22edo

**22edo**
Prime factorization	2 × 11
Step size	54.5455¢
Fifth	13\22 (709.1¢)
Major 2nd	4\22 (218.2¢)
Semitones (A1:m2)	3:1 (163.6¢ : 54.5¢)
Consistency limit	11

22 equal divisions of the octave (22edo), or 22(-tone) equal temperament (22tet, 22et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or about 54.5 cents. Because it distinguishes 10/9 and 9/8, it is not a meantone system.

Theory

History

The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the music theory of India, Bosenquet 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 19 equal temperament, and J. Murray Barbour in his classic survey of tuning history, Tuning and Temperament.

Overview to JI approximation quality

The 22et 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 31 equal temperament does much better, 22et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-odd-limit consistently. Furthermore, 22et, 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, such as 22-tone guitars and the like.

22et can also be treated as adding harmonics 3 and 5 to 11et's 2.7.9.11.15.17 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 fairly 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.

22et is very close to an extended "quarter-comma superpyth", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma 64/63 instead of the syntonic comma 81/80. Because of this it has nearly pure septimal major thirds (9/7).

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
Error	relative (%)	+0	+13	-8	+24	-11	-41	+8	-45	+48	+12	+1
Steps (reduced)		22 (0)	35 (13)	51 (7)	62 (18)	76 (10)	81 (15)	90 (2)	93 (5)	100 (12)	107 (19)	109 (21)

Properties of 22 equal temperament

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, and therefore is not a system of meantone temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, 31edo, … do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as 34edo, 41edo and 53edo.

The diatonic scale it produces is instead derived from superpyth temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, 5L 2s), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22et. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

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 flat minor whole tone of 10/9, two of which is a slightly 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 is notable for being the 5-limit temperament lowest in badness which is not approximated by the familiar 12-tone equal temperament, 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 3 1 3 3 3 3 3 3 (and their respective modes).

The 164¢ "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.

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.

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.

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.

Subset edos

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*	Ups and Downs Notation
0	0.000	1/1	perfect unison	P1	D
1	54.545	36/35, 34/33, 33/32, 32/31	minor 2nd	m2	Eb
2	109.091	18/17, 17/16, 16/15, 15/14	upminor 2nd	^m2	^Eb
3	163.636	12/11, 11/10, 10/9	downmajor 2nd	vM2	vE
4	218.182	9/8, 17/15, 8/7	major 2nd	M2	E
5	272.737	20/17, 7/6	minor 3rd	m3	F
6	327.273	6/5, 17/14, 11/9	upminor 3rd	^m3	^F
7	381.818	5/4, 96/77	downmajor 3rd	vM3	vF#
8	436.364	14/11, 9/7, 22/17	major 3rd	M3	F#
9	490.909	4/3	perfect fourth	P4	G
10	545.455	15/11, 11/8	up-4th, dim 5th	^4, d5	^G, Ab
11	600.000	7/5, 24/17, 17/12, 10/7	downaug 4th, updim 5th	vA4, ^d5	vG#, ^Ab
12	654.545	16/11, 22/15	aug 4th, down-5th	A4, v5	G#, vA
13	709.091	3/2	perfect 5th	P5	A
14	763.636	17/11, 14/9, 11/7	minor 6th	m6	Bb
15	818.182	8/5, 77/48	upminor 6th	^m6	^Bb
16	872.727	18/11, 28/17, 5/3	downmajor 6th	vM6	vB
17	927.273	17/10, 12/7	major 6th	M6	B
18	981.818	7/4, 30/17, 16/9	minor 7th	m7	C
19	1036.364	9/5, 11/6, 20/11	upminor 7th	^m7	^C
20	1090.909	28/15, 15/8, 32/17, 17/9	downmajor 7th	vM7	vC#
21	1145.455	31/16, 64/33, 33/17, 35/18	major 7th	M7	C#
22	1200.000	2/1	perfect octave	P8	D

* some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

Notations

Superpyth/Porcupine Notation, Porcupine Notation and Pentatonic 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.

Another possible 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. This is the only way to use a heptatonic notation without additional accidentals. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D. The natural notes represent a chain of 2nds ABCDEFG.

Yet another notation is pentatonic. The degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. This is the only way to use a chain-of-fifths notation without additional accidentals. The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D. The natural notes represent a chain of 5ths FCGDA.

