods used by those who have been working in this field for over fifteen years longer than he.

And his exercises in proving the likes of Bach are "fractal" are, I am afraid, nothing more than an academic elucidation of the principle of "clutching at straws".

Anonymous said...

As if Bach was aware of fractals. Pure coincidence, nothing more.

And I agree with the previous comment. On his web site Brothers dismisses all Fractal Music other than his own as "the worst kind of nonsense", and regards himself as a founder of a "fledgling" genre.

The truth is myself and others have been making fractal music (that fully satisfies his limited definition of the term) since the early 1990s.

Brothers regards himself as being on some kind of crusade to rectify "misconceptions" of what Fractal Music is. The only misconception here is that he thinks people weren't using the techniques he has "invented" as much as fifteen years ago. I published my first album using these techniques in 1998, for example.

This man is a Snake Oil seller of the highest order. You only have to look at his ridiculous gadget patents to see that.

Frankly I am shocked that Fractal Music has produced yet another Bandwagon Jumper, it has produced several. But I am shocked that Benoit Mandelbrot would condescend to be associated with him in public.

Anonymous said...


My apologies, my previous comment was posted anonymously in error. And I obviously omitted a "not" from the opening phrase of my final paragraph. Both a result of me being new to this site, I was expecting a preview.

Phil Thompson
Organised Chaos: The Music of Chaos Theory (1991-2008)

No doubt Mr.Brothers would claim somehow that this music is not fractal according to his "classification". The truth is that it is, it actually exceeds his requirements in ways his own methods cannot produce.

But in 1998 my material was already so more elaborate and sophisticated than his own recent examples that I doubt he will have the ear to be able to perceive it as such.

So I'm afraid his own web-site deserves the award of "nonsense of the worst kind" if any.

Robert Walker said...

I was interested to hear about this research, since my Fractal Tune Smithy program makes "motivic scaling canons"

Many of the pieces that come with Fractal Tune Smithy are strict mensuration canons, and also fractals according to Brothers' definition. All the pieces in the Sloth Canons folder for instance. It has been available for download since the late 1990s, and I first thought of the idea in the 1980s. For some reason it has a low google rating for "fractal music" but a high google rating for "fractal tune".

A strict musical fractal would have to be self similar in the time and pitch dimension at arbitrarily small pitch and time scales. You can do that too with Tune Smithy, and I have a an example on the web site based on a Cantor set here:
That really is a strict musical fractal, based on the Cantor set.

I would agree that there is something fractal going on in much of ordinary music. For instance music often falls into call / answer type bars, then those phrases into four bar patterns, then eight and often to higher numbers too, and that fractal structure seems to appeal to us, for reasons unknown.

I think music should count as fractal even when one can't pick up the fractal easily by ear. Visual fractals are less easily obscured, we pick up the fractals in a fractal landscape and immediately see the "fractal structure" of the clouds, mountains etc, even though it would require intricate analysis to work out what the fractals were that were used to construct them.

In music it is more elusive. You have a feeling of an overall unity and structure to the fractal music even when you can't hear them. You are affected by it and intuit it rather than hear it directly. Many of the tunes that you can make with Tune Smithy are so transformed that it is impossible to pick up the underlying sloth canon by ear, but it is still what gives the music its feeling of unity and structure I believe.

So - I think in music particularly, it is hard to say where strict fractal music ends and fractally inspired music begins. So I feel the use of the word "fractal music" to mean "fractally inspired music" which has become common usage is appropriate here. Meaning, that you aren't expected to be able to pick out the underlying fractal by ear - but the music is constructed in some way using mathematical fractals. If one wants to refer to motivic scaling canons in particular, why not call them just "motivic scaling canons".

See also my post at

Robert Walker said...

Hi there,

I'd also like to mention the fibonacci rhythms and tonescapes that come with Tune Smithy, which have a very direct conection with this interesting analysis of a fractal A B type structure in the Bach 'Cello Suite.

The idea of the fibonacci rhythm dates back to David Canright and his paper on Fibonacci Gamelan Patterns, published in 1/1 in 1990:

FTS also has fractal tonescapes using David Canright's fractal gamelan patterns for the rhythm, and pitches based on the tiles in a row of a Penrose tiling in the pitch domain. These tunes go up in pitch on every long beat and down in pitch on every short beat, or vice versa, with the interval size adjusted to keep the pitch as steady as possible over long time intervals.

A generalisation allows three beat rhythms and tonescapes with a similar structure but with three rhythm units and three interval sizes. These tunes are fractal in both the pitch and time domain, giving another type of motivic canon type fractal.

The pitch domain fibonacci tonescape uses an idea due to Erv Wilson.
See the last page of:

Anonymous said...

