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mathoverflowUser
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I would like to add one technique based on a positive definite kernel over the natural numbers:

$$k(a,b) = \frac{\gcd(a,b)^2}{ab}$$ and $$k(a,b) = \frac{\min(a,b)}{\max(a,b)}$$

Details of this method are described here.

Music generated by this method might be found here as an example:

https://www.youtube.com/watch?v=EbrEBeDqq24

or here

https://musescore1983.bandcamp.com/track/for-us

Disclaimer: I made all this... :-)

Edit: I have done a website to compose "parametric" music, which is based on positive definite kernels over the natural numbers. Here is an example, which I call

"Thinking and Inventing"

Parameters:

octaves = 3;4;3;5;2;3

rests = 0,1;0,1;0,1;0,1;0,1;0,1

durations: 0.125,0.25;0.25,0.125;0.25,0.5;0.5,0.25;0.125,0.5;0.5,0.125

neighbors = 5

cycles = 60

reverse = [x]

weights = 2.0,3.0,4.0,1.6;2.0,3.0,4.0,1.6;2.0,3.0,4.0,1.6;2.0,3.0,4.0,1.6;2.0,3.0,4.0,1.6;2.0,3.0,4.0,1.6

Audio:

https://drive.google.com/file/d/1dVRajNbXIxaCB6ehXbaP4uCQ4HJjUP6w/view?usp=sharing

Score:

https://drive.google.com/file/d/1S2PlKXOac0eXITD5P1H73PMp28ptx8UG/view?usp=sharing

If you want to try it out yourself: The easiest way to start is to change the number of neighbors to say 4,6,8,10,15 etc. and see the result.

mathoverflowUser
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