Degree	Cents	Superpyth/Porcupine Notation		Porcupine			Pentatonic
0	0	Natural Unison	1	perfect unison	P1	D	perfect unison	P1	D
1	55	s-minor second	sm2	aug unison	A1	D#	aug unison	A1	D#
2	109	p-diminished second	pd2	dim 2nd	d2	Eb	double-aug unison, double-dim sub3rd	AA1, dds3	Dx, Fb3
3	164	p-minor second	pm2	perfect 2nd	P2	E	dim sub3rd	ds3	Fbb
4	218	(s/p) Major second	M2	aug 2nd	A2	E#	minor sub3rd	ms3	Fb
5	273	s-minor third	sm3	dim 3rd	d3	Fb	major sub3rd	Ms3	F
6	327	p-minor third	pm3	minor 3rd	m3	F	aug sub3rd	As3	F#
7	382	p-Major third	pM3	major 3rd	M3	F#	double-aug sub3rd, double-dim 4thoid	AAs3, dd4d	Fx, Gbb
8	436	s-Major third	sM3	aug 3rd, dim 4th	A3, d4	Fx, Gb	dim 4thoid	d4d	Gb
9	491	Natural Fourth	4, N4	minor 4th	m4	G	perfect 4thoid	P4d	G
10	545	p-Major fourth, s-dim fifth	pM4, sd5	major 4th	M4	G#	aug 4thoid	A4d	G#
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
12	655	p-minor Fifth, s-aug Fourth	pm5, sA4	minor 5th	m5	Ab	dim 5thoid	d5d	Ab
13	709	Natural Fifth	5, N5	major 5th	M5	A	perfect 5thoid	P5d	A
14	764	s-minor sixth	sm6	aug 5th, dim 6th	A5, d6	A#, Bbb	aug 5thoid	A5d	A#
15	818	p-minor sixth	pm6	minor 6th	m6	Bb	double-aug 5thoid, double-dim sub7th	AA5d, dds7	Ax, Cb3
16	873	p-Major sixth	pM6	major 6th	M6	B	dim sub7th	ds7	Cbb
17	927	s-Major sixth	sM6	aug 6th	A6	B#	minor sub7th	ms7	Cb
18	982	(s/p) minor seventh	m7	dim 7th	d7	Cb	major sub7th	Ms7	C
19	1036	p-Major seventh	pM7	perfect 7th	P7	C	aug sub7th	As7	C#
20	1091	p-Augmented seventh	pA7	aug 7th	A7	C#	double-aug sub7th, double-dim octave	AAs7, dd8	Cx, Dbb
21	1145	s-Major seventh	sM7	dim 8ve	d8	Db	dim octave	d8	Db
22	1200	Octave	8	perfect octave	P8	D	perfect octave	P8	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.

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
minor	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
major	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

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)

For a more complete list, see 22edo Chord Names and Ups and Downs Notation #Chords and Chord Progressions.

JI approximation

15-odd-limit interval mappings

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

15-odd-limit intervals by direct approximation (even if inconsistent)
Interval, complement	Error (abs, ¢)	Error (rel, %)
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 by patent val mapping
Interval, complement	Error (abs, ¢)	Error (rel, %)
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

Selected 17-limit intervals

See also: 22edo Solfege

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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.

Commas

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

Prime limit	Ratio^[1]	Monzo	Cents	Color name	Name
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
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
7	245/243	[0 -5 1 2⟩	14.19	Zozoyo	Sensamagic
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
7	6144/6125	[11 1 -3 -2⟩	5.36	Saruru-atrigu	Porwell
7	65625/65536	[-16 1 5 1⟩	2.35	Lazoquinyo	Horwell
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
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
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Bilorugu	Kalisma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap
31	125/124	[-2 0 3 0 0 0 0 0 0 0 -1⟩	13.91	Thiwutriyo	Twizzler

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

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

Scales

Scales are written be steps in degrees of 22edo. MOS scales are listed in their symmetric mode if one exists, and otherwise in the "brightest" mode - the mode with the highest average pitch height / the lexicographically highest mode.

MOS scales

See also 22edo Modes, 22edo tetrachords

Porcupine[7] - 3334333
Porcupine[8] - 33333331
Porcupine[15] - 121212121212121
Orwell[5] - 55255
Orwell[9] - 232323232
Orwell[13] - 2122122212212
Magic[7] - 1616161
Magic[10] - 5115115111
Magic[13] - 1141114111411
Magic[16] - 3111131111311111
Magic[19] - 1112111112111112111
Superpyth[5] - pentatonic - 45454
Superpyth[7] - diatonic - 4144414
Superpyth[12] - chromatic - 313131131311
Superpyth[17] - hyperchromatic - 12111211211211121
Pajara[10] - symmetric decatonic - 2232222322
Pajara[12] - 222221222221
Hedgehog[6] - 353353
Hedgehog[8] - 33323332
Hedgehog[14] - 21212122121212
Astrology[6] - 434434
Astrology[10] - 3131331313
Astrology[16] - 2121121121211211
Doublewide[4] - 5656
Doublewide[6] - 551551
Doublewide[10] - 4141141411
Doublewide[14] - 31131113113111
Doublewide[18] - 211121111211121111