I'd like to say for the record that Mr. Thompson's angry comments and the somewhat less volatile comments of the first anonymous poster have no basis in reality. These individuals clearly know nothing of my connection to fractal music and my history with Benoit Mandelbrot. It appears they know only slightly more about my research.

I may be wrong, but my guess is that, as of their postings, neither person actually read the paper that Ivars Peterson wrote about.

It further appears that Mr. Thompson did not read my Web site carefully enough and thus misattributed remarks and unnecessarily worked himself to a fever pitch. Nowhere have I ever claimed to be a "founder of a fledgling genre." What I do claim is that the field of "fractal music analysis" is young and that this is where I am devoting my efforts. Toward this end, I often compose pieces that illustrate a specific fractal characteristic. I should also point out that all of my current research has passed muster with some the undeniably greatest minds in the field of fractal geometry.

Ironically, one of the goals of my work is to be able to illustrate the extent to which some musicians across the ages (like the talented and accomplished Mr. Thompson), may have a natural predilection for composing music that possess truly and mathematically demonstrable fractal characteristics.

Finally, it may be that Mr. Thompson owes not me, but Mr. Peterson an apology. After all, it seems he is claiming that the highly accomplished and universally respected Mr. Peterson has been duped into wasting his valuable time on "snake oil."

Anonymous said...

Hello again. I would like to thank Robert Walker, who offered a subsequent post to Mr. Thompson's. He was kind enough to contact me directly with his questions and with examples of his substantial work generating fractal music.

Anonymous said...

Re: Harlan Brothers' response to my comment.

" In recent years, Harlan united his love of mathematics and music by helping to forge the fledgling field of fractal music. "

…yet this field has existed since the early 1990s or before. “Fledgling”?

Until recently Mr.Brothers also featured the following words on PDF documents for visitors of his web-site to download, print and distribute:

“All books and web-sites on fractal music are best avoided as the worst kind of nonsense.”

On another forum he has recently attempted to deny responsibility for this utterly arrogant and deeply offensive remark by blaming his colleagues at Yale. Whether they are Yale’s words, or his, he was fully endorsing them by featuring them on his web-site and frankly, this remark is both inaccurate, and disgusting.

I resent Mr.Brothers’ insinuation that I am some kind of “hot head”, I am merely appalled at his repugnant level of ignorance and arrogance towards those who have done much to establish this field long before he appeared to take an interest in it.

His web-site also insinuates that all fractal music related web-sites listed in Google are guilty of promoting “misconceptions” that render the phrase “relatively meaningless.” And then goes on to highlight “common errors” –
and makes observations that myself and others in the genre have been aware of for at least a decade, errors we have not been making - as he would have known had he bothered to ask one of us before making such a claim.

And then he goes on to advocate techniques for making fractal music as if they are his discovery, when the fact is I have been using these techniques since 1984 and as shown above, they have featured in the software and music of others also, for many years before he began to use them.

Re: Bach. If you read the following article regarding some of my previous work by Dr.David Whitehouse, Science Editor, BBC, you will find such “research” is hardly a revelation…

As someone who has been tasked by Benoit Mandelbrot to undertake a “rigorous treatment” of fractal music and has allegedly been doing so since 2004, I find it very curious that Mr.Brothers appears to have been blissfully unaware of my name or work until I posted my initial comment to this blog. Whilst I am under no illusion that I am world famous my name and music is VERY well known in the field, and has been for over a decade. Long before Mr.Brothers appeared to show any interest in this subject. And I’m sure that he would have found my name mentioned MANY times if he had actually read just a few of the web-sites he has been dismissing out of hand as “the worst kind of nonsense.” His interpretation of “rigorous treatment” therefore falls far short of mine, and I doubt this is what Benoit Mandelbrot had in mind when he tasked Mr.Brothers to perform this task.

Finally, my “Snake Oil” remark was regarding his ridiculous patents as much as his attitude to the fractal music field. But I’m sure when you consider the above readers will understand where I was coming from.

As a result of this disgrace I have now abandoned using fractal formulae in my music permanently, because I am disgusted by Harlan Brothers and Yale for their scandalous attitude regarding others in this field, and no longer wish my name or work to be associated with anything they are doing in the name of “fractal music”, even in error.

With respect to Mr.Peterson, who I'm sure is acting in all good faith,

Phil Thompson

Anonymous said...

Mr. Thompson should be aware of the fact that the use of the word "fledgling" on my Web site refers to "fractal music analysis" and not "fractal music." Even if he were not, I state plainly in my introductory page on the subject that I believe fractal music is at least six centuries old. It is therefore not a credible claim on Mr. Thompson's part to say that somehow I think fractal music is new. In his determination to prove his false assertion, it seems he has scoured the Web to find a third party error on another site. I thank him for finding the error and I have requested that it be corrected by adding the word "analysis."

