# 22edo

(Redirected from 22-edo)
 ← 21edo 22edo 23edo →
Prime factorization 2 × 11
Step size 54.5455¢
Fifth 13\22 (709.091¢)
Semitones (A1:m2) 3:1 (163.6¢ : 54.55¢)
Consistency limit 11
Distinct consistency limit 5
Special properties
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22 equal divisions of the octave (abbreviated 22edo or 22ed2), also called 22-tone equal temperament (22tet) or 22 equal temperament (22et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 22 equal parts of about 54.5 ¢ each. Each step represents a frequency ratio of 21/22, or the 22nd root of 2. Because it distinguishes 10/9 and 9/8, it is not a meantone system.

## Theory

### Prime harmonics

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)

### History

The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist R. H. M. Bosanquet. Inspired by the division of the octave into 22 unequal parts in the music theory of India, Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after 19edo, and J. Murray Barbour in his classic survey of tuning history, Tuning and Temperament.

### Overview to JI approximation quality

The 22edo system is in fact the third equal division, after 12 and 19, which is capable of approximating the 5-limit to within a TE error of 4 cents/oct. While not an integral or gap EDO it at least qualifies as a zeta peak. Moreover, there is more to it than just the 5-limit; unlike 12 or 19, it is able to approximate the 7- and 11-limit to within 3 cents/oct of error. While 31edo does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the 11-odd-limit consistently. Furthermore, 22edo, unlike 12 and 19, is not a meantone system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

22edo can also be treated as adding harmonics 3 and 5 to 11edo's 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 subgroup temperament. Let us also mind its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma (64/63), and since it tunes the septimal supermajor third (9/7) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended quarter-comma meantone.

### Subsets and supersets

As 22 is divisible by 11, a 22edo instrument can play any music in 11edo, in the same way that 12edo can play 6edo (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In Sagittal notation, 11 can be notated as every other note of 22.

## Intervals

Degree Cents Approximate Ratios[note 1] Ups and Downs Notation SKULO notation (K = 1) Audio
0 0.000 1/1 perfect unison P1 D perfect unison P1 D
1 54.545 36/35, 34/33, 33/32, 32/31 up-unison, minor 2nd ^1, m2 ^D, Eb comma-wide unison, minor 2nd K1, m2 KD, Eb
2 109.091 18/17, 17/16, 16/15, 15/14 downaug 1sn, upminor 2nd vA1, ^m2 vD#, ^Eb classic minor 2nd Km2 KEb
3 163.636 12/11, 11/10, 10/9 aug 1sn, downmajor 2nd A1, vM2 D#, vE classic/comma-narrow major 2nd kM2 kE
4 218.182 9/8, 17/15, 8/7 major 2nd M2 E major 2nd M2 E
5 272.727 20/17, 7/6 minor 3rd m3 F minor 3rd m3 F
6 327.273 6/5, 17/14, 11/9 upminor 3rd ^m3 ^F classic minor 3rd Km3 KF
7 381.818 5/4, 96/77 downmajor 3rd vM3 vF# classic major 3rd kM3 kF#
8 436.364 14/11, 9/7, 22/17 major 3rd M3 F# major 3rd M3 F#
9 490.909 4/3 perfect 4th P4 G perfect 4th P4 G
10 545.455 15/11, 11/8 up-4th, dim 5th ^4, d5 ^G, Ab comma-wide 4th K4 KG
11 600.000 7/5, 24/17, 17/12, 10/7 downaug 4th, updim 5th vA4, ^d5 vG#, ^Ab comma-narrow augmented 4th
comma-wide diminished 5th
kA4
Kd5
kG#, KAb
12 654.545 16/11, 22/15 aug 4th, down-5th A4, v5 G#, vA comma-narrow 5th k5 kA
13 709.091 3/2 perfect 5th P5 A perfect 5th P5 A
14 763.636 17/11, 14/9, 11/7 minor 6th m6 Bb minor 6th m6 Bb
15 818.182 8/5, 77/48 upminor 6th ^m6 ^Bb classic minor 6th Km6 KBb
16 872.727 18/11, 28/17, 5/3 downmajor 6th vM6 vB classic major 6th kM6 kB
17 927.273 17/10, 12/7 major 6th M6 B major 6th M6 B
18 981.818 7/4, 30/17, 16/9 minor 7th m7 C minor 7th m7 C
19 1036.364 9/5, 11/6, 20/11 upminor 7th, dim 8ve ^m7, d8 ^C, Db classic minor 7th Km7 kC
20 1090.909 28/15, 15/8, 32/17, 17/9 downmajor 7th, updim 8ve vM7, ^d8 vC#, ^Db classic major 7th kM7 kC#
21 1145.455 31/16, 64/33, 33/17, 35/18 major 7th, down 8ve M7, v8 C#, vD major 7th / comma-narrow 8ve M7 / k8 C#, kD
22 1200.000 2/1 perfect octave P8 D perfect 8ve P8 D