Other scales

Pentachordal decatonic - Pajara[10] 2|2 (2) #8 - 2232223222
Hexachordal dodecatonic - Pajara[12] 3|2 (2) b10 or Pajara[12] 2|3 (2) #4 - 222122221222
Zarlino/Ptolemy diatonic, "just" major, Ma grama - 4324342
inverse of Zarlino/Ptolemy diatonic, natural minor - 4234243
tetrachordal major, Sa grama - 4324432
inverse of tetrachordal major, "just"/tetrachordal minor - 4234234
"just" major pentatonic - 43636
"just" minor pentatonic (inverse of "just" major pentatonic) - 63463
Porcupine bright major #7 - Porcupine[7] 6|0 #7 - 4333342
Porcupine bright major #6 #7 - Porcupine[7] 6|0 #6 #7 - 4333432
Porcupine bright minor #2 - Porcupine[7] 4|2 #2 4243333 (mode of bright major #7)
Porcupine dark minor #2 - Porcupine[7] 3|3 #2 4234333 (inverse of bright major #6 #7)
Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7 4333324
Superpyth harmonic minor - Superpyth[7] 2|4 #7 - 4144171
Superpyth harmonic major - Superpyth[7] 5|1 b6 - 4414171 (inverse of harmonic minor)
Superpyth melodic minor - Superpyth[7] 5|1 b3 - 4144441
Superpyth double harmonic major - Superpyth[7] 5|1 b2 b6 - 1714171
"just" harmonic minor - 4234252
"just" harmonic major - 4324252
"just" melodic minor - 4234342
Marvel double harmonic major - 2524252
Chromatic tetrachord octave species - 2254225, 5224522, 2524252
Enharmonic trichord octave species - 27427, 72472
Enharmonic tetrachord octave species - 1174117, 7114711, 1714171
Syntonic dipentatonic / blackdye - 1323132313
Marvel hexatonic - 526252
232242232 (Marvel detempering of Negri[9] and August[9])
2.3.7.11 4-SN octatonic - 32353231

Staff notation

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:

Ups and Downs Notation

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)

Music

See also: Category:22edo tracks

Stephen Weigel

Emancipate Pitch!

Metaclown

Couples' Therapy

Claudi Meneghin

Canon 2 in 1 upon a ground (22edo)^{[dead link]}

Paul Erlich

TIBIA
- Ups and Downs score of Tibia in G page 1, page 2
Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity - YouTube

Paul Erlich and Ara Sarkissian

Glassic
Decatonic Swing (jazz)

Joel Grant Taylor

12-22hexachordal Dirge^{[dead link]}

Jake Freivald

Igliashon Jones

Dragged by a Storm Across the Desert Years^{[dead link]} (synth with electric guitar)
Numerology^{[dead link]} (progressive metal)
Revenge of the inorganic compounds^{[dead link]} (progressive metal)

Randy Winchester

Comets Over Flatland 17

Gene Ward Smith and Modest Mussorgsky

Night on Porcupine Mountain

Mike Battaglia

Improvisation in 22-equal temperament - YouTube

Mats Öljare

Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - MIDI files
- Sagittal score of Sunday Pipes

The Stern Brocot Band

Yak Butter, 1L 6s MOS, compressed period/generator

Redrick Sultan

Recurring Mimosa^{[dead link]} (SoundCloud user page)

Chris Vaisvil

My Crazy Aunt Sophie (play). Blatantly xenharmonic piano.
Phobos Light in hedgehog[14] tuned to 22edo.
The Capture and Release of the Fairy (play)
From the Sky Islands They Came (play)
Smoke Filled Bar (play)
The Saharan Pump
22 edo electric guitar duet
For the Sunset - 22edo rock ensemble

Lillian Hearne

Alex Ness

Rose, liz, printemps, verdure (in 22edo with stretched octaves)

Jake Huryn

Palinkalin Viharo (Flowers in the Mist)^{[dead link]} (Score^{[dead link]}); uses 11edo machine[6], 22edo orwell[9]

Diamond Doll

Ray Perlner

Octatonic Groove (22 EDO version)

Andrew Heathwaite

where words are said to mean, a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This & the following pieces by Andrew are for 22-tone guitar & voice.
I've come with a bucket of roses (orwell[9]: 3 2 3 2 3 2 3 2 2).
one drop of rain (orwell[9]).
being a (porcupine[8]: 3 1 3 3 3 3 3).
my own house (a pelog-flavored subset of orwell[9]: 3 2 7 3 7).

Brendan Byrnes (site)

Tracks of ILEVENS - all their tracks on SoundCloud are tagged with 22edo
Short piece and demonstration (video) (electric guitar)
22 EDO Guitar Etude^{[dead link]} (Bandcamp user page)
Llurion (on YouTube) from his 2017 album Neutral Paradise
Hysteria (on YouTube) from his 2017 album Neutral Paradise
Unreachable Island from his 2020 album Realism

Circular17

MÜÜR (site)

References

Barbour, James Murray, Tuning and temperament, a historical survey, East Lansing, Michigan State College Press, 1953 [c1951]
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

External links

[1] Ratios longer than 10 digits are presented by placeholders with informative hints

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