22 equal divisions of the octave (22edo), or 22-tone equal temperament (22tet), 22 equal temperament (22et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 22 equal parts of about 54.5 ¢ each. Because it distinguishes 10/9 and 9/8, it is not a meantone system.
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 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 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 11edo's 22.214.171.124.15.17 subgroup, making it a (rather accurate) 126.96.36.199.11.17 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 188.8.131.52.184.108.40.206.
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).
Subsets and supersets
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.
- See also: 22edo solfege
|Degree||Cents||Approximate Ratios*||Ups and Downs Notation|
|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||272.727||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#|
|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|
|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#|
* some simpler ratios, ordered by increasing size, based on treating 22-edo as a 220.127.116.11.11.17 subgroup temperament; other approaches are possible.
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.
|Interval, complement||Error (abs, ¢)||Error (rel, %)|
|Interval, complement||Error (abs, ¢)||Error (rel, %)|
Selected 17-limit intervals
Septimal vs syntonic comma
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, and 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).
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.
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 are easier to distinguish in 22.
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.
Regular temperament properties
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|
|18.104.22.168||50/49, 64/63, 245/243||[⟨22 35 51 62]]||-1.80||2.85||5.23|
|22.214.171.124.11||50/49, 55/54, 64/63, 99/98||[⟨22 35 51 62 76]]||-1.11||2.90||5.33|
|126.96.36.199.11.17||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 188.8.131.52.11.17 subgroup, and the next equal temperament that does better in this subgroup is 46.
|Min. size||Max. size||Wart notation||Map|
|21.5000||21.5353||22bccdddeeeeff||⟨22 34 50 60 74 80]|
|21.5353||21.5505||22bccdddeeff||⟨22 34 50 60 75 80]|
|21.5505||21.7492||22bccdeeff||⟨22 34 50 61 75 80]|
|21.7492||21.7542||22bdeeff||⟨22 34 51 61 75 80]|
|21.7542||21.7671||22bdee||⟨22 34 51 61 75 81]|
|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]|
|22.2629||22.2946||22cddef||⟨22 35 52 63 77 82]|
|22.2946||22.3980||22cddefff||⟨22 35 52 63 77 83]|
|22.3980||22.4025||22bbcddefff||⟨22 36 52 63 77 83]|
|22.4025||22.5000||22bbcddeeefff||⟨22 36 52 63 78 83]|
|3||(22 digits)||[35 -22⟩||156.98|
|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||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|
|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
- List of 22et rank two temperaments by badness
- List of 22et rank two temperaments by complexity
- List of edo-distinct 22et rank two temperaments
|1||5\22||Orwell (22) / blair (22) / winston (22f)|
|1||7\22||Magic / telepathy|
|1||9\22||Superpyth / suprapyth|
|2||1\22||Shrutar / hemipaj|
|2||2\22||Srutal / pajara / pajarous|
|2||3\22||Hedgehog / echidna|
|2||5\22||Doublewide / fleetwood|
See 22edo modes.
See 22edo tetrachords.
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 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.
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.
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.
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).
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.