## Notation

### Ups and downs notation

Standard Pythagorean chain-of-fifths notation can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that E♭ and D♯ are different notes and that E♭ is significantly lower in pitch than D♯.

Notation of 22edo
Degree Cents Ups and Downs Notation
Diatonic Interval Names Note Names
on D
0 0.00 Perfect unison (P1) D
1 54.545 Minor second (m2)
Up-unison (^1)
Eb
^D
2 109.091 Upminor 2nd (^m2)
Down-augmented unison (vA1)
Diminished third (d3)
^Eb
vD#
Fb
3 163.636 Downmajor second (vM2)
Augmented unison (A1)
vE
D#
4 218.182 Major second (M2)
Up-augmented unison (^A1)
Downminor third (vm3)
E
^D#
vF
5 272.727 Upmajor second (^M2)
Minor third (m3)
^E
F
6 327.273 Upminor third (^m3)
Diminished fourth (d4)
^F
Gb
7 381.818 Downmajor third (vM3)
Augmented second (A2)
Up-diminished fourth (^d4)
vF#
E#
^Gb
8 436.364 Major third (M3)
Up-augmented second (^A2)
Down-fourth (v4)
F#
^E#
vG
9 490.909 Perfect fourth (P4) G
10 545.455 Up-fourth (^4)
Diminished fifth (d5)
^G
Ab
11 600.000 Down-augmented fourth (vA4)
Up-diminished fifth (^d5)
vG#
^Ab
12 654.545 Augmented fourth (A4)
Down-fifth (v5)
G#
vA
13 709.091 Perfect fifth (P5) A
14 763.636 Up-fifth (^5)
Minor sixth (m6)
^A
Bb
15 818.182 Down-augmented fifth (vA5)
Upminor sixth (^m6)
vA#
^Bb
16 872.727 Augmented fifth (A5)
Downmajor sixth (vM6)
A#
vB
17 927.273 Major sixth (M6)
Up-augmented fifth (^A5)
Downminor seventh (vm7)
B
^A#
vC
18 981.818 Minor seventh (m7)
Upmajor sixth (^M6)
Down-diminished octave (vd8)
C
^B
vDb
19 1036.364 Upminor seventh (^m7)
Diminished octave (d8)
^C
Db
20 1090.909 Downmajor seventh (vM7)
Up-diminished octave (^d8)
Augmented sixth (A6)
vC#
^Db
B#
21 1145.455 Major seventh (M7)
Down-octave (v8)
C#
vD
22 1200.000 Perfect octave (P8) D

Treating ups and downs as "fused" with sharps and flats, and never appearing separately:

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.

Alternatively, arrow accidentals from Helmholtz–Ellis notation can be used instead of independent ups and downs:

 Step Offset Sharp Symbol Flat Symbol 0 1 2 3 4 5 6 7

Shown below is Paul Erlich's "Tibia" in G, with independent ups and downs.

### Superpyth/Porcupine Notation

Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.

### Porcupine Notation

Porcupine Notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.

The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.

### Pentatonic Notation

In Pentatonic Notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals.

The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.

### Decatonic Notation

The Decatonic Notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.

Chain 1: C G D A E

Chain 2: γ δ α ε β

The alphabet is, in ascending order: C δ D ε E γ G α A β C

In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.

### Sagittal Notation

When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:

This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.

The division of the apotome into three syntonic commas also indicates 22's tempering out of the porcupine comma (which is equivalent to three syntonic commas minus a Pythagorean apotome).

