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Theory


In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or 54.55 cents. Because it distinguishes 10/9 and 9/8, it's good for 5-limit.

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''.

The 22-et 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-limits to within 3 cents/oct of error. While 31 equal temperament does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit consistently. Furthermore, 22-et, 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.

22-et can also be treated as adding harmonics 3 and 5 to 11-EDO'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 it's 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.

22-et 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).

Intervalic Naming Systems

The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents procupine 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.

Intervals by degree (Superpyth/Porcupine)

Degree
Name and Abbreviation
Cents
Approximate
Ratios*
0
Natural Unison, 1
0
1/1
1
s-minor second, sm2
54.55
33/32, 34/33, 32/31
2
p-diminished second, pd2
109.09
18/17, 17/16, 16/15, 15/14
3
p-minor second, pm2
163.64
11/10, 10/9, 32/29
4
(s/p) Major second, M2
218.18
9/8, 8/7, 17/15
5
s-minor third, sm3
272.73
7/6, 20/17
6
p-minor third, pm3
327.27
6/5, 17/14, 11/9, 29/24
7
p-Major third, pM3
381.82
5/4
8
s-Major third, sM3
436.36
9/7, 14/11, 22/17
9
Natural Fourth, 4, N4
490.91
4/3
10
p-Major Fourth, pM4
s-dim fifth
545.45
11/8, 15/11
11
Augmented Fourth, A4,
Half-Octave, HO
600
7/5, 10/7, 17/12, 24/17
12
p-minor Fifth, pm5
s-aug fourth
654.55
16/11, 22/15
13
Natural Fifth, 5, N5
709.09
3/2
14
s-minor sixth, sm6
763.64
11/7, 14/9, 17/11
15
p-minor sixth, pm6
818.18
8/5
16
p-Major sixth, pM6
872.73
5/3, 18/11, 28/17
17
s-Major sixth, sM6
927.27
12/7, 17/10
18
(s/p) minor seventh, m7
981.82
7/4, 16/9, 30/17
19
p-Major seventh, pM7
1036.36
20/11, 9/5, 29/16
20
p-Augmented Seventh
1090.91
15/8, 32/17, 17/9, 28/15
21
s-Major Seventh, sM7
1145.45
33/17, 64/33, 31/16
22
Octave, 8
1200
2/1

22edo intervals can also be notated using ups and downs. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D.

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.

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.

Intervals by degree (Ups and Downs, Porcupine and Pentatonic)

Degree
Size (Cents)
Ups and downs
Porcupine
Pentatonic
0
0
perfect unison
P1
D
perfect unison
P1
D
perfect unison
P1
D
1
55
minor 2nd
m2
Eb
aug unison
A1
D#
aug unison
A1
D#
2
109
upminor 2nd
^m2
Eb^
dim 2nd
d2
Eb
double-aug unison,
double-dim sub3rd
AA1,
dds3
Dx,
Fb3
3
164
downmajor 2nd
vM2
Ev
perfect 2nd
P2
E
dim sub3rd
ds3
Fbb
4
218
major 2nd
M2
E
aug 2nd
A2
E#
minor sub3rd
ms3
Fb
5
273
minor 3rd
m3
F
dim 3rd
d3
Fb
major sub3rd
Ms3
F
6
327
upminor 3rd
^m3
F^
minor 3rd
m3
F
aug sub3rd
As3
F#
7
382
downmajor 3rd
vM3
F#v
major 3rd
M3
F#
double-aug sub3rd,
double-dim 4thoid
AAs3,
dd4d
Fx,
Gbb
8
436
major 3rd
M3
F
aug 3rd, dim 4th
A3, d4
Fx, Gb
dim 4thoid
d4d
Gb
9
491
perfect fourth
P4
G
minor 4th
m4
G
perfect 4thoid
P4d
G
10
545
up-4th, dim 5th
^4, d5
G^, Ab
major 4th
M4
G#
aug 4thoid
A4d
G#
11
600
downaug 4th,
updim 5th
vA4, ^d5
G#v,
Ab^
aug 4th,
dim 5th
A4, d5
Gx,
Abb
double-aug 4thoid,
double-dim 5thoid
AA4d,
dd5d
Gx,
Abb
12
655
aug 4th, down-5th
A4, v5
G#, Av
minor 5th
m5
Ab
dim 5thoid
d5d
Ab
13
709
perfect 5th
P5
A
major 5th
M5
A
perfect 5thoid
P5d
A
14
764
minor 6th
m6
Bb
aug 5th, dim 6th
A5, d6
A#, Bbb
aug 5thoid
A5d
A#
15
818
upminor 6th
^m6
Bb^
minor 6th
m6
Bb
double-aug 5thoid,
double-dim sub7th
AA5d,
dds7
Ax,
Cb3
16
873
downmajor 6th
vM6
Bv
major 6th
M6
B
dim sub7th
ds7
Cbb
17
927
major 6th
M6
B
aug 6th
A6
B#
minor sub7th
ms7
Cb
18
982
minor 7th
m7
C
dim 7th
d7
Cb
major sub7th
Ms7
C
19
1036
upminor 7th
^m7
C^
perfect 7th
P7
C
aug sub7th
As7
C#
20
1091
downmajor 7th
vM7
C#v
aug 7th
A7
C#
double-aug sub7th,
double-dim octave
AAs7,
dd8
Cx,
Dbb
21
1145
major 7th
M7
C#
dim 8ve
d8
Db
dim octave
d8
Db
22
1200
perfect octave
P8
D
perfect octave
P8
D
perfect octave
P8
D

