I am interested in the notions of polyrhythm, cross-rhythm, and hypermeter, including acquiring more mathematically based concepts that can aid in quality music production. I have done extensive reading of papers which claim to unlock the mysteries of what makes a rhythm "good". The most generic, by far, is the paper by GodFried T. Toussaint, which identifies the Euclidean algorithm as being central to the intuitive human generation of good rhythm. The Euclidean algorithm has other applications such as computing the greatest common divisor between two numbers, so it is quite amazing that this has a strong connection to polyrhythm.

An additional useful notion is that of interlocking rhythm or cross-rhythm. Again, the mathematical workhorse for generating "groovy" cross-rhythms is Euclid's algorithm, however, now it is about superimposing two Euclidean rhythms in such a way that the result is pleasing to the ear.

What the Euclidean concept omits to include are the notions of hypermeter. All polyrhythms/cross rhythms generated by this method are essentially for one measure (if we think in terms of score 4/4 notation). The rhythm is pleasant for one measure, but looping it endlessly can by no means generate an interesting track.

I am particularly interested in electronic music such as house and tech-house. A common structure is hypermeter where a 4-loop is superimposed on an 8 loop which is superimposed on a 16 loop. Indeed one could think of the entire track as being a loop in and of itself which can be repeated.

The Euclidean concept does not account for such structures, yet they are extremely common in electronic music. Are there formalized mathematical structures that tie the Euclidean algorithm to hypermetric structures? If not what other mathematical concepts are relevant to such structures?

Note I am mostly interested in working with 4/4 tracks which have a 3:4 polyrhythm. Basically most pop/electronic music uses this paradigm.

  • 2
    Take a look at the Schillinger system for one person's view of a "unified field theory of rhythm": en.wikipedia.org/wiki/Schillinger_System fransabsil.nl/archpdf/rhythm.pdf
    – lightning
    Aug 3, 2018 at 19:26
  • Personally, I don't use anything mathematically sophisticated. I just take a large non-prime number (for example, 60) and find its factors (2*2*3*5). I then break 60 equally long notes into groups based on those factors. With 60 you could do 2, or 3, or 5, or 4, or 15, etc. If you want rocket science, you could look into Indian polyrhythms or any academic research into Frank Zappa, or Mr. Bungle, or Meshuggah.
    – Pyromonk
    Aug 4, 2018 at 0:33
  • @lightning: About 4 years ago I delved deeply into Schillinger, ordering the original (rare) books from the library and studying the book of rhythm. I recall a few strong arguments, including the idea of symmetry (such as in rhythms like 3+1+2+2+2+2+1+3, and applying operators on rhythms such as permutation, etc. However I remember no mention of Euclid's algorithm. Thanks for the document, it seems to be more formalized and better explained than the original itself. It might be time for me to go back to Schillinger and see if its worth anything again
    – user32882
    Aug 4, 2018 at 8:42
  • Try studying Indian percussion.
    – user50691
    Dec 29, 2018 at 14:52

1 Answer 1


the Indian rhythm system called konnokol gives a very good approach on applying rhythm in a musical and oral context. It connects the improvisation by using spoken rhythms in combination with keeping control with the clapping of your hands.

For example try this:

Speak "ta ki ta" repeatedly representing 3 notes with equal length. 
clap on every 1st, 4th, 7th etc. note that you speak. You place accents this way.

you will get a 3/4 beat, looking like this:

>        >        >        >        >
ta ki ta ta ki ta ta ki ta ta ki ta ta

Now, change the accentuation every four notes:

>           >           >           >
ta ki ta ta ki ta ta ki ta ta ki ta ta

This way konnokol can get you any rhythm and feel in all metres and note counts.

  • While this is a nice overview of the Indian rhythm system, it does not by any means answer the original question.... But thanks for your input
    – user32882
    Jun 24, 2019 at 16:24

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