22edo

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22edo

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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

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22 equal temperament

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 2^1/22, or the 22nd root of 2. Because it distinguishes 10/9 and 9/8, it is not a meantone system.

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 supposed 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.

Theory

22edo is the third edo, after 12 and 19, which is capable of approximating the 5-limit to within a Tenney–Euclidean error of 4 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the 7- and 11-limit to within 3 cents of error, and in fact 22 is the smallest edo to represent the 11-odd-limit consistently, though 31edo is considerably more accurate.

Possibly the most striking characteristic of 22edo to those not used to it is that it does not temper out 81/80 (the syntonic comma), and instead maps it to one step. Additionally, it is a superset of 11edo and is close to 24edo, having only 2 fewer steps than it, and thus behaves like 11edo and 13edo in that melodic movements similar to 12edo can quickly arrive at an unfamiliar place. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

22edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out 64/63, equating the pythagorean minor seventh with 7/4, and supporting superpyth. In that manner, 22edo can be thought of as widening the gap of 49/48 between septimal intervals like 7/6 and 8/7 to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in 25/24 being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out 50/49, so the 7/5 and 10/7 are equated to the 600 ¢ half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and 12/7. Reasonably, 36/35 is also tempered to 1 step just like 25/24 and 49/48.

22edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5–6 ¢ flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a neutral third dividing the perfect fifth in two, which means 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 ¢ sharp, but also because 22edo's step is just short of being small enough to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 24edo, 27edo, and 31edo all include fully). Because 22edo does not contain "neutral" intervals, 11/9 is mapped to the same interval as 6/5 and 12/11 is mapped to the submajor second, inflating 243/242 to a full step.

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.

22edo is also the third-smallest edo (after 10edo and 15edo) that maintains 25% or lower relative error on all of the first eight harmonics of the harmonic series.

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)

As a tuning of other temperaments

Observance of 81/80

22edo, unlike 12 and 19, 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 in comparison to 5-limit JI and many more accurate temperaments such as 34edo, 41edo, and 53edo, allowing many opportunities for alternate interpretations of these intervals. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to the 5L 2s mos as in meantone systems. Instead, it is a ternary scale, having the nicetone pattern.

Superpyth temperament

The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from superpyth temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic mos has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 is tempered out rather than the syntonic comma of 81/80, which one of 22et's core features.

Superpyth temperament equates the Pythagorean sevenths (such as A–G and C–B♭ in chain-of-fifths notation) to harmonic sevenths instead of 5-limit minor sevenths (approximating 7/4 instead of 9/5). Due to the sharper fifths, the diatonic scale is more uneven than in meantone systems and 12edo. In addition to the more uneven diatonic scale, 22edo has a quasi-equal pentatonic scale (the major whole tone and subminor third are rather close in size). The step patterns of the pentatonic and diatonic scales in 22et are 4 – 4 – 5 – 4 – 5 and 4 – 4 – 1 – 4 – 4 – 4 – 1 respectively. In superpyth (and thus in 22edo and technically 12edo), the 1–5/4–3/2–16/9 dominant seventh chord and an otonal tetrad are represented by the same chord.

Porcupine temperament

22edo additionally tempers out the porcupine comma or maximal diesis of 250/243 (S10²⋅S11), 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.

Porcupine temperament allows the 5-limit diatonic scale (the zarlino scale), present as 4 – 3 – 2 – 4 – 3 – 4 – 2 and tuned particularly accurately in 22edo, to be notated with only 1 set of accidentals (conventionally sharps and flats) representing both the syntonic comma and the classical chromatic semitone, as the difference between them (250/243) is tempered out.

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. Porcupine's generator 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).

Pajara temperament

A third important temperament that 22edo supports is pajara. In the 5-limit, 2048/2025 (diaschisma) is tempered out, meaning that the 5-limit tritones are equated to one another and to the semioctave. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, 50/49 (jubilisma) is tempered out, meaning that the tritones 7/5 and 10/7 are equated to the semioctave, and consequently 64/63 is tempered out as in superpyth—5/4 is a semioctave away from 7/4. Since 50/49 is tempered out, the 25/24 and 49/48 intervals are equated to a single interval, and it functions as a chroma in the 2L 8s mos. This suggests the use of a decatonic notation system, where 7/6 and 8/7 are the same number of scale degrees, and 7/4 is a major interval. Thus the 1–5/4–3/2–7/4 major tetrad has 5/4 and 7/4 as major intervals, and replacing them with the corresponding minor intervals gives us the 1–6/5–3/2–12/7 subharmonic sixth chord or minor tetrad. Pajara temperament is also supported by 12edo, as it also tempers out 50/49 and 64/63.

The decatonic scales of pajara have been considered by many to be a system in the 7-limit analogous to the diatonic scale of meantone temperament in the 5-limit, as described in Paul Erlich's paper Tuning, Tonality and 22-Tone Temperament.