We also have, from the appendix to The Sagittal Songbook by Jacob A. Barton, this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

### Comparison of 22edo notation systems

Degree Cents Superpyth/Porcupine Notation Porcupine Pentatonic Decatonic Ups and Downs SKULO interval names
0 0 Natural Unison 1 perfect unison P1 D perfect unison P1 D natural 1st N1 C perfect unison P1 D perfect unison P1 D
1 55 s-minor second sm2 aug unison A1 D# aug unison A1 D# flat 2nd f2 C#, δb up-unison, minor 2nd ^1, m2 ^D, Eb comma-wide unison, minor 2nd K1, m2 KD, Eb
2 109 p-diminished second pd2 dim 2nd d2 Eb double-aug unison,
double-dim sub3rd
AA1,
dds3
Dx,
Fb3
natural 2nd N2 δ downaug 1sn, upminor 2nd vA1, ^m2 vD#, ^Eb classic minor 2nd Km2 KEb
3 164 p-minor second pm2 perfect 2nd P2 E dim sub3rd ds3 Fbb sharp 2nd, flat 3rd s2, f3 δ#, Db aug 1sn, downmajor 2nd A1, vM2 D#, vE classic/comma-narrow major 2nd kM2 kE
4 218 (s/p) Major second M2 aug 2nd A2 E# minor sub3rd ms3 Fb natural 3rd N3 D major 2nd M2 E major 2nd M2 E
5 273 s-minor third sm3 dim 3rd d3 Fb major sub3rd Ms3 F sharp 3rd s3 D# minor 3rd m3 F minor 3rd m3 F
6 327 p-minor third pm3 minor 3rd m3 F aug sub3rd As3 F# flat 4th f4 εb upminor 3rd ^m3 ^F classic minor 3rd Km3 KF
7 382 p-Major third pM3 major 3rd M3 F# double-aug sub3rd,
double-dim 4thoid
AAs3,
dd4d
Fx,
Gbb
natural 4th N4 ε downmajor 3rd vM3 vF# classic major 3rd kM3 kF#
8 436 s-Major third sM3 aug 3rd, dim 4th A3, d4 Fx, Gb dim 4thoid d4d Gb sharp 4th, flat 5th s4, f5 ε#, Eb major 3rd M3 F# major 3rd M3 F#
9 491 Natural Fourth 4, N4 minor 4th m4 G perfect 4thoid P4d G natural 5th N5 E perfect 4th P4 G perfect 4th P4 G
10 545 p-Major fourth, s-dim fifth pM4, sd5 major 4th M4 G# aug 4thoid A4d G# sharp 5th, flat 6th s5, f6 E#, γb up-4th, dim 5th ^4, d5 ^G, Ab comma-wide 4th K4 KG
11 600 p-Augmented Fourth,
p-diminished Fifth,
Half-Octave
A4, HO aug 4th,
dim 5th
A4, d5 Gx,
Abb
double-aug 4thoid,
double-dim 5thoid
AA4d,
dd5d
Gx,
Abb
natural 6th N6 γ downaug 4th, updim 5th vA4, ^d5 vG#, ^Ab comma-narrow augmented 4th

