22edo: Difference between revisions
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{{ | {{Interwiki | ||
| en = 22edo | |||
| de = 22-EDO | | de = 22-EDO | ||
| es = 22 EDO | | es = 22 EDO | ||
| ja = 22平均律 | | ja = 22平均律 | ||
| Line 13: | Line 13: | ||
== Theory == | == Theory == | ||
22edo is the third edo, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents of error, and in fact 22 is the smallest edo to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is considerably more accurate. | |||
Possibly the most striking characteristic of 22edo to those not used to it is that it does | Possibly the most striking characteristic of 22edo to those not used to it is that it does ''not'' [[tempering out|temper out]] [[81/80]] (the syntonic comma), and instead maps it to one step. Additionally, it is a superset of 11edo and is close to [[24edo]], having only 2 fewer steps than it, and thus behaves like [[11edo]] and [[13edo]] in that melodic movements similar to 12edo can quickly arrive at an unfamiliar place. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars. | ||
22edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and | 22edo's approximation to the [[7/1|7th harmonic]] is about 13 cents sharp, somewhat similar to 12edo's approximation to the [[5/1|5th harmonic]]. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and [[support]]ing [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in [[25/24]] being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] are equated to the 600{{c}} half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to 1 step just like 25/24 and 49/48. | ||
22edo's approximation of the 11-limit is somewhat contentious: | 22edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5–6{{c}} flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] dividing the perfect fifth in two, which means 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7{{c}} sharp, but also because 22edo's step is just short of being small enough to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which [[24edo]], [[27edo]], and 31edo all include fully). Because 22edo does not contain "neutral" intervals, [[11/9]] is mapped to the same interval as 6/5 and [[12/11]] is mapped to the submajor second, inflating [[243/242]] to a full step. | ||
Since 22edo's fifth is sharp of just by approximately one | Since 22edo's fifth is sharp of just by approximately one quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]]. | ||
22edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | 22edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | ||
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=== As a tuning of other temperaments === | === As a tuning of other temperaments === | ||
==== Observance of 81/80 ==== | ==== Observance of 81/80 ==== | ||
22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of | 22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of these intervals. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to the [[5L 2s]] [[mos]] as in meantone systems. Instead, it is a ternary scale, having the [[nicetone]] pattern. | ||
==== Superpyth temperament ==== | ==== Superpyth temperament ==== | ||
The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic | The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic mos has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 is tempered out rather than the syntonic comma of 81/80, which one of 22et's core features. | ||
