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Article 249 of comp.music:
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>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
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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