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Let me apologize first as I see this may be way off topic. Still it is a really fun question I've been meaning to ask a few fellow grads/faculty members, and so I think it's worth a shot here.

I'm interested in suggestions for using math formulas or concepts in coding algorithmic music.

In Stephen Cope's Workshop in Algorithmic Computer Music in 2004 I was introduced to the art of algorithmic composition, through coding LISP to generate midi compositions, frequently using markov chains to weight transitions, from the large scale harmonic progression and rhythmic structure to the individual notes and their time values.

We played a bit with simple math functions for generating simple pieces. One of these I wrote "sonified" the towers of Hanoi. The movement of the kth largest disk generated a bleep of frequency N(2/3)^k, for N some (high) starting frequency. Since 2/3 is roughly the ratio to get the next lower 5th, the I was able to stay roughly in the 12 tone equal temperament, while superimposing the same pulse at (2/3)^k the (tempo and wave) frequency. The piece wasn't particularly interesting musically, but conceptually fun.

In the workshop many other math themes are explored such as cellular automata, genetic algorithms, Brownian motion. I've been thinking since about interesting curves on the orbifold $T^n/\Sigma_n$ ($n$ continuous voices modulo the octave and modulo their labeling), and also about energy functions which give harmonic progressions as geodesics. (Perhaps harmonic functions would be applicable here, after all!)

I wonder what specific examples others have for making interesting pieces of music (art), or vague examples for that matter.

I'm happy to close this off, too, if no one is interested. Sorry for the softy.

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    $\begingroup$ Check out the Journal of Mathematics and Music: informaworld.com/smpp/title~content=t741809807~db=all , also look at the list of editors, look up their home pages and see what they're up to. There's plenty of responses to your question in there. $\endgroup$
    – Ryan Budney
    Commented Dec 1, 2010 at 6:13
  • $\begingroup$ tones.wolfram.com might be of interest as well. $\endgroup$
    – J. M. isn't a mathematician
    Commented Dec 1, 2010 at 6:31
  • $\begingroup$ I have a score for Fermat's Last Fugue, but the character limit for this comment is too small to describe it. (Sorry, someone had to say it.) Seriously though, a long time ago my friend came up with an audio-visual demo of the towers of Hanoi with 7 disks on his computer. It was impressive to me at the time, but for "sonification" he just had a simple scale. I suggested instead (least significant bit to most significant bit) A-B-E-F-C(up octacve)-D-A(down octave). Second voice (rest)-E-A-A-G-B-A. Third voice (rest)-(rest)-C-D-E-G-C. The song still sticks with me as a binary count mnemonic. $\endgroup$
    – Daniel Mehkeri
    Commented Dec 5, 2010 at 19:19

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Sonic versions of sorting algorithms have long been used to help students gain an intuitive feeling for the differences between the algorithms (say, insertion sort vs. bubble sort), including their running times. When combined with visuals, the impact is quite dramatic. Here is an example at Synth Music and Electronics.

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There is Per Norgard's "infinity series" which he used in his Symphony no. 2.

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I messed around with this when I was a C++ TA. I came up with "musical sorting algorithms", "musical Gauss Seidel" and a terrible sounding FFT. Details: http://www.math.ucla.edu/~rcompton/art.html

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A few years ago, Dylan Thurston programmed the following for me (he did it with Haskore).

Take a 1-dimensional aperiodic tiling (Penrose tiling?) and reproduce it in the times axis. Here's a link to the sound file: http://www.staff.science.uu.nl/~henri105/fibonacci.wav. I sounds like:

(short)(long)(short)(long)(long)(short)(long)(short)(long)(long)(short)(long)(long)...

where (short)=1 and (long)=golden number.

I then made some feeble attempts at composing something that follows that basic rythm.

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  • $\begingroup$ It's a great idea. Unfortunately I've tried to open it in 2 browsers, and get error, does the file open for you? $\endgroup$
    – AndrewLMarshall
    Commented Dec 7, 2010 at 8:27
  • $\begingroup$ Is this different than a random sequence of 1's and $\phi$'s ? That generated, say, by intersecting and irrational slope with the coordinate grid with 1 for horizontal and $\phi$ for vertical intercepts? $\endgroup$
    – AndrewLMarshall
    Commented Dec 7, 2010 at 8:30
  • $\begingroup$ @AndrewMarshall: I made the file executable. Maybe that was the problem. @AndrewMarshall: Your description produces the same sequence of 1's and $\phi$'s. $\endgroup$
    – André Henriques
    Commented Dec 8, 2010 at 21:00
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OEIS, The Movie find information at http://oeis.org/Seis.html

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Disclaimer: I will report from my point of view, which might be boring to some.

I have been experimenting with algorithmic music from the mathematical point of view since Covid, not so much from music point of view, and I would like to point to a positive definite kernel for measuring the consonance of two pitches: $$k(a,b) = \frac{\gcd(a,b)^2}{ab}$$ The description can be find here. From the "sonification of math" point of view, I would like to mention the #InfinitePiChallenge where there are given infinitely many formulas for calculating $\pi$ and the challenge is to pick one formula and sonify it. I did that with a few formulas, two of which I like to highlight:

Convergence I, #InfinitePiChallenge for Violin and Cello

Convergence II, #InfinitePiChallenge for Piano

Here you can find a graph theoretic approach to music generation with prime numbers.

And here you can find an experiment with roots of unity sonified with supercollider.

Do not forget the subtle sonification of some prime numbers, which I have forgotten how I sonified it.

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