Superpyth | Superpyth temperament equates the Pythagorean sevenths (such as A–G and C–B♭ in [[chain-of-fifths notation]]) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximating [[7/4]] instead of [[9/5]]). Due to the sharper fifths, the diatonic scale is more uneven than in meantone systems and 12edo. In addition to the more uneven diatonic scale, 22edo has a quasi-equal pentatonic scale (the major whole tone and subminor third are rather close in size). The step patterns of the pentatonic and diatonic scales in 22et are {{dash|4, 4, 5, 4, 5}} and {{dash|4, 4, 1, 4, 4, 4, 1}} respectively. In superpyth (and thus in 22edo and technically 12edo), the [[36:45:54:64|1–5/4–3/2–16/9]] dominant seventh chord and an otonal tetrad are represented by the same chord. | ||
==== Porcupine temperament ==== | ==== Porcupine temperament ==== | ||
22edo additionally tempers out the porcupine comma or maximal diesis of [[250/243]] (S10<sup>2</sup> | 22edo additionally tempers out the porcupine comma or maximal diesis of [[250/243]] ([[S-expression|S10<sup>2</sup>⋅S11]]), which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a very flat minor whole tone of ~[[10/9]] (usually tuned slightly flat of [[11/10]]), two of which is a 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 temperament allows the 5-limit diatonic scale (the [[zarlino]] scale), present as 4 | Porcupine temperament allows the 5-limit diatonic scale (the [[zarlino]] scale), present as {{nowrap|{{dash|4, 3, 2, 4, 3, 4, 2}}}} and tuned particularly accurately in 22edo, to be notated with only 1 set of accidentals (conventionally sharps and flats) representing both the syntonic comma and the classical chromatic semitone, as the difference between them (250/243) is tempered out. | ||
It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. Porcupine's generator forms | It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. Porcupine's generator forms mos scales of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3}} (and their respective modes). | ||
==== Pajara temperament ==== | ==== Pajara temperament ==== | ||
A third important temperament that 22edo supports is pajara. In the 5-limit, [[2048/2025]] ( | A third important temperament that 22edo supports is [[pajara]]. In the 5-limit, [[2048/2025]] (diaschisma) is tempered out, meaning that the 5-limit tritones are equated to one another and to the [[semioctave]]. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, [[50/49]] (jubilisma) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are equated to the semioctave, and consequently 64/63 is tempered out as in superpyth—5/4 is a semioctave away from 7/4. Since 50/49 is tempered out, the 25/24 and 49/48 intervals are equated to a single interval, and it functions as a chroma in the [[2L 8s]] mos. This suggests the use of a decatonic notation system, where 7/6 and 8/7 are the same number of scale degrees, and 7/4 is a major interval. Thus the [[4:5:6:7|1–5/4–3/2–7/4]] major tetrad has 5/4 and 7/4 as major intervals, and replacing them with the corresponding minor intervals gives us the [[70:84:105:120|1–6/5–3/2–12/7]] subharmonic sixth chord or minor tetrad. Pajara temperament is also supported by [[12edo]], as it also tempers out 50/49 and 64/63. | ||
The decatonic scales of pajara have been considered by many to be a system in the 7-limit analogous to the diatonic scale of meantone temperament in the 5-limit, as described in Paul Erlich's paper [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf Tuning, Tonality and 22-Tone Temperament]. | |||
==== Additional commas ==== | ==== Additional commas ==== | ||
Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} ( | Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} ({{S|15}}, [[marvel comma]]), so that the marvel augmented triad is a chord of 22et. A 7-limit comma not tempered out by 12et which 22et does temper out is [[1728/1715]], the orwell comma; therefore, the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third (5 degrees) as a generator, and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2}}. While orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]], 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish. | ||
