An isochronous rhythm places its onsets at perfectly regular intervals — the trivial case where k divides n

A rhythm is isochronous when its onsets are spaced perfectly evenly, which for a Euclidean rhythm E(k,n) happens exactly when k divides n without remainder — e.g. E(3,12) = [x…x…x…], a plain grid with period 3. Isochronous rhythms are the ‘obvious’ Euclidean rhythms and are common worldwide, but musically uninteresting because every inter-onset interval is identical: four-on-the-floor E(4,4) is the limiting case. The interest in Euclidean rhythm generation comes precisely from the NON-isochronous case, where k and n are relatively prime and the onsets cannot all be equally spaced, forcing an uneven-but-balanced distribution. A subtlety: even isochronous meters in real performance carry slight deviations from perfect regularity, and there is psychological evidence these micro-deviations aid beat-tracking rather than harm it.

Examples

E(3,12) = [x…x…x…] is isochronous (period 3). E(4,4) = [xxxx] four-on-the-floor. Contrast E(3,8) = [x..x..x.], non-isochronous because gcd(3,8)=1.

Assessment

State the divisibility condition on k and n that makes E(k,n) isochronous. Given E(4,12), decide whether it is isochronous and give its period.