Additional commas

Both 22edo and 12edo also temper out (50/49)/(64/63) = 225/224 (S15, marvel comma), so that the marvel augmented triad is a chord of 22et. A 7-limit comma not tempered out by 12et which 22et does temper out is 1728/1715, the orwell comma; therefore, the orwell tetrad is also a chord of 22et. The orwell temperament uses the septimal subminor third (5 degrees) as a generator, and forms mos scales with step patterns 2 – 3 – 2 – 3 – 2 – 3 – 2 – 3 – 2 and 2 – 1 – 2 – 2 – 1 – 2 – 2 – 1 – 2 – 2 – 1 – 2 – 2. While orwell can be tuned more accurately in other temperaments, such as 31edo, 53edo, and 84edo, 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish.

Subsets, supersets, and inheritances

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 quartertones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In particular, 22edo can be roughly conceptualized as 24 but with only two types of thirds rather than three. In Sagittal notation, 11 can be notated as every other note of 22.

22 inherits 11edo's 11/8 and 7/4, and inherits 2edo's tritone, which is mapped in both systems to 7/5 and 10/7.

Other features

The 163.6 ¢ "flat minor whole tone" or "submajor second" 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.

Higher-limit interpretations

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. Also note that 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 the 2.3.5.7.11.17.29.31 subgroup.

Intervals

See also: 22edo solfege

See also: SKULO interval names § Alternatives

Degree	Cents	Approximate Ratios	Audio	Ups and downs notation (EUs: v3A1 and ^^d2)			SKULO notation (K = 1)
0	0.0	1/1		perfect unison	P1	D	perfect unison	P1	D
1	54.5	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	109.1	18/17, 17/16, 16/15, 15/14		downaug 1sn, upminor 2nd	vA1, ^m2	vD#, ^Eb	classic minor 2nd	Km2	KEb
3	163.6	12/11, 11/10, 10/9		aug 1sn, downmajor 2nd	A1, vM2	D#, vE	classic/comma-narrow major 2nd	kM2	kE
4	218.2	9/8, 17/15, 8/7		major 2nd	M2	E	major 2nd	M2	E
5	272.7	20/17, 7/6		minor 3rd	m3	F	minor 3rd	m3	F
6	327.3	6/5, 17/14, 11/9		upminor 3rd	^m3	^F	classic minor 3rd	Km3	KF
7	381.8	5/4, 96/77		downmajor 3rd	vM3	vF#	classic major 3rd	kM3	kF#
8	436.4	14/11, 9/7, 22/17		major 3rd	M3	F#	major 3rd	M3	F#
9	490.9	4/3		perfect 4th	P4	G	perfect 4th	P4	G
10	545.5	15/11, 11/8		up-4th, dim 5th	^4, d5	^G, Ab	comma-wide 4th	K4	KG
11	600.0	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	654.5	16/11, 22/15		aug 4th, down-5th	A4, v5	G#, vA	comma-narrow 5th	k5	kA
13	709.1	3/2		perfect 5th	P5	A	perfect 5th	P5	A
14	763.6	17/11, 14/9, 11/7		minor 6th	m6	Bb	minor 6th	m6	Bb
15	818.2	8/5, 77/48		upminor 6th	^m6	^Bb	classic minor 6th	Km6	KBb
16	872.7	18/11, 28/17, 5/3		downmajor 6th	vM6	vB	classic major 6th	kM6	kB
17	927.3	17/10, 12/7		major 6th	M6	B	major 6th	M6	B
18	981.8	7/4, 30/17, 16/9		minor 7th	m7	C	minor 7th	m7	C
19	1036.4	9/5, 11/6, 20/11		upminor 7th, dim 8ve	^m7, d8	^C, Db	classic minor 7th	Km7	kC
20	1090.9	28/15, 15/8, 32/17, 17/9		downmajor 7th, updim 8ve	vM7, ^d8	vC#, ^Db	classic major 7th	kM7	kC#
21	1145.5	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	1200.0	2/1		perfect octave	P8	D	perfect 8ve	P8	D

Notation

Stein–Zimmermann–Gould notation

Since a sharp raises by three steps, 22edo is a good candidate for Stein–Zimmermann–Gould notation, using sharps and flats with arrows similar to 29edo:

Step offset	0	1	2	3	4	5	6	7
Sharp symbol								
Flat symbol								

If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with double arrows.