=== Subsets, supersets, and inheritances === | === Subsets, supersets, and inheritances === | ||
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 | 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 quartertones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In particular, 22edo can be roughly conceptualized as 24 but with only two types of thirds rather than three. In [[Sagittal notation]], 11 can be notated as every other note of 22. | ||
22 inherits 11edo's [[11/8]] and [[7/4]], and inherits [[2edo]]'s tritone, which is mapped in both systems to [[7/5]] and [[10/7]]. | 22 inherits 11edo's [[11/8]] and [[7/4]], and inherits [[2edo]]'s tritone, which is mapped in both systems to [[7/5]] and [[10/7]]. | ||
=== Other features === | === Other features === | ||
The 163.6{{c}} "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. | The 163.6{{c}} "flat minor whole tone" or "submajor second" 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. | ||
=== Higher-limit interpretations === | === Higher-limit interpretations === | ||
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! Cents | ! Cents | ||
! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref> | ! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref> | ||
! Audio | |||
! colspan="3" | [[Ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2) | ! colspan="3" | [[Ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2) | ||
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}} | ! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}} | ||
|- | |- | ||
| 0 | | 0 | ||
| 0.0 | | 0.0 | ||
| [[1/1]] | | [[1/1]] | ||
| [[File:0-0.000c_P1.mp3]] | |||
| perfect unison | | perfect unison | ||
| P1 | | P1 | ||
| Line 83: | Line 86: | ||
| P1 | | P1 | ||
| D | | D | ||
|- | |- | ||
| 1 | | 1 | ||
| 54.5 | | 54.5 | ||
| [[36/35]], [[34/33]], [[33/32]], [[32/31]] | | [[36/35]], [[34/33]], [[33/32]], [[32/31]] | ||
| [[File:0-54.545c_22edo.mp3]] | |||
| up-unison, minor 2nd | | up-unison, minor 2nd | ||
| ^1, m2 | | ^1, m2 | ||
| Line 94: | Line 97: | ||
| K1, m2 | | K1, m2 | ||
| KD, Eb | | KD, Eb | ||
|- | |- | ||
| 2 | | 2 | ||
| 109.1 | | 109.1 | ||
| [[18/17]], [[17/16]], [[16/15]], [[15/14]] | | [[18/17]], [[17/16]], [[16/15]], [[15/14]] | ||
| [[File:0-109.091c_11edo.mp3]] | |||
| downaug 1sn, upminor 2nd | | downaug 1sn, upminor 2nd | ||
| vA1, ^m2 | | vA1, ^m2 | ||
| Line 105: | Line 108: | ||
| Km2 | | Km2 | ||
| KEb | | KEb | ||
|- | |- | ||
| 3 | | 3 | ||
| 163.6 | | 163.6 | ||
| [[12/11]], [[11/10]], [[10/9]] | | [[12/11]], [[11/10]], [[10/9]] | ||
| [[File:0-163.636c_22edo.mp3]] | |||
| aug 1sn, downmajor 2nd | | aug 1sn, downmajor 2nd | ||
| A1, vM2 | | A1, vM2 | ||
| Line 116: | Line 119: | ||
| kM2 | | kM2 | ||
| kE | | kE | ||
|- | |- | ||
| 4 | | 4 | ||
| 218.2 | | 218.2 | ||
| [[9/8]], [[17/15]], [[8/7]] | | [[9/8]], [[17/15]], [[8/7]] | ||
| [[File:0-218.182c_11edo.mp3]] | |||
| major 2nd | | major 2nd | ||
| M2 | | M2 | ||
| Line 127: | Line 130: | ||
| M2 | | M2 | ||
| E | | E | ||
|- | |- | ||
| 5 | | 5 | ||
| 272.7 | | 272.7 | ||
| [[20/17]], [[7/6]] | | [[20/17]], [[7/6]] | ||
| [[File:0-272.727c_22edo.mp3]] | |||
| minor 3rd | | minor 3rd | ||
| m3 | | m3 | ||
| Line 138: | Line 141: | ||
| m3 | | m3 | ||
| F | | F | ||
|- | |- | ||
| 6 | | 6 | ||
| 327.3 | | 327.3 | ||
| [[6/5]], [[17/14]], [[11/9]] | | [[6/5]], [[17/14]], [[11/9]] | ||
| [[File:0-327.273c_11edo.mp3]] | |||
| upminor 3rd | | upminor 3rd | ||
| ^m3 | | ^m3 | ||
| Line 149: | Line 152: | ||
| Km3 | | Km3 | ||
| KF | | KF | ||
|- | |- | ||
| 7 | | 7 | ||
| 381.8 | | 381.8 | ||
| [[5/4]], [[96/77]] | | [[5/4]], [[96/77]] | ||
| [[File:0-381.818c_22edo.mp3]] | |||
| downmajor 3rd | | downmajor 3rd | ||
| vM3 | | vM3 | ||
| Line 160: | Line 163: | ||
| kM3 | | kM3 | ||
| kF# | | kF# | ||
