72edo

The high divisibility of Pythagorean intervals in 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.

Approximation of prime harmonics in 72edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-2.0	-3.0	-2.2	-1.3	-7.2	-5.0	+2.5	+5.1	+3.8	+5.0
	Relative (%)	0.0	-11.7	-17.9	-13.0	-7.9	-43.2	-29.7	+14.9	+30.4	+22.5	+29.8
Steps (reduced)		72 (0)	114 (42)	167 (23)	202 (58)	249 (33)	266 (50)	294 (6)	306 (18)	326 (38)	350 (62)	357 (69)

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.

Notation

Ups and downs

72edo may be notated with ups and downs combined with neutral diatonic notation; this means that all notes can be reached in at most one neutral diatonic accidental and one ups-and-downs accidental.

Just notation systems

72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo and prime-factor Sagittal, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS and Ben Johnston.

Steps	Neutral FJS	Sagittal/Color/FJS/HEJI
0	natural
1	5-comma up
2	7-comma up
3	semisharp	11 M-diesis up
4	7-comma down sharp
5	5-comma down sharp
6	sharp

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.