Imagine that we have a rhythm consisting of an 8 pulse structure with 3 onsets which we want to have as evenly spaced as possible in between the silent pulses. One way of accomplishing such a task would be to organise the pulses in such a way that all the onsets are at the start and all the silent pulses are at the end of the rhythm before we start arranging them in a better way so our starting point would then be:
If we define the number of pulses in this example as x and the number of silent pulses as y then x mod y = 8 mod 5 = 3 will give us the number of onsets in this structure. Comparing this to Euclid’s algorithm example from part 3 of the Tresillo blog series we can see that this is equivalent to steps 1 and 2 in the algorithm.
If we now try to put one silent pulse after every sounded pulse then we end up with the following three groupings of sounded and silent pulses as well as a remainder of two silent pulses:
This is therefore equivalent to 5 groupings overall (x) and 3 sounded and silent pulse groupings (y) with the remainder being 5 mod 3 = 2 so 2 is the number of silent pulses we have not yet attributed (z). Therefore at this stage we obtain the same answers as for steps 3 and 4 in Euclid’s algorithm.
Finally, if we now distribute the final two silent onsets we receive 2 equivalent groupings of three pulses (y) and 1 grouping of two pulses (z) so we are left with 3 groupings overall (x) which is equivalent to steps 5 and 6 in the process illustrated in the last post of this series:
It is here that we can see some remarkable discoveries: Firstly, by trying to evenly distribute 3 onsets among 5 silent pulses we have effectively created the Tresillo rhythm which is prevalent or at least implied in any salsa song as part of the son clave. Secondly, the process we have used to try and distribute the onsets evenly among the silent pulses is equivalent to Euclid’s algorithm as demonstrated by this example which is why Toussaint gave this type of rhythm the name Euclidean rhythm.
Toussaint demonstrates that many of the world’s most popular rhythms are in fact Euclidean rhythms although arguable the Tresillo rhythm is the most famous example. A Euclidean rhythm can be completely defined as a function (E) of the number of pulses (p) and the number of onsets (o) and is usually denoted E(o,p). Therefore, the Tresillo rhythm can simply be written as a Euclidean rhythm with the shorthand notation E(3,8).
