The Strange Beauty of Fractal Music

Fractals are abstract mathematical objects which, as such, cannot be seen or heard, but their structure can be used to model computer generation of images that may be visually very attractive. Fractal images have been popular for more than a decade, but I think we are just scratching the surface of the possibilities of fractal music. The compositions on this page aim to explore some of these possibilities, showing that recursive algorithms can be employed to create music of the same strange beauty as the richly decorated fractal picture at the side .

(NEW) Check here the music in MP3 format:    The CD

The pieces are listed newest first. They cover a variety of styles, since "fractal music" doesn't mean a musical genre but rather a method at the composer's disposal for his or her own expressive purposes. In the end, as it happens with any other method of composition, only the musical merits of the result should actually count.

These are strict canons in three voices at the octave in quintuple meter, and should be listened to in sequence, as three movements of a single composition. Pitch-bend is used to get the tones of an Arabic scale (maqam Segah) of the Middle Ages, and instruments were chosen in order to evoke an Oriental atmosphere. Of course, these canons don't pretend to pass for Arabic music (as much as my fractal "jazz" pieces below aren't meant as real jazz), but the result sounded interesting to me. The piece was composed with the help of Mandelbrot Music, by Yo Kubota.

A calm soliloquy for voice accompanied by chromatic harp, strings and soft percussion. It divides into three sections that differ in thematic material but not in tempo, rhythm, instrumentation or general atmosphere, so that the piece always conveys the same mood. The stones are, of course, solitary tombstones washed by a thin rain, on a cold autumn afternoon. The feeling, however, isn't of sadness, just peace and meditation.

This piece is set for jazz ensemble and has a jazz feel overall but the frequent articulation of the two upper voices (flute and vibraphone) in strict imitative patterns, both in direct and inverted forms, isn't, of course, characteristic of jazz. But I find this kind of "contrapuntal jazz" fascinating, and if you liked this piece you may want to hear the jazz arrangement I made of Shostakovich's G sharp minor fugue at my GeoCities page.

If pressed I would describe this piece as a rondo, just because of the recurring F major chorus. There are, however, no clear cut sections, and contrasts are based on texture, not on thematic material. In fact everything is derived from that introductory chorus, whose swinging presence is felt everywhere. I plan to add other movements later to complete the quintet.

This piece belies the idea that the beauty of fractal based music is necessarily strange. There's nothing audibly "mathematical" about it, and its three movements are reminiscent of a certain kind of early 20th Century French chamber music, hence a title in French seemed appropriate...

This was my first successful attempt to put together a fractal composition on a bigger scale with some coherent structure, and it opened my eyes to the organizational capabilities of a program like David Singer's FMusic. It has the form of a loose sonata-allegro, exploring all the MIDI patches for strings, either solo or tutti, and is one of my favorite pieces on this page.

This collection comprises five short pieces in quite different styles, although nos. 2 and 4 have traits in common. A thematic and structural simplicity prevails, but some of them present a refined instrumental texture. They were all composed with the help of Lars Kindermann's MusiNum.

This is a short piece for string orchestra. An atonal, expressionist pathos evolves freely and is sustained throughout, although two thirds toward the end the atmosphere lightens up a little and some luminous glimpses of tonality are momentarily caught before the music returns to the dark atmosphere of the beginning. Composed with the help of David T. Strohbeen's Art Song 2.3

These were my very first experiences in fractal composition; form and style are rather undisciplined, and the result is a kind of aural wallpaper, delectable but with little substance. On the other hand, they would be nearer, perhaps, to what is popularly thought as "fractal music". Mood I was accidentally deleted and exists no more. Beware of Mood II - it lasts more than 30 min - but it may be interesting to listen to a long piece that is always different and always the same, and could go on like that forever. Mood III was originally even longer, but I pruned it considerably, much to its benefit.

Unless otherwise indicated, all pieces were composed with the help of FMusic, by David Singer, a program able to control rather complex structures and with which I am more familiar. FMusic is currently in version 1.9. A new version (2.0) is on the way.

Mandelbrot Music produces music following the sinuosities of the Mandelbrot "mountains" along straight lines. It also allows the mapping of the numeric data onto a wide variety of musical scales. I used it in "Three Canons", my most recent fractal composition.

MusiNum is a cute and intuitive program, ideal for beginners but also very versatile. I used it to compose the five pieces of Set no. 1.

Rhapsody for Strings is the only piece I composed with Art Song. This program is full of resources and I certainly intend to explore it more deeply.

You can find these programs following the links below:

FMusic 1.9   ©1996-1999 by David H. Singer

Mandelbrot Music    ©1997 by Yo Kubota

MusiNum: The Music in the Numbers    ©1995-1998 by Lars Kindermann

Art Song 2.3    ©1995-1996 by David T. Strohbeen

All MIDI files on this page are in GM format and optimally balanced for the SB AWE32 sound card with the 8MB GM E-Mu sound fon. Full copyright of the compositions is retained by me and by the authors of the software employed.

