22edo

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22edo
Prime factorization 2 × 11
Step size 54.545¢
Fifth 13\22 = 709.091¢
Major 2nd 4\22 = 218¢
Minor 2nd 1\22 = 55¢
Augmented 1sn 3\22 = 164¢

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.

Theory

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17
error absolute (¢) 0 +7.2 -4.5 +13.0 -5.9 -22.3 +4.1
relative (%) 0 +13 -8 +24 -11 -40 +7
nearest edomapping 22 13 7 18 10 15 2
fifthspan 0 +1 +9 -2 -6 -9 -10

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).

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 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 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 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 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 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 22edo 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 22edo 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.

In the 11-limit, 22edo tempers out 117440512/117406179, 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.

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.

Notation

Superpyth/Porcupine Notation

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 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.

Degree Name and Abbreviation Cents Approximate Ratios*
0 Natural Unison, 1 0.000 1/1
1 s-minor second, sm2 54.545 36/35, 34/33, 33/32, 32/31
2 p-diminished second, pd2 109.091 18/17, 17/16, 16/15, 15/14
3 p-minor second, pm2 163.636 12/11, 11/10, 10/9
4 (s/p) Major second, M2 218.182 9/8, 17/15, 8/7
5 s-minor third, sm3 272.737 20/17, 7/6
6 p-minor third, pm3 327.273 6/5, 17/14, 11/9
7 p-Major third, pM3 381.818 5/4, 96/77
8 s-Major third, sM3 436.364 14/11, 9/7, 22/17
9 Natural Fourth, 4, N4 490.909 4/3
10 p-Major Fourth, pM4, s-dim fifth 545.455 15/11, 11/8
11 Augmented Fourth, A4, Half-Octave, HO 600.000 7/5, 24/17, 17/12, 10/7
12 p-minor Fifth, pm5, s-aug fourth 654.545 16/11, 22/15
13 Natural Fifth, 5, N5 709.091 3/2
14 s-minor sixth, sm6 763.637 17/11, 14/9, 11/7
15 p-minor sixth, pm6 818.182 8/5, 77/48
16 p-Major sixth, pM6 872.727 18/11, 28/17, 5/3
17 s-Major sixth, sM6 927.273 17/10, 12/7
18 (s/p) minor seventh, m7 981.818 7/4, 30/17, 16/9
19 p-Major seventh, pM7 1036.364 9/5, 11/6, 20/11
20 p-Augmented Seventh 1090.909 28/15, 15/8, 32/17, 17/9
21 s-Major Seventh, sM7 1145.455 31/16, 64/33, 33/17, 35/18
22 Octave, 8 1200.000 2/1

* 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.

Ups and Downs, Porcupine and Pentatonic Notations

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.

Degree Cents Ups and Downs Notation 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 vE 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 vF# 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 vG#,
^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#, vA 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 vB 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 vC# 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 name 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

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

See also 22 EDO Chords, Chords of orwell.

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 vE G Cv C downmajor or C down
ru 14:18:21 0-8-13 C E G C C major or C

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).

0-8-13-18 = C E G Bb = C7 = "C seven"

0-7-13-18 = C vE G Bb = Cv,7 = "C down add-seven"

0-8-13-21 = C E G B = CM7 = "C major seven"

0-7-13-20 = C vE G vB = CvM7 = "C downmajor seven"

0-3-13 = C vD G = Cv2 or C(v2) = "C down-two"

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 = Cdim

0-5-11 = C Eb ^Gb = Cdim(^5)

0-5-12 = C Eb vG = Cm(v5)

0-5-10-15 = C Eb Gb Bbb = Cdim7

0-5-11-14 = C Eb ^Gb vBbb = 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)

0-5-13-17 = C Eb G A = Cm6

0-8-13-17 = C E G A = C6

0-8-13-16 = C E G vA = C,v6 = C add down-six

0-7-13-17 = C vE G A = C6(v3)

0-7-13-16 = C vE G vA = Cv6

0-5-13-18 = C Eb G Bb = Cm7

0-6-13-19 = C ^Eb G ^Bb = C^m7

0-5-13-16 = C Eb G vA = Cm,v6 = "C minor add down-6"

0-8-13-19 = C E G ^Bb = C,^7 = "C add up-seven"

Sometimes doubled ups/downs are unavoidable:

0-6-12-15 = C ^Eb vG vvA = Cm6(^3,v5,vv6), or C ^Eb ^^Gb Bbb = Cdim7(^3,^^5)

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

Just approximation

Selected just intervals by error

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.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
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
Interval, complement Error (abs, ¢)
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

Selected 17-limit intervals

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

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 22et.

3-limit 5-limit 7-limit 11-limit 2.3.5.7.11.17
Octave stretch (¢) -2.25 -0.86 -1.80 -1.11 -1.09
Error absolute (¢) 2.25 2.70 2.85 2.90 2.65
relative (%) 4.12 4.94 5.23 5.33 4.87
  • 22et has a lower relative error than any previous ET in the 11-limit. The next ET that does better in this subgroup is 31.
  • 22et is most prominent in the 2.3.5.7.11.17 subgroup, and the next ET that does better in this is 46.

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

Important MOSes include:

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 Orwell/blair/orson
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

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] 4|4(2) #8 - 2232223222
  • 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
  • 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 - Superpyth[7] 5|1 b2 b6 - 1714171
  • "just" harmonic minor - 4234252
  • "just" harmonic major - 4324252
  • "just" melodic minor - 4234342
  • "just" double harmonic - 2524252

Commas

22 EDO 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(s)
5 250/243 [1 -5 3⟩ 49.17 Triyo Maximal diesis, Porcupine comma
5 3125/3072 [-10 -1 5⟩ 29.61 Laquinyo Small diesis, Magic comma
5 2048/2025 [11 -4 -2⟩ 19.55 Sagugu Diaschisma
5 (14 digits) [-21 3 7⟩ 10.06 Lasepyo Semicomma, Fokker comma
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 Tritonic diesis, Jubilisma
7 64/63 [6 -2 0 -1⟩ 27.26 Ru Septimal comma, Archytas' comma, Leipziger Komma
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, Orwell comma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Septimal kleisma, 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, Gauss' comma
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
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Staff Notation

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

Internal links

External links

References

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

Music

By Andrew Heathwaite

By Brendan Byrnes

By Johann alias Circular17

By MÜÜR

By Sevish

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