Comparison of 22edo notation systems
|Degree||Cents||Superpyth/Porcupine Notation||Porcupine||Pentatonic||Decatonic||Sagittal||Ups and Downs|
|0||0||Natural Unison||1||perfect unison||P1||D||perfect unison||P1||D||natural 1st||N1||C||perfect unison||P1||D|
|1||55||s-minor second||sm2||aug unison||A1||D#||aug unison||A1||D#||flat 2nd||f2||C#, δb||minor 2nd||m2||Eb|
|2||109||p-diminished second||pd2||dim 2nd||d2||Eb||double-aug unison,
|natural 2nd||N2||δ||upminor 2nd||^m2||^Eb|
|3||164||p-minor second||pm2||perfect 2nd||P2||E||dim sub3rd||ds3||Fbb||sharp 2nd, flat 3rd||s2, f3||δ#, Db||downmajor 2nd||vM2||vE|
|4||218||(s/p) Major second||M2||aug 2nd||A2||E#||minor sub3rd||ms3||Fb||natural 3rd||N3||D||major 2nd||M2||E|
|5||273||s-minor third||sm3||dim 3rd||d3||Fb||major sub3rd||Ms3||F||sharp 3rd||s3||D#||minor 3rd||m3||F|
|6||327||p-minor third||pm3||minor 3rd||m3||F||aug sub3rd||As3||F#||flat 4th||f4||εb||upminor 3rd||^m3||^F|
|7||382||p-Major third||pM3||major 3rd||M3||F#||double-aug sub3rd,
|natural 4th||N4||ε||downmajor 3rd||vM3||vF#|
|8||436||s-Major third||sM3||aug 3rd, dim 4th||A3, d4||Fx, Gb||dim 4thoid||d4d||Gb||sharp 4th, flat 5th||s4, f5||ε#, Eb||major 3rd||M3||F#|
|9||491||Natural Fourth||4, N4||minor 4th||m4||G||perfect 4thoid||P4d||G||natural 5th||N5||E||perfect fourth||P4||G|
|10||545||p-Major fourth, s-dim fifth||pM4, sd5||major 4th||M4||G#||aug 4thoid||A4d||G#||sharp 5th, flat 6th||s5, f6||E#, γb||up-4th, dim 5th||^4, d5||^G, Ab|
p-diminished Fifth Half-Octave
|A4, HO||aug 4th,
|natural 6th||N6||γ||downaug 4th, updim 5th||vA4, ^d5||vG#, ^Ab|
|12||655||p-minor Fifth, s-aug Fourth||pm5, sA4||minor 5th||m5||Ab||dim 5thoid||d5d||Ab||sharp 6th, flat 7th||s6, f7||γ#, Gb||aug 4th, down-5th||A4, v5||G#, vA|
|13||709||Natural Fifth||5, N5||major 5th||M5||A||perfect 5thoid||P5d||A||natural 7th||N7||G||perfect 5th||P5||A|
|14||764||s-minor sixth||sm6||aug 5th, dim 6th||A5, d6||A#, Bbb||aug 5thoid||A5d||A#||sharp 7th||s7||G#||minor 6th||m6||Bb|
|15||818||p-minor sixth||pm6||minor 6th||m6||Bb||double-aug 5thoid,
|flat 8th||f8||αb||upminor 6th||^m6||^Bb|
|16||873||p-Major sixth||pM6||major 6th||M6||B||dim sub7th||ds7||Cbb||natural 8th||N8||α||downmajor 6th||vM6||vB|
|17||927||s-Major sixth||sM6||aug 6th||A6||B#||minor sub7th||ms7||Cb||sharp 8th, flat 9th||s8, f9||α#, Ab||major 6th||M6||B|
|18||982||(s/p) minor seventh||m7||dim 7th||d7||Cb||major sub7th||Ms7||C||natural 9th||N9||A||minor 7th||m7||C|
|19||1036||p-Major seventh||pM7||perfect 7th||P7||C||aug sub7th||As7||C#||sharp 9th, flat 10th||s9, f10||A#, βb||upminor 7th||^m7||^C|
|20||1091||p-Augmented seventh||pA7||aug 7th||A7||C#||double-aug sub7th,
|natural 10th||N10||β||downmajor 7th||vM7||vC#|
|21||1145||s-Major seventh||sM7||dim 8ve||d8||Db||dim octave||d8||Db||sharp 10th||s10||β#, Cb||major 7th||M7||C#|
|22||1200||Octave||8||perfect octave||P8||D||perfect octave||P8||D||natural 11th||N11||C||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> 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|
- 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)
Further discussion of 22edo chord naming:
- 22edo Chord Names
- 22 EDO Chords
- Ups and Downs Notation #Chords and Chord Progressions
- Chords of orwell
- Main article: 22edo/Music
- See also: Category:22edo tracks
- Lumatone mapping for 22edo
- William Lynch's Thoughts on Septimal Harmony and 22 EDO
- 22edo/Eliora's Approach
- 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
- 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