Chord Names


22edo chords can be named using ups and downs notation.

0-8-13 = C E G = C = "C" or "C major"
0-7-13 = C Ev G = C.v = "C downmajor" or "C dot down"
The period is needed because "Cv", spoken as "C down", is either a note, or a major chord Cv Ev Gv.
0-6-13 = C Eb^ G = C.^m = "C upminor"
0-5-13 = C Eb G = Cm = "C minor"
The period isn't needed in the last chord name because there's no ups or downs immediately after the note name.

0-8-13-18 = C E G Bb = C7 = "C seven"
0-7-13-18 = C Ev G Bb = C7(v3) = "C seven, down third"
0-8-13-21 = C E G B = CM7 = "C major seven"
0-7-13-20 = C Ev G Bv = C.vM7 = "C downmajor seven" (the down symbol affects both the 3rd and the 7th)

0-3-13 = C Dv G = Csusv2
0-4-13 = C D G = Csus2
0-5-13 = C Eb G = Cm
0-6-13 = C Eb^ G = C.^m
0-7-13 = C Ev G = C.v
0-8-13 = C E G = C
0-9-13 = C F G = Csus4
0-10-13 = C F^ G = Csus^4

0-5-10 = C Eb Gb = Cdim
0-5-11 = C Eb Gb^ = Cdim(^5)
0-5-12 = C Eb Gv = Cm(v5)

0-5-10-15 = C Eb Gb Bbb = Cdim7
0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)
0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)
0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)
0-5-13-17 = C Eb G A = Cm6

Sometimes doubled ups/downs are unavoidable:
0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)

0-8-13-17 = C E G A = C6
0-8-13-16 = C E G Av = C(v6)
0-7-13-17 = C Ev G A = C6(v3)
0-7-13-16 = C Ev G Av = C.v6 (the down symbol applies to both the 3rd and the 6th)

0-5-13-18 = C Eb G Bb = Cm7
0-6-13-19 = C Eb^ G Bb^ = C.^m7
0-7-13-20 = C Ev G Bv = C.vM7
0-8-13-21 = C E G B = CM7

0-5-13-16 = C Eb G Av = Cm(v6)
0-8-13-19 = C E G Bb^ = C(^7)
0-7-13-18-26 = C Ev G Bb D = C9(v3)
0-7-13-18-26-32 = C Ev G Bb D F^ = C9(v3,^11)

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 22edo (ordered by absolute error).
Interval, complement
Error (abs., in cents)
9/7, 14/9
1.280
11/10, 20/11
1.368
16/15, 15/8
2.640
5/4, 8/5
4.496
7/6, 12/7
5.856
11/8, 16/11
5.863
4/3, 3/2
7.136
15/11, 22/15
8.504
15/14, 28/15
10.352
6/5, 5/3
11.631
8/7, 7/4
12.992
12/11, 11/6
12.999
9/8, 16/9
14.272
13/11, 22/13
16.482
7/5, 10/7
17.488
13/10, 20/13
17.850
18/13, 13/9
17.928
10/9, 9/5
18.767
14/11, 11/7
18.856
14/13, 13/7
19.207
11/9, 18/11
20.135
16/13, 13/8
22.346
15/13, 26/15
24.986
13/12, 24/13
25.064

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See also: 22edo Solfege, 22edo Tetrachords, 22 EDO Chords, 22edo Modes

Properties of 22 equal temperament


Possibly the most striking characteristic of 22-et 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 12-EDO, 19-EDO, 31-EDO, ... 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 22-EDO. 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 12-equal and 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 22-EDO supports porcupine temperament. The generator for porcupine is 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 22-EDO. It forms MOS's of 7 and 8, which in 22-EDO 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 22-EDO, 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 22-EDO 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.

