22edo

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Theory[edit]

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 not meantone.

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[edit]

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)[edit]

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. The natural notes represent the conventional chain of 5ths FCGDAEB.

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.

Intervals by degree (Ups and Downs, Porcupine and Pentatonic)[edit]

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

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

quality color monzo format examples
minor zo {a, b, 0, 1} 7/6, 7/4
" fourthward wa {a, b}, 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}, b > 1 9/8, 27/16
" ru {a, b, 0, -1} 9/7, 12/7

Chord Names[edit]

All 22edo chords can be named using ups and downs notation. 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 Ev G C.v C downmajor or C dot down
ru 14:18:27 0-8-13 C E G C C major or C

For C.v, the period is needed because "Cv", spoken as "C down", is either a note, or a major chord Cv Ev Gv.

The period isn't needed in Cm 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 applies to both the 3rd and the 7th)

0-3-13 = C Dv G = C(v2)

0-4-13 = C D G = C2

0-9-13 = C F G = C4

0-10-13 = C F^ G = C(^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(^5) (the up symbol applies to both the 3rd and the 7th)

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

0-8-13-21 = C E G B = CM7

0-7-13-20 = C Ev G Bv = C.vM7

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[edit]

The following table shows how some prominent just intervals are represented in 22edo (ordered by absolute error).

Best direct mapping, even if inconsistent[edit]

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

Patent val mapping[edit]

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
10/9, 9/5 18.767
14/11, 11/7 18.856
11/9, 18/11 20.135
16/13, 13/8 22.346
15/13, 26/15 24.986
13/12, 24/13 29.482
14/13, 13/7 35.338
18/13, 13/9 36.618

File:22ed2-001e.svg

22ed2-001e.svg

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

Properties of 22 equal temperament[edit]

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 even 19edo), 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[edit]

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

Period Generator Temperaments
1 22\22 1\22 Sensa/chromo/ceratitid
1 22\22 3\22 Porcupine
1 22\22 5\22 Orson/orwell/blair
1 22\22 7\22 Magic/telepathy
1 22\22 9\22 Superpyth/suprapyth
2 11\22 1\22 Shrutar/hemipaj/comic
2 11\22 2\22 Srutal/pajara/pajarous
2 11\22 3\22 Hedgehog/echidna
2 11\22 4\22 Astrology/wizard/antikythera
2 11\22 5\22 Doublewide/fleetwood
11 2\22 1\22 Hendecatonic/undeka

Commas[edit]

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
4294967296/4271484375 | 32 -7 -9 > 9.49 Escapade Comma
9010162353515625/9007199254740992 | -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[edit]

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[edit]

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[edit]

The decatonic system is an approach of notation 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

Decatonic Alphabet[edit]

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.

Internal links[edit]

External links[edit]

References[edit]

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

Music[edit]

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