Furthermore, I have not recently changed any downloadable material regarding my Fractal Music Workshops as anyone with Adobe Reader can verify. Nowhere does his "quote" appear. The following quote does appear on the Yale site for fractal geometry (

"Beware of books or webpages about fractal music: many of these are the worst kind of nonsense."

Comparing this actual quote with Mr. Thompson's version of the quote:

“All books and web-sites on fractal music are best avoided as the worst kind of nonsense”

one can glean some insight into the nature of the problem here.

Regardless of the provenance of the actual quote, the truth is that, from a mathematical perspective, there is "nonsense" out there.

As to the general tone of Mr. Thompson's approach, readers can come to their own conclusions by going to:

The history of our exchange begins with Mr. Thompson's post number 11805. Another member was thoughtful enough to contact me directly and apprise me of the situation. For those who don't wish to slog through the entire ordeal, my good faith responses appear here:


I later offered posts 11846 and 11853 as part of a continuing discussion.

Post 11822 speaks specifically of my both my knowledge of and respect for Mr. Thompson’s music. Nonetheless, he has somehow convinced himself that my comments regarding "misconceptions" are directed at him.

It should be clear to him by now that my "rigorous treatment of fractal music" deals with developing the math and methods of analysis. The word "rigorous" here refers to a precise and mathematically consistent approach to the subject. This is not the same as a treatment or survey of everything that is currently referred to as "fractal music." To help insure that there are no further misunderstandings, I now use the phrase "mathematically rigorous."

The point here is that Mr. Thompson's music, or that of anyone else, may or may not possess fractal characteristics in the sense intended by Mandelbrot and generally applied to images and objects of nature. Without models, tools, and techniques for analysis, particularly for pre-Mandelbrot music, it is impossible to know with certainty.

My Web site is a work in progress. In addition to providing an introduction, background information, and examples of different types of musical power-law relationships, it references well-known modern composers who have shared or published their methods for producing music that has a mathematically demonstrable form of fractal structure.

The fact that Mr. Thompson's name does not currently appear on my Web site is neither a personal slight nor an inadvertent oversight. Whether or not he thinks (or previously thought) that it is important to differentiate between music that is simply fractal-based and music that demonstrably exhibits fractal structure, such distinctions are fundamental to the many researchers like myself who wish to learn about the ways that fractal geometry manifests itself in human endeavors.

I welcome discussion of the subject of fractal music and its analysis and I have enjoyed discourse with mathematicians and musicians around the world. However, such discussion must be civil, respectful, and rational.

The invective that Mr. Thompson has directed at me, along with his less-than-flattering assessment of my character, are easily dispelled by visiting my Web site: At this point, Mr. Thompson and I have had sufficient communication on this subject for us, or anyone else, to understand our respective points of view.

As I have written before (see post 11822 at cnfractal_music), I believe that Mr. Thompson is talented and accomplished and that he and I have more in common than he wishes to acknowledge. It would be a welcome change if we could build on our commonalities and our shared love of music and mathematics.

Harlan Brothers

Anonymous said...

I actually think the existence of fractals in famous pieces of music, past and present, is undeniable.

Obvious symmetry may not be present, however, through my study of sheet music and musical performance, I've discovered that "classic" songs almost always incorporate some form of imitation counterpoint.

In other words, all the rhythms and melodic content within a song are just altered manifestations pattern established in the intro or soon there after.

puremusic said...

Thank you to all who have posted informative comments. I'm convinced that much more goes on in the human psyche, than we have conscious access to - both as creators and as appreciators. The mathematical patterns are, indeed, remarkable.

But I stopped reading when the posts turned, apparently, into a battle of the egos. Can't you indulge in this kind of thing more privately? Ah, but then I don't suppose you'd have an audience?

Here's to the music!

Unknown said...

Bravo puremusic! Agreed!

Anonymous said...

Yeah, there's an intermediate position between "Bach knew fractal theory" and "researchers find whatever they want in pure coincidence". There's a fractal dimension to the phenomenology of listening. As we listen to music, we are following musical activity of multiple cohabiting timescales: we hear a given motive "by itself" as it is introduced, but we also hear it as part of broader phrases, until, at the extreme end, you get someone like Adorno who wanted every minute gesture to be justified in light of the development of the total work (in his essay on popular music). So doesn't it make sense that a composer who is already very sensitive to architectural, structural aspects, would be able to write something like this?

That fractals are innate, rather than knowable only through the technical developments of a certain period, makes sense when you consider that Mandelbrot didn't discover fractals. Fractals have signified the infinite in the finite in indigenous African designs since long before the 20th century--search for "African Fractals". And traditional Javanese gamelan music displays structural and melodic scaling on at least as many levels as this Bach example.

Of course, none of these are truly fractal: by definition, a fractal displays detail on all (infinite) levels.