comma-wide diminished 5th

kA4

Kd5

kG#, KAb
12 655 p-minor Fifth, s-aug Fourth pm5, sA4 minor 5th m5 Ab dim 5thoid d5d Ab sharp 6th, flat 7th s6, f7 γ#, Gb aug 4th, down-5th A4, v5 G#, vA comma-narrow 5th k5 kA
13 709 Natural Fifth 5, N5 major 5th M5 A perfect 5thoid P5d A natural 7th N7 G perfect 5th P5 A perfect 5th P5 A
14 764 s-minor sixth sm6 aug 5th, dim 6th A5, d6 A#, Bbb aug 5thoid A5d A# sharp 7th s7 G# minor 6th m6 Bb minor 6th m6 Bb
15 818 p-minor sixth pm6 minor 6th m6 Bb double-aug 5thoid,
double-dim sub7th
AA5d,
dds7
Ax,
Cb3
flat 8th f8 αb upminor 6th ^m6 ^Bb classic minor 6th Km6 KBb
16 873 p-Major sixth pM6 major 6th M6 B dim sub7th ds7 Cbb natural 8th N8 α downmajor 6th vM6 vB classic major 6th kM6 kB
17 927 s-Major sixth sM6 aug 6th A6 B# minor sub7th ms7 Cb sharp 8th, flat 9th s8, f9 α#, Ab major 6th M6 B major 6th M6 B
18 982 (s/p) minor seventh m7 dim 7th d7 Cb major sub7th Ms7 C natural 9th N9 A minor 7th m7 C minor 7th m7 C
19 1036 p-Major seventh pM7 perfect 7th P7 C aug sub7th As7 C# sharp 9th, flat 10th s9, f10 A#, βb upminor 7th, dim 8ve ^m7, d8 ^C, Db classic minor 7th Km7 kC
20 1091 p-Augmented seventh pA7 aug 7th A7 C# double-aug sub7th,
double-dim octave
AAs7,
dd8
Cx,
Dbb
natural 10th N10 β downmajor 7th, updim 8ve vM7, ^d8 vC#, ^Db classic major 7th kM7 kC#
21 1145 s-Major seventh sM7 dim 8ve d8 Db dim octave d8 Db sharp 10th s10 β#, Cb major 7th, down 8ve M7, v8 C#, vD major 7th / comma-narrow 8ve M7 / k8 C#, kD
22 1200 Octave 8 perfect octave P8 D perfect octave P8 D natural 11th N11 C perfect octave P8 D perfect 8ve P8 D

## Approximation to JI

Selected 17-limit intervals approximated in 22edo

### Interval mappings

The following tables show how 15-odd-limit intervals are represented in 22edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 22edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 2.3
11/10, 20/11 1.368 2.5
15/8, 16/15 2.640 4.8
5/4, 8/5 4.496 8.2
7/6, 12/7 5.856 10.7
11/8, 16/11 5.863 10.7
3/2, 4/3 7.136 13.1
15/11, 22/15 8.504 15.6
15/14, 28/15 10.352 19.0
5/3, 6/5 11.631 21.3
7/4, 8/7 12.992 23.8
11/6, 12/11 12.999 23.8
9/8, 16/9 14.272 26.2
13/11, 22/13 16.482 30.2
7/5, 10/7 17.488 32.1
13/10, 20/13 17.850 32.7
13/9, 18/13 17.928 32.9
9/5, 10/9 18.767 34.4
11/7, 14/11 18.856 34.6
13/7, 14/13 19.207 35.2
11/9, 18/11 20.135 36.9
13/8, 16/13 22.346 41.0
15/13, 26/15 24.986 45.8
13/12, 24/13 25.064 46.0
15-odd-limit intervals in 22edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 2.3
11/10, 20/11 1.368 2.5
15/8, 16/15 2.640 4.8
5/4, 8/5 4.496 8.2
7/6, 12/7 5.856 10.7
11/8, 16/11 5.863 10.7
3/2, 4/3 7.136 13.1
15/11, 22/15 8.504 15.6
15/14, 28/15 10.352 19.0
5/3, 6/5 11.631 21.3
7/4, 8/7 12.992 23.8
11/6, 12/11 12.999 23.8
9/8, 16/9 14.272 26.2
13/11, 22/13 16.482 30.2
7/5, 10/7 17.488 32.1
13/10, 20/13 17.850 32.7
9/5, 10/9 18.767 34.4
11/7, 14/11 18.856 34.6
11/9, 18/11 20.135 36.9
13/8, 16/13 22.346 41.0
15/13, 26/15 24.986 45.8
13/12, 24/13 29.482 54.0
13/7, 14/13 35.338 64.8
13/9, 18/13 36.618 67.1

## Defining features

### Septimal vs. syntonic comma

Possibly the most striking characteristic of 22edo to those not used to it is that it does not temper out the syntonic comma of 81/80. This means that 22 is not a system of meantone temperament, and as such it distinguishes a number of 3-limit and 5-limit intervals that meantone tunings (most notably 12edo, 19edo, 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo and 27edo in comparison to 5-limit JI and many more accurate temperaments such as 34edo, 41edo, and 53edo, allowing many opportunities for alternate interpretations of their harmony.

The diatonic scale it produces is instead derived from superpyth temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, 5L 2s), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A–G♮ and C–B♭ are harmonic sevenths instead of 5-limit minor sevenths (approximately 7/4 instead of 9/5), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are 4–4–5–4–5 and 4–4–1–4–4–4–1, respectively.

