22edo

22edo, or 22 equal divisions of the octave, is the equal tuning featuring steps of (1200/22) ~= 54.5 cents, 22 of which stack to the perfect octave 2/1. It is not a meantone system, but it is a functional 11-limit system, with 3 at ~709 cents, 5 at ~382 cents, 7 at ~982 cents, and 11 at ~545 cents.

Theory

JI approximation

22edo is comparable in accuracy in the 7-limit to 12edo in the 5-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 22edo as 4-3-2-4-3-4-2. However, it also features a MOS diatonic of 4-4-1-4-4-4-1, characteristic of superpyth systems. 22edo can also be seen as an 11-limit system, however due to equating 11/9 with 6/5, it lacks much of the function of prime 11.

Approximation of prime harmonics in 22edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+7.1	-4.5	+13.0	-5.9	-22.3	+4.1	-24.8	+26.3	+6.8	+0.4
	Relative (%)	0.0	+13.1	-8.2	+23.8	-10.7	-41.0	+7.6	-45.4	+48.2	+12.4	+0.8
Steps (reduced)		22 (0)	35 (13)	51 (7)	62 (18)	76 (10)	81 (15)	90 (2)	93 (5)	100 (12)	107 (19)	109 (21)

Thirds in 22edo

Quality	Subminor	Pentaminor	Pentamajor	Supermajor
Cents	273	327	382	436
Just interpretation	7/6	6/5	5/4	9/7

Diatonic thirds are bolded.

Chords

Because it approximates the 7-limit, 22edo supports the harmonic tetrad 4:5:6:7, tuned as [0 7 13 18], and because 5/4 and 7/4 are separated by a perfect semioctave, it also supports an alteration shared with any jubilic temperament in which the 5 and 7 are both flattened by a chroma, resulting in the "minor harmonic tetrad" [0 6 13 17], approximating [1/1 6/5 3/2 12/7]. As a consequence, the distance between 5/4 and 6/5 is narrowed, and the distance between 7/4 and 12/7 is widened.

Scales

A scale in 22edo with similar properties to 12edo's diatonic that takes advantage of the important structural role of the semioctave in the aforementioned tetrads is jaric (2L 8s), with the tuning 2-2-2-2-3-2-2-2-2-3. This means that 22edo can be usefully thought of as not just adding more qualities to existing ordinals, but adding three new ordinals with their own qualities, roughly surrounding 8/7 (the "unilatus"), the semioctave, and 7/4 (the "antilatus"). This has the function of giving the simplest 7-limit intervals their own category separate from sixths and sevenths, much as the simplest 5-limit intervals have their own diatonic category in the form of thirds.

From a diatonic perspective, 22edo has four varieties of third: subminor (7/6, 5\22), pentaminor (6/5, 6\22), pentamajor (5/4, 7\22), and supermajor (9/7, 8\22).

Regular temperaments

22edo shares Superpyth temperament with 27edo, Pajara temperament with 12edo, and Porcupine temperament with 15edo. It shares Keemic temperament with 15edo and 19edo.

Notation

22edo may be notated with diamond-mos notation (or another KISS notation) for jaric, or with ups and downs notation for diatonic. Simple Pythagorean notation may also be used, as in 12edo, although it does not support commatic alterations of interval categories (as is common in microtonal music in order to notate, for example, supermajor triads, or in 22edo's case pentamajor triads).