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.

Note: Interval names in this article use the ADIN system.

Theory

Edostep interpretations

22edo's edostep has the following interpretations in the 2...11.17 subgroup:

• 36/35 (the difference between 5/4 and 9/7)
• 81/80 (the difference between 10/9 and 9/8)
• 25/24 (the difference between 5/4 and 6/5)
• 49/48 (the difference between 8/7 and 7/6)
• 33/32 (the difference between 4/3 and 11/8)
• 34/33 (the difference between 17/16 and 33/32)

22edo tempers out the following commas:

• 64/63 (the difference between 8/7 and 9/8)
• 121/120 (the difference between 12/11 and 11/10)
• 250/243 (making 10/9 half of 6/5)

JI approximation

22edo is comparable in accuracy in the 7-limit to 12edo in the 5-limit. Even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic.

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)

From a diatonic perspective, 22edo has four varieties of third: subminor at 273c (7/6, 5\22), Nearminor/classical minor at 327c (6/5, 6\22), Nearmajor/classical major at 382c (5/4, 7\22), and supermajor at 436 (9/7, 8\22). Only the first and last are available in the diatonic scale generated by the perfect fifth.

Scales

22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

Pythagorean diatonic

This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)

Zarlino diatonic

A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. This encounters problems with existing familiarity with notation (for example, one of D-G and G-C must now be a flat 5th, called a "wolf fifth" and representing 16/11 as opposed to 3/2), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.

Pajara

Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.

And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.

To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.

Symmetric scale

One possible scale of 22edo, as mentioned previously, is the Pajara[10] decatonic scale, represented as ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤. This scale can be explored here. Below is a chart of its five modes, ordered by rotation. Some names are from Paul Erlich.

	Chart	2	3	4	6	8	9	Mode on fifth	Mode on fourth
Dynamic minor	├─┴─┴─┴─┴──┴─┴─┴─┴─┴──┤	minor	minor	dim	perfect	minor	minor
Static minor	├─┴─┴─┴──┴─┴─┴─┴─┴──┴─┤	minor	minor	perfect	perfect	minor	major
Static major	├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤	minor	major	perfect	perfect	major	major
Dynamic major	├─┴──┴─┴─┴─┴─┴──┴─┴─┴─┤	major	major	perfect	perfect	major	major
Augmented	├──┴─┴─┴─┴─┴──┴─┴─┴─┴─┤	major	major	perfect	aug	major	major

Pentachordal scale

This scale is constructed from two identical "pentachords" and the semioctave, and is represented as ├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:

	Chart	2	3	4	6	8	9	Mode on fifth	Mode on fourth
Bediyic	├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤	minor	minor	dim	perfect	minor	minor	Hininic	-
Alternate minor (Skoronic)	├─┴─┴─┴─┴──┴─┴─┴─┴──┴─┤	minor	minor	dim	perfect	minor	major	Aujalic	-
Moriolic	├─┴─┴─┴──┴─┴─┴─┴──┴─┴─┤	minor	minor	perfect	perfect	major	major	Mielauic	Hininic
Standard major (Staimosic)	├─┴─┴──┴─┴─┴─┴──┴─┴─┴─┤	minor	major	perfect	perfect	major	major	Prathuic	Aujalic
Sebaic	├─┴──┴─┴─┴─┴──┴─┴─┴─┴─┤	major	major	perfect	aug	major	major	-	Mielauic
Awanic	├──┴─┴─┴─┴──┴─┴─┴─┴─┴─┤	major	major	perfect	aug	major	major	-	Prathuic
Standard minor (Hininic)	├─┴─┴─┴──┴─┴─┴─┴─┴─┴──┤	minor	minor	perfect	perfect	minor	minor	Moriolic	Bediyic
Aujalic	├─┴─┴──┴─┴─┴─┴─┴─┴──┴─┤	minor	major	perfect	perfect	minor	major	Staimosic	Skoronic
Alternate major (Kielauic)	├─┴──┴─┴─┴─┴─┴─┴──┴─┴─┤	major	major	perfect	perfect	major	major	Sebaic	Moriolic
Prathuic	├──┴─┴─┴─┴─┴─┴──┴─┴─┴─┤	major	major	perfect	perfect	major	major	Awanic	Staimosic

Some names are from Paul Erlich.

Blackdye

Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an aberrisma, which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between.

Other scales

Name	Chart	Notes
Onyx	├──┴──┴──┴───┴──┴──┴──┤	Approximate Greek scale (equable diatonic), basic MOS of Porcupine.
Gramitonic (4L5s)	├──┴─┴──┴─┴──┴─┴──┴─┴─┤	Basic MOS of Orwell temperament.
Zarlino diatonic	├─┴───┴──┴───┴─┴───┴──┤	Greek scale (intense diatonic). Zarlino rank-3 diatonic.
Mosdiatonic	├┴───┴───┴───┴┴───┴───┤	Greek scale (Pythagorean or Archytas diatonic). Basic MOS of Superpyth.
Zarlino pentatonic	├─────┴──┴───┴─────┴──┤	One possible pentatonic analog to the Zarlino diatonic.
Pentic	├────┴───┴───┴────┴───┤	Basic MOS of Superpyth

Comparison of notation systems

In all proposed notation systems except Mosdiatonic, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. For Mosdiatonic, a sharp corresponds to +3 EDO steps while a flat corresponds to -3. The Mosdiatonic system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step. Unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff).