### Porcupine comma

It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo supports porcupine temperament. The generator for porcupine is a very flat minor whole tone of ~10/9 (usually tuned slightly flat of 11/10), two of which is a sharp ~6/5, and three of which is a slightly flat ~4/3, implying the existence of an equal-step tetrachord, which is characteristic of porcupine. It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. It forms mos scales of 7 and 8, which in 22edo are tuned respectively as 4–3–3–3–3–3–3 and 1–3–3–3–3–3–3–3 (and their respective modes).

### 5-limit commas

Other 5-limit commas 22edo tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12et or 22et, the diatonic tritone 45/32, which is a major third above a major whole tone representing 9/8, is equated to its inverted form, 64/45. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

### 7-limit commas

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both 50/49, (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the septimal kleisma, so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the orwell comma; and the orwell tetrad is also a chord of 22et.

### 11-limit commas

In the 11-limit, 22edo tempers out the quartisma, leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with 24edo, is surprisingly not shared with a number of other relatively small edos such as 17edo, 26edo and 34edo. In fact, not even the famous 53edo has this property – although it should be noted that the related 159edo does.

### Other features

The 163.6¢ "flat minor whole tone" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the 11-limit: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.

22edo also supports the orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms mos scales with step patterns 3–2–3–2–3–2–3–2–2 and 1–2–2–1–2–2–1–2–2–1–2–2–2. Harmonically, orwell can be tuned more accurately in other temperaments, such as 31edo, 53edo and 84edo. But 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In Sagittal notation, 11 can be notated as every other note of 22.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [35 -22 [22 35]] −2.25 2.25 4.12
2.3.5 250/243, 2048/2025 [22 35 51]] −0.86 2.70 4.94
2.3.5.7 50/49, 64/63, 245/243 [22 35 51 62]] −1.80 2.85 5.23
2.3.5.7.11 50/49, 55/54, 64/63, 99/98 [22 35 51 62 76]] −1.11 2.90 5.33
2.3.5.7.11.17 50/49, 55/54, 64/63, 85/84, 99/98 [22 35 51 62 76 90]] −1.09 2.65 4.87

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is 31. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is 46.

### Uniform maps

13-limit uniform maps between 21.5 and 22.5
Min. size Max. size Wart notation Map
21.5000 21.5353 22bccdddeeeeff 22 34 50 60 74 80]
21.5353 21.5505 22bccdddeeff 22 34 50 60 75 80]
21.5505 21.7492 22bccdeeff 22 34 50 61 75 80]
21.7492 21.7542 22bdeeff 22 34 51 61 75 80]
21.7542 21.7671 22bdee 22 34 51 61 75 81]
21.7671 21.8244 22dee 22 35 51 61 75 81]
21.8244 21.9067 22d 22 35 51 61 76 81]
21.9067 22.0244 22 22 35 51 62 76 81]
22.0244 22.1135 22f 22 35 51 62 76 82]
22.1135 22.1798 22ef 22 35 51 62 77 82]
22.1798 22.2629 22cef 22 35 52 62 77 82]
22.2629 22.2946 22cddef 22 35 52 63 77 82]
22.2946 22.3980 22cddefff 22 35 52 63 77 83]
22.3980 22.4025 22bbcddefff 22 36 52 63 77 83]
22.4025 22.5000 22bbcddeeefff 22 36 52 63 78 83]

### Commas

22et tempers out the following commas. (Note: This assumes the val 22 35 51 62 76 81].)

