72edo

72edo, or 72 equal divisions of the octave, is an equal tuning system with a step size of exactly ⅙ of a semitone, or 16⅔ cents. It is a remarkably accurate model of 11-limit just intonation.

Theory

72edo is a superset of 12edo, sharing the same perfect fifth, major second, and other 3-limit intervals, as well as the same approximations of prime harmonics 17 and 19. It also inherits 24edo's prime 11 and 36edo's primes 7 and 13. However, its approximation of prime harmonic 5 is unique compared to all edos lower than it.

Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three 8/7s, leading to 36edo's Slendric scales. Combining these divisions yields Miracle scales, dividing the fifth into six 16/15s.

72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a 2.3.7-subgroup (septal) interval. This makes extending standard diatonic notation straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.

Octave stretch

72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

Use in software

72edo can be achieved (at least to the nearest cent or so) with most 12edo software instruments by using six instances of it that are all detuned ⅙ of a semitone from each other. In some DAWs, all six instances can be controlled from a single MIDI track on the same piano roll view by routing one MIDI channel to each instance.