From gos.ukc.ac.uk!harrier.ukc.ac.uk!ukc!mcvax!uunet!lll-winken!csd4.milw.wisc.edu!mailrus!uflorida!haven!umbc3!motteler Mon Apr 17 18:04:59 BST 1989 Article 249 of comp.music: Path: gos.ukc.ac.uk!harrier.ukc.ac.uk!ukc!mcvax!uunet!lll-winken!csd4.milw.wisc.edu!mailrus!uflorida!haven!umbc3!motteler >From: motteler@umbc3.UMBC.EDU (Howard E. Motteler) Newsgroups: rec.music.classical,comp.music Subject: Re: Fractal Music Summary: basic concepts are simple Keywords: fractal Voss Gardner Message-ID: <1867@umbc3.UMBC.EDU> Date: 5 Apr 89 09:33:00 GMT References: <1683@ncar.ucar.edu> <7840@boulder.Colorado.EDU> Reply-To: motteler@umbc3.umbc.edu (Howard E. Motteler) Followup-To: rec.music.classical Organization: University of Maryland, Baltimore County Lines: 93 Xref: gos.ukc.ac.uk rec.music.classical:2474 comp.music:249 In article <7840@boulder.Colorado.EDU> eesnyder@boulder.Colorado.EDU (Eric E. Snyder) writes: >All this talk about Fractal Music..... Where can I find some? It's easily generated in the privacy of your own home. The following ideas are mostly taken from one of Martin Gardner's "Mathematical Games" columns in Scientific American, from the late seventies. Gardner got his information from Richard F. Voss of the Thomas J. Watson Research Center. (Sorry I don't have the exact date; all I have is a Xerox of the article.) I'll stick with tone sequences, though the same ideas can be applied to other musical parameters. Suppose we chose a scale--diatonic, chromatic, whatever, preferably over at least a couple of octaves, and want to generate a "random" sequence of tones, drawn from this scale. One approach would be to take a spinner or dice, with as many faces as we have tones, and simply spin or roll repeatedly, to get a succession of tones. Voss calls this a "white melody," in analogy to white noise. The key property of this scheme is that successive notes are totally independent. An alternate approach would be to pick a starting point, and use a spinner numbered, e.g., -2, -1, 0, +1, +2, or whatever range is desired. Successive spins give the direction to go, for the next note: up 2, down 1, etc. Voss calls this a "brown melody," in analogy with Brownian motion. With this scheme, successive notes depend very strongly on previous notes. White and Brownian noise are both "scaling noise," and so in some sense, either white or brown music is fractal music, although "fractal music" is generally taken to be a sort of compromise between the two. The idea of "fractal music" is to get a sequence that is neither uncorrelated nor too strongly correlated. (Technically, white noise has a spectral density of 1/(f^0), brownian noise a spectral density of 1/(f^2), and what we would like is a tone generation scheme that corresponds to noise with a spectral density of 1/f.) The following scheme will come very close to generating a "fractal melody," in this case 16 notes long, drawn from a scale of 16 notes. Suppose we have 3 dice, red, green and blue. Our 16 notes will correspond to the values 3 thru 18, that we can get from the dice. Write out the numbers 0 to 7 twice in binary, and label the binary digits R, G, and B: step R G B 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1 8-15 Repeat the above bit pattern To start, roll all 3 dice and add their sum; this is the first note. Then roll only the B die, leaving the other two alone, and again add all three; this is the second note. Then roll the G and B dice, leaving the R die alone, add all three for the third note. In general, successive notes are obtained by rolling only those dice corresponding to digit changes-- the B die is rolled every time, the G die every other time, etc. The dice in the higher bit positions will contribute to long term variations, and the dice in the low bit positions give short term variation. Voss claims that after quite a few tests, most listeners found white music too random, brown music too correlated, and the 1/f music "just about right." Regarding this, Gardner says It is commonplace in musical criticism to say that we enjoy good music because it offers a mixture of order and surprise. How could it be otherwise? Surprise would not be surprise if there were not sufficient order for us to anticipate what is likely to come next. If we guess too accurately, say in listening to a tune that is no more than walking up and down the keyboard in one step intervals, there is no surprise at all. Good music, like a person's life or the pageant of history, is a wondrous mixture of expectation and unanticipated turns. There is nothing new about this insight, but what Voss has done is to suggest a mathematical measure for the mixture. He (Gardner) also says, towards the end of the article No one pretends, of course, that stochastic 1/f music, even with added transition and rejection rules, can compete with the music of good composers. Well, he hadn't met Skip Agrade, yet! -- Howard E. Motteler | Dept. of Computer Science motteler@umbc3.umbc.edu | UMBC, Catonsville, MD 21228