Kite's ups and downs notation

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

Step offset	0	1	2	3	4	5	6	7
Sharp symbol
Flat symbol

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

#	Cents	Kite's ups and downs notation
		Diatonic interval names	Note names
0	0.0	Perfect unison (P1)	D
1	54.5	Minor second (m2) Up unison (^1)	Eb ^D
2	109.1	Upminor second (^m2) Downaugmented unison (vA1) Diminished third (d3)	^Eb vD# Fb
3	163.6	Downmajor second (vM2) Augmented unison (A1)	vE D#
4	218.2	Major second (M2) Upaugmented unison (^A1) Downminor third (vm3)	E ^D# vF
5	272.7	Upmajor second (^M2) Minor third (m3)	^E F
6	327.3	Upminor third (^m3) Diminished fourth (d4)	^F Gb
7	381.8	Downmajor third (vM3) Augmented second (A2) Updiminished fourth (^d4)	vF# E# ^Gb
8	436.4	Major third (M3) Upaugmented second (^A2) Down fourth (v4)	F# ^E# vG
9	490.9	Perfect fourth (P4)	G
10	545.5	Up fourth (^4) Diminished fifth (d5)	^G Ab
11	600.0	Downaugmented fourth (vA4) Updiminished fifth (^d5)	vG# ^Ab
12	654.5	Augmented fourth (A4) Down fifth (v5)	G# vA
13	709.1	Perfect fifth (P5)	A
14	763.6	Up fifth (^5) Minor sixth (m6)	^A Bb
15	818.2	Downaugmented fifth (vA5) Upminor sixth (^m6)	vA# ^Bb
16	872.7	Augmented fifth (A5) Downmajor sixth (vM6)	A# vB
17	927.3	Major sixth (M6) Upaugmented fifth (^A5) Downminor seventh (vm7)	B ^A# vC
18	981.8	Minor seventh (m7) Upmajor sixth (^M6) Downdiminished octave (vd8)	C ^B vDb
19	1036.4	Upminor seventh (^m7) Diminished octave (d8)	^C Db
20	1090.9	Downmajor seventh (vM7) Updiminished octave (^d8) Augmented sixth (A6)	vC# ^Db B#
21	1145.5	Major seventh (M7) Down octave (v8)	C# vD
22	1200.0	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.

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

• 
  Tibia in G (page 1)
• 
  Tibia in G (page 2)

Sagittal notation

This notation uses the same sagittal sequence as edos 15 and 29, is a subset of the notations for edos 44 and 66, and is a superset of the notation for 11edo.

Evo flavor

Revo flavor

When 22edo is treated as generated by a cycle of its fifths, the natural notes 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:

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.

A score video demonstrating this type of notation using redefined sharp and flat symbols is available: Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic) (2020) by John Moriarty. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.

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.

Comparison of 22edo notation systems

Degree	Cents	Superpyth/porcupine		Porcupine (Onyx)			Porcupine (Zarlino)			Pentatonic			Decatonic			Ups and downs			SKULO interval names
0	0	Natural unison	1	perfect unison	P1	D	perfect unison	P1	C	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#	augmented unison	A1	C#	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	minor second	m2	Db	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	narrow major second	nM2	D	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#	wide major second	WM2	D#	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	wolf third	w3	Ebb	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	minor third	m3	Eb	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#	major third	M3	E	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	augmented third	A3	E#	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 fourth	P4	F	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#	wolf fourth	w4	F#	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	augmented fourth, diminished fifth	A4, d5	F##, Gbb	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	wolf fifth	w5	Gb	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 fifth	P5	G	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	diminished sixth	d6	Abb	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	minor sixth	m6	Ab	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	major sixth	M6	A	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#	wolf sixth	w6	A#	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	narrow minor seventh	nm7	Bbb	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	wide minor seventh	Wm7	Bb	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#	major seventh	M7	B	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	diminished octave	d8	Cb	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	C	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

15-odd-limit intervals by 22f 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
7/5, 10/7	17.488	32.1
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
13/12, 24/13	25.064	46.0
15/13, 26/15	29.559	54.2
13/8, 16/13	32.200	59.0
13/10, 20/13	36.695	67.3
13/11, 22/13	38.063	69.8

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 does best 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 21.8 and 22.2

Min. size	Max. size	Wart notation	Map
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]

Commas

22et tempers out the following commas. This assumes the val ⟨22 35 51 62 76 81].

Prime limit	Ratio	Monzo	Cents	Color name	Name
3	(22 digits)	[35 -22⟩	156.98	Trisawa	22-comma
5	20480/19683	[12 -9 1⟩	68.72	Sayo	Superpyth comma
5	250/243	[1 -5 3⟩	49.17	Triyo	Porcupine comma
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
5	(32 digits)	[-53 10 16⟩	0.57	Quadla-quadquadyo	Kwazy 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

• List of 22et rank two temperaments by badness
• List of 22et rank two temperaments by complexity
• List of edo-distinct 22et rank two temperaments

Periods per 8ve	Generator	Temperaments
1	1\22	Escapade / escaped 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

Octave stretch or compression

22edo can benefit from slightly compressing the octave, especially when using it as an 7-limit equal temperament. With the right amount of compression we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11.

Good compressed-22 options include: 80zpi or 57ed6.

Scales

Main article: 22edo modes

See also: List of MOS scales in 22edo

Tetrachords

Main article: 22edo tetrachords

Chords

Main article: 22edo chords

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)

Scordatura piano

Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by Juhani Nuorvala's Improvisations on a piano tuned to 22edo (2026).

Keyboards

A potential layout for a 22edo keyboard with both split black and white keys.

Lumatone mappings for 22edo are available.

Music

Main article: 22edo/Music

See also: Category:22edo tracks

See also

• Unque's approach
• William Lynch's approach
• Eliora's approach

Further reading

• Sword, Ron. Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave. 2011.
• Erlich, Paul, Tuning, Tonality, and Twenty-Two Tone Temperament
• "Porcupine Music" - Website Focused on the Development of 22 EDO music
• 11-limit comma lists of selected microtonal EDOs
• Joseph Monzo's visualizations of 22edo scale generation from temperaments

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