Fractal geometry is a relatively new and complex branch of mathematics, which has many applications in science and art. A good introductory reading is The Beauty of Fractals, by H. O. Peitgen and P. H. Richter (Springer-Verlag Berlin Heidelberg, 1986), with 100+ fractal images and an accessible technical  exposition.

The sci.fractals FAQ  is a good reference for people interested in the subject, and a useful guide to the ever growing amount of informations about fractals available on the Internet.

The best practical way to learn about fractals is to get Fractint, an awesome freeware program for creating fractal images such as the one at the top of this page.

For a wealth of specific information about fractal music, and a large number of interesting links, visit David Strohbeen's Fractal Music Lab.

Fractal Vibes is another important site dedicated to fractals and fractal music, put together by Phil Jackson (formerly "Phil's Fractal Pages"), It has many fractal compositions by Phil Jackson, several galleries of fractal images and a lot of interesting links. It also hosts a mailing list for people interested in the subject.

Phil Thompson's Organised Chaos page has quickly become one of the most complete references on fractal music in the Web. Check also there the progress of the Strange Attractions CD, featuring compositions by ten fractal composers, including my "Old Stones" above.

For all those interested in the general topic of music and mathematics, a  visit to  "The Sound of Mathematics" page by Daniel Cummerow is a real must.

Very original and interesting fractal music based on the Mandelbrot set can be found at the Strange Music Archives, by Yo Kubota.

These are some extracts from my postings on rec.music.compose on the subject of fractal music composition.

[About the "legitimacy" of generative music]

This is a very interesting question. I myself was thinking of asking something of the sort here, because I am very curious about how composers consider this form of composition.

I am definitely a bit uneasy about the status of this kind of music. Since I started writing "generative music" with the help of a few computer programs I carefully kept these pieces separated from my "regular" compositions, on a different Web page, and clearly identified them as such. So, in my mind, no matter how much I try to convince myself that this is just a compositional method like any other, I feel still unsure if this is really so (although I am certain that some of my fractal pieces - like the Sonatina for flute, clarinet and piano - could plausibly foil a Turing test anytime...).

On the other hand, I don't follow the rule that "the whole idea behind generative music is that the composer sets some parameters and then starts a system that produces music without any further human intervention." It's true that I take a lot of time fine-tuning the parameters for a particular piece, but the final result isn't by any means sacred, and I don't hesitate to intervene whenever I think it is appropriate (the "artist's discernment" is the measure of this appropriateness...). I wouldn't go as far as to say that the output of the programs is treated just as raw material, or building blocks, since I always stick to its general outlines in terms of micro- and even macro-structure. But If something pleases me I may put it in evidence and even duplicate it in some other points for structure's sake. If some passages don't work well, I just cut them out. So, my pieces probably shouldn't be called "generative music" after all...

[About generative music being the product of the "algorithm author", not of the person who tweaks the parameters]

This is undoubtedly right in reference to "cool" commercial toys like SSEYO KOAN, which are explicitly advertised as "compose music without knowing anything about composition" (or something in that line). But there are other quite different kinds of programs.

Lars Kindermann's MusiNum is an interesting case because it is simple enough to be used as an immediately entertaining toy by laypersons but has enough depth to allow exploration by more mathematically *and* musically endowed people. Quite impressive results can be obtained, but they demand time, work and brains.

David Singer's FMusicgoes one step further because it has so many possibilities to choose, employ and mix algorithms that the simple notion of "tweaking parameters" isn't faithful to what it demands from the user. You can organize and plan whole sections, link sections recursively to each other, in short, impose a chosen structure to the piece, both at the macro as at the micro levels. Further, it isn't planned to be easy to operate, and acceptable results will only appear after a lot of work dedicated to understand the capabilities of the program (which means acquiring notions of how L-systems and cellular automata work, and especially how to make use of these mathematical structures for expressive musical ends).

On the farther end of the spectre there are programs like David Strobeehn's String Rewrite and Text2MIDI. The first acts just as an ambient for recursive generation of strings of characters starting from axioms and rules defined by the user. The strings then are loaded to the second program which has the capability to translate them into MIDI file, also according to user-defined rules. So, this is a clear case in which one can't speak at all of "parameter tweaking" but of plain invention and implementing of algorithms.

Between KOAN-like toys and D. Strobeehn's Spartan tools there is a large gray area in which we can't say simply that the generated music is a product of, or "belongs" to the creator of the program. Consider that these very creators are frequently surprised at the results users extracted from their programs - they by no means were able to predict these results nor would (necessarily) be able to produce those results themselves. I propose that there is indeed a space for a real creative and musical effort from the user in these cases. There is, here, place for composing (sort of) and not just blind button-pushing.

I'm an engineer turned philosopher. It's been a long way from electronics to the philosophy of logic and mathematics, but throughout it music remained a lifelong interest. I had classical training in theory and composition when young, but it was only recently that I went back to composing, after discovering MIDI and recursion based composition software.

See also:

My MIDI and music page, including my non-fractal compositions.

My page with the complete collection of Shostakovich's Preludes and Fugues opus 87 in MIDI format.