Euclidean rhythms are exactly the maximally even rhythms — those maximizing pairwise inter-onset distance on the circle of time

Plot a rhythm as k points on a circle of n equally spaced positions. It is maximally even if the sum of all pairwise straight-line distances between the onset points is as large as possible. Demaine et al. (2005) proved a rhythm is maximally even if and only if it is a Euclidean rhythm or a rotation of one. This is why Euclidean patterns feel balanced without being uniform: the algorithm pushes each onset as far from the others as the grid allows. Isochronous rhythms achieve maximal evenness trivially; the property is interesting when k and n are relatively prime. Maximal evenness also explains why Indian talas mostly fall outside the family — of the 35 Suladi talas only 4 are maximally even. The principle unifies the geometric, algorithmic, and cross-cultural threads of the paper.

Examples

E(5,8) maximizes inter-onset distance for 5 onsets over 8 steps; compare the clustered [xxxxx…] which does not. Only 4 of 35 Suladi talas are maximally even.

Assessment

Explain why ‘maximally even’ and ‘as evenly spaced as possible’ name the same property. Given gcd(6,9)=3, decide whether E(6,9) is maximally even and justify.