Prime
limit
Ratio[note 2] Monzo Cents Color name Name
3 (22 digits) [35 -22 156.98 Trisawa 22-comma
5 250/243 [1 -5 3 49.17 Triyo Porcupine comma, maximal diesis
5 3125/3072 [-10 -1 5 29.61 Laquinyo Magic comma
5 2048/2025 [11 -4 -2 19.55 Sagugu Diaschisma
5 (14 digits) [-21 3 7 10.06 Lasepyo Semicomma
5 (20 digits) [32 -7 -9 9.49 Sasa-tritrigu Escapade comma
7 50/49 [1 0 2 -2 34.98 Biruyo Jubilisma
7 64/63 [6 -2 0 -1 27.26 Ru Septimal comma
7 875/864 [-5 -3 3 1 21.90 Zotriyo Keema
7 2430/2401 [1 5 1 -4 20.79 Quadru-ayo Nuwell comma
7 245/243 [0 -5 1 2 14.19 Zozoyo Sensamagic comma
7 1728/1715 [6 3 -1 -3 13.07 Triru-agu Orwellisma
7 225/224 [-5 2 2 -1 7.71 Ruyoyo Marvel comma
7 10976/10935 [5 -7 -1 3 6.48 Trizo-agu Hemimage comma
7 6144/6125 [11 1 -3 -2 5.36 Saruru-atrigu Porwell comma
7 65625/65536 [-16 1 5 1 2.35 Lazoquinyo Horwell comma
7 (12 digits) [-6 -8 2 5 1.12 Quinzo-ayoyo Wizma
11 99/98 [-1 2 0 -2 1 17.58 Loruru Mothwellsma
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11 121/120 [-3 -1 -1 0 2 14.37 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.86 Lorugugu Valinorsma
11 896/891 [7 -4 0 1 -1 9.69 Saluzo Pentacircle comma
11 65536/65219 [16 0 0 -2 -3 8.39 Satrilu-aruru Orgonisma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 540/539 [2 3 1 -2 -1 3.21 Lururuyo Swetisma
11 4000/3993 [5 -1 3 0 -3 3.03 Triluyo Wizardharry comma
11 9801/9800 [-3 4 -2 -2 2 0.18 Bilorugu Kalisma
13 65/64 [-6 0 1 0 0 1 26.84 Thoyo Wilsorma
13 78/77 [1 1 0 -1 -1 1 22.34 Tholuru Negustma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap comma, biome comma
13 31213/31104 [-7 -5 0 4 0 1 6.06 Thoquadzo Praveensma
31 125/124 [-2 0 3 0 0 0 0 0 0 0 -1 13.91 Thiwutriyo Twizzler comma

### Rank-2 temperaments

Periods
per octave
Generator Temperaments
1 1\22 Sensa
Chromo
Ceratitid
1 3\22 Porcupine
1 5\22 Orwell (22) / blair (22) / winston (22f)
1 7\22 Magic / telepathy
1 9\22 Superpyth / suprapyth
2 1\22 Shrutar / hemipaj
Comic
2 2\22 Srutal / pajara / pajarous
2 3\22 Hedgehog / echidna
2 4\22 Astrology
Antikythera
Wizard
2 5\22 Doublewide / fleetwood
11 1\22 Undeka
Hendecatonic

See 22edo modes.

## Tetrachords

See 22edo tetrachords.

## Chord names

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality Color name Monzo Format Examples
minor zo [a b 0 1> 7/6, 7/4
fourthward wa [a b> where b < -1 32/27, 16/9
upminor gu [a b -1> 6/5, 9/5
downmajor yo [a b 1> 5/4, 5/3
major fifthward wa [a b> where b > 1 9/8, 27/16
ru [a b 0 -1> 9/7, 12/7

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

Color of the 3rd JI Chord Notes as edosteps Notes of C chord Written name Spoken name
zo 6:7:9 0-5-13 C Eb G Cm C minor
gu 10:12:15 0-6-13 C ^Eb G C^m C upminor
yo 4:5:6 0-7-13 C vE G Cv C downmajor or C down
ru 14:18:21 0-8-13 C E G C C major or C

Examples:

• 0-4-13 = C D G = C2
• 0-9-13 = C F G = C4
• 0-10-13 = C ^F G = C^4 or C(^4)
• 0-5-10 = C Eb Gb = Cd = Cdim
• 0-5-11 = C Eb ^Gb = Cd(^5)
• 0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

## Notes

1. Based on treating 2.3.5.7.11.17 subgroup as a temperament; other approaches are also possible.
2. Ratios longer than 10 digits are presented by placeholders with informative hints.

## References

1. Barbour, James Murray, Tuning and temperament, a historical survey, East Lansing, Michigan State College Press, 1953 [c1951]
2. Bosanquet, R.H.M. On the Hindoo division of the octave, with additions to the theory of higher orders, Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, Hindu Music from Various Authors, Chowkhamba Sanskrit Series, Varanasi, India, 1965