Outside of the ADIN system, the "Nearminor" and "Nearmajor" intervals may be called "classic", "classical", or "ptolemaic" major/minor.

Edostep	Cents	Interval name (ADIN)	Just Intonation	Pajara	Mosdiatonic	Vector's Blackdye/Zarlino
0	0	Perfect 1sn	1/1	0	C	C
1	55	Subminor 2nd	36/35, 25/24	1b	^C, Db	C#
2	109	Nearminor 2nd	16/15	1	vC#, ^Db	Db
3	164	Nearmajor 2nd	10/9	1#	C#, vD	D
4	218	Supermajor 2nd	9/8, 8/7	2	D	D#
5	273	Subminor 3rd	7/6	2#	^D, Eb	Ebb / Dx
6	327	Nearminor 3rd	6/5	3b	vD#, ^Eb	Eb
7	382	Nearmajor 3rd	5/4	3	D#, vE	E
8	436	Supermajor 3rd	9/7	4b	E	E#
9	491	Perfect 4th	4/3	4	F	F
10	545	Near 4th	11/8	4#	^F, Gb	F#
11	600	Tritone	7/5, 10/7	5	vF#, ^Gb	Gbb / Fx
12	655	Near 5th	16/11	6b	F#, vG	Gb
13	709	Perfect 5th	3/2	6	G	G
14	764	Subminor 6th	14/9	6#	Ab	G#
15	818	Nearminor 6th	8/5	7	vG#, ^Ab	Ab
16	873	Nearmajor 6th	5/3	7#	G#, vA	A
17	927	Supermajor 6th	12/7	8b	A	A#
18	982	Subminor 7th	7/4, 16/9	8	^A, Bb	Bbb / Ax
19	1036	Nearminor 7th	9/5	9b	vA#, ^Bb	Bb
20	1091	Nearmajor 7th	15/8	9	A#, vB	B
21	1145	Supermajor 7th	48/25, 35/18	9#	B	Cb
22	1200	Octave	2/1	10	C	C

Triads and tetrads

Triads bounded by P5

Name	1	2	Bounding interval	Edostep	Chart
Sus4 triad	Perfect 4th	Supermajor 2nd	Perfect 5th	[0 9 13]	├────────┴───┴────────┐
Supermajor triad	Supermajor 3rd	Subminor 3rd	Perfect 5th	[0 8 13]	├───────┴────┴────────┐
Nearmajor triad	Nearmajor 3rd	Nearminor 3rd	Perfect 5th	[0 7 13]	├──────┴─────┴────────┐
Nearminor triad	Nearminor 3rd	Nearmajor 3rd	Perfect 5th	[0 6 13]	├─────┴──────┴────────┐
Subminor triad	Subminor 3rd	Supermajor 3rd	Perfect 5th	[0 5 13]	├────┴───────┴────────┐
Sus2 triad	Supermajor 2nd	Perfect 4th	Perfect 5th	[0 4 13]	├───┴────────┴────────┐

Tetrads with P5th

Harmonic tetrads

These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in Pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.

Name	1	2	3	4	Bounding interval 1	Bounding interval 2	Bounding interval 3	Edostep	Chart
Nearmajor harmonic tetrad	Nearmajor 3rd	Nearminor 3rd	Subminor 3rd	Supermajor 2nd	Perfect 5th	Subminor 7th	Perfect 8ve	[0 7 13 18]	├──────┴─────┴────┴───┤
Nearminor harmonic tetrad	Nearminor 3rd	Nearmajor 3rd	Supermajor 2nd	Subminor 3rd	Perfect 5th	Supermajor 6th	Perfect 8ve	[0 6 13 17]	├─────┴──────┴───┴────┤

Diatonic tetrads

These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.

Name	1	2	3	Bounding interval 1	Bounding interval 2	Edostep	Chart
Supermajor diatonic tetrad	Supermajor 3rd	Subminor 3rd	Supermajor 3rd	Perfect 5th	Supermajor 7th	[0 8 13 21]	├───────┴────┴───────┴┤
Nearmajor diatonic tetrad	Nearmajor 3rd	Nearminor 3rd	Nearmajor 3rd	Perfect 5th	Nearmajor 7th	[0 7 13 20]	├──────┴─────┴──────┴─┤
Nearminor diatonic tetrad	Nearminor 3rd	Nearmajor 3rd	Nearminor 3rd	Perfect 5th	Nearminor 7th	[0 6 13 19]	├─────┴──────┴─────┴──┤
Subminor diatonic tetrad	Subminor 3rd	Supermajor 3rd	Subminor 3rd	Perfect 5th	Subminor 7th	[0 5 13 18]	├────┴───────┴────┴───┤
Sus2 diatonic tetrad	Supermajor 2nd	Perfect 4th	Supermajor 2nd	Perfect 5th	Supermajor 6th	[0 4 13 17]	├───┴────────┴───┴────┤