22-EDO also supports 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 22-equal 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 22-EDO tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12-et or 22-et, 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 22-et is a magic system, where five major thirds make up a perfect fifth.

In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both 50/49, (the jubilee comma), and 64/63, (the 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 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the orwell comma; and the orwell tetrad is also a chord of 22-et.

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). 11-equal is interesting for sounding melodically very similar to 12-equal (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, 11 can be notated as every other note of 22.

Rank Two 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 octave
Generator
Temperaments
1
1\22
Sensa/chromo/ceratitid
1
3\22
Porcupine
1
5\22
Orson/orwell/blair
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/wizard/antikythera
2
5\22
Doublewide/fleetwood
11
1\22
Hendecatonic/undeka

Commas

22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.)
Rational
Monzo
Size (Cents)
Name 1
Name 2
Name 3
250/243
| 1 -5 3 >
49.17
Maximal Diesis
Porcupine Comma

3125/3072
| -10 -1 5 >
29.61
Small Diesis
Magic Comma

2048/2025
| 11 -4 -2 >
19.55
Diaschisma


2109375/2097152
| -21 3 7 >
10.06
Semicomma
Fokker Comma

9193891/9143623
| 32 -7 -9 >
9.49
Escapade Comma


4758837/4757272
| -53 10 16 >
0.57
Kwazy


50/49
| 1 0 2 -2 >
34.98
Tritonic Diesis
Jubilisma

64/63
| 6 -2 0 -1 >
27.26
Septimal Comma
Archytas' Comma
Leipziger Komma
875/864
| -5 -3 3 1 >
21.90
Keema


2430/2401
| 1 5 1 -4 >
20.79
Nuwell


245/243
| 0 -5 1 2 >
14.19
Sensamagic


1728/1715
| 6 3 -1 -3 >
13.07
Orwellisma
Orwell Comma

225/224
| -5 2 2 -1 >
7.71
Septimal Kleisma
Marvel Comma

10976/10935
| 5 -7 -1 3 >
6.48
Hemimage


6144/6125
| 11 1 -3 -2 >
5.36
Porwell


65625/65536
| -16 1 5 1 >
2.35
Horwell


420175/419904
| -6 -8 2 5 >
1.12
Wizma


99/98
| -1 2 0 -2 1 >
17.58
Mothwellsma


100/99
| 2 -2 2 0 -1 >
17.40
Ptolemisma


121/120
| -3 -1 -1 0 2 >
14.37
Biyatisma


125/124
|-4 0 3 0 ... -1>
13.91
Twizzler


176/175
| 4 0 -2 -1 1 >
9.86
Valinorsma


896/891
| 7 -4 0 1 -1 >
9.69
Pentacircle


65536/65219
| 16 0 0 -2 -3 >
8.39
Orgonisma


385/384
| -7 -1 1 1 1 >
4.50
Keenanisma


540/539
| 2 3 1 -2 -1 >
3.21
Swetisma


4000/3993
<| 5 -1 3 0 -3 >
3.03
Wizardharry


9801/9800
| -3 4 -2 -2 2 >
0.18
Kalisma
Gauss' Comma

91/90
| -1 -2 -1 1 0 1 >
19.13
Superleap



How to Notate 22edo in Sagittal


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:
22edo.png
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).

How to notate 22edo with ups and downs


Treating ups and downs as "fused" with sharps and flats, and never appearing separately:
Tibia 22edo ups and downs guide 1.png
Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:
Tibia 22edo ups and downs guide 2.png
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.
Tibia 22edo guide D major.png
Paul Erlich's "Tibia" in G, with independent ups and downs:

Tibia in G for the book-1.png

Tibia in G for the book-2.png

The Decatonic System

The decatonic system is an approach of notation based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system bases music into a 10 tone scale rather than 7.
This approach requires an entire re-learning of chords, intervals, and notation but the advantage is that it allows 22 EDO to be notated using only one pair of accidentals, as well asgives the opportunity to escape a heptatonic thinking pattern.



Decatonic Alphabet

The system is based on two chains of fifths. One represented by latin letters, the other greek. The two chains can be looked at as two juxtaposed pentatonic scales.

Chain 1: C G D A EChain 2: γ δ α ε β

The alphabet is 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 counter parts. For example G D is a fifth as well as γ δ.

External links


Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''

"Porcupine Music" - Website Focused on the Development of 22 EDO music

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


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