|- | |- | ||
| 8 | | 8 | ||
| 436.4 | | 436.4 | ||
| [[14/11]], [[9/7]], [[22/17]] | | [[14/11]], [[9/7]], [[22/17]] | ||
| [[File:0-436.364c_11edo.mp3]] | |||
| major 3rd | | major 3rd | ||
| M3 | | M3 | ||
| Line 171: | Line 174: | ||
| M3 | | M3 | ||
| F# | | F# | ||
|- | |- | ||
| 9 | | 9 | ||
| 490.9 | | 490.9 | ||
| [[4/3]] | | [[4/3]] | ||
| [[File:0-490.909c_22edo.mp3]] | |||
| perfect 4th | | perfect 4th | ||
| P4 | | P4 | ||
| Line 182: | Line 185: | ||
| P4 | | P4 | ||
| G | | G | ||
|- | |- | ||
| 10 | | 10 | ||
| 545.5 | | 545.5 | ||
| [[15/11]], [[11/8]] | | [[15/11]], [[11/8]] | ||
| [[File:0-545.455c_11edo.mp3]] | |||
| up-4th, dim 5th | | up-4th, dim 5th | ||
| ^4, d5 | | ^4, d5 | ||
| Line 193: | Line 196: | ||
| K4 | | K4 | ||
| KG | | KG | ||
|- | |- | ||
| 11 | | 11 | ||
| 600.0 | | 600.0 | ||
| [[7/5]], [[24/17]], [[17/12]], [[10/7]] | | [[7/5]], [[24/17]], [[17/12]], [[10/7]] | ||
| [[File:0-600.000c_2edo.mp3]] | |||
| downaug 4th, updim 5th | | downaug 4th, updim 5th | ||
| vA4, ^d5 | | vA4, ^d5 | ||
| Line 204: | Line 207: | ||
| kA4<br />Kd5 | | kA4<br />Kd5 | ||
| kG#, KAb | | kG#, KAb | ||
|- | |- | ||
| 12 | | 12 | ||
| 654.5 | | 654.5 | ||
| [[16/11]], [[22/15]] | | [[16/11]], [[22/15]] | ||
| [[File:0-654.545c_11edo.mp3]] | |||
| aug 4th, down-5th | | aug 4th, down-5th | ||
| A4, v5 | | A4, v5 | ||
| Line 215: | Line 218: | ||
| k5 | | k5 | ||
| kA | | kA | ||
|- | |- | ||
| 13 | | 13 | ||
| 709.1 | | 709.1 | ||
| [[3/2]] | | [[3/2]] | ||
| [[File:0-709.091c_22edo.mp3]] | |||
| perfect 5th | | perfect 5th | ||
| P5 | | P5 | ||
| Line 226: | Line 229: | ||
| P5 | | P5 | ||
| A | | A | ||
|- | |- | ||
| 14 | | 14 | ||
| 763.6 | | 763.6 | ||
| [[17/11]], [[14/9]], [[11/7]] | | [[17/11]], [[14/9]], [[11/7]] | ||
| [[File:0-763.636c_11edo.mp3]] | |||
| minor 6th | | minor 6th | ||
| m6 | | m6 | ||
| Line 237: | Line 240: | ||
| m6 | | m6 | ||
| Bb | | Bb | ||
|- | |- | ||
| 15 | | 15 | ||
| 818.2 | | 818.2 | ||
| [[8/5]], [[77/48]] | | [[8/5]], [[77/48]] | ||
| [[File:0-818.182c_22edo.mp3]] | |||
| upminor 6th | | upminor 6th | ||
| ^m6 | | ^m6 | ||
| Line 248: | Line 251: | ||
| Km6 | | Km6 | ||
| KBb | | KBb | ||
|- | |- | ||
| 16 | | 16 | ||
| 872.7 | | 872.7 | ||
| [[18/11]], [[28/17]], [[5/3]] | | [[18/11]], [[28/17]], [[5/3]] | ||
| [[File:0-872.727c_11edo.mp3]] | |||
| downmajor 6th | | downmajor 6th | ||
| vM6 | | vM6 | ||
| Line 259: | Line 262: | ||
| kM6 | | kM6 | ||
| kB | | kB | ||
|- | |- | ||
| 17 | | 17 | ||
| 927.3 | | 927.3 | ||
| [[17/10]], [[12/7]] | | [[17/10]], [[12/7]] | ||
| [[File:0-927.273c_22edo.mp3]] | |||
| major 6th | | major 6th | ||
| M6 | | M6 | ||
| Line 270: | Line 273: | ||
| M6 | | M6 | ||
| B | | B | ||
|- | |- | ||
| 18 | | 18 | ||
| 981.8 | | 981.8 | ||
| [[7/4]], [[30/17]], [[16/9]] | | [[7/4]], [[30/17]], [[16/9]] | ||
| [[File:0-981.818c_11edo.mp3]] | |||
| minor 7th | | minor 7th | ||
| m7 | | m7 | ||
| Line 281: | Line 284: | ||
| m7 | | m7 | ||
| C | | C | ||
|- | |- | ||
| 19 | | 19 | ||
| 1036.4 | | 1036.4 | ||
| [[9/5]], [[11/6]], [[20/11]] | | [[9/5]], [[11/6]], [[20/11]] | ||
| [[File:0-1036.364c_22edo.mp3]] | |||
| upminor 7th, dim 8ve | | upminor 7th, dim 8ve | ||
| ^m7, d8 | | ^m7, d8 | ||
| Line 292: | Line 295: | ||
| Km7 | | Km7 | ||
| kC | | kC | ||
|- | |- | ||
| 20 | | 20 | ||
| 1090.9 | | 1090.9 | ||
| [[28/15]], [[15/8]], [[32/17]], [[17/9]] | | [[28/15]], [[15/8]], [[32/17]], [[17/9]] | ||
| [[File:0-1090.909c_11edo.mp3]] | |||
| downmajor 7th, updim 8ve | | downmajor 7th, updim 8ve | ||
| vM7, ^d8 | | vM7, ^d8 | ||
| Line 303: | Line 306: | ||
| kM7 | | kM7 | ||
| kC# | | kC# | ||
|- | |- | ||
| 21 | | 21 | ||
| 1145.5 | | 1145.5 | ||
| [[31/16]], [[64/33]], [[33/17]], [[35/18]] | | [[31/16]], [[64/33]], [[33/17]], [[35/18]] | ||
| [[File:0-1145.455c_22edo.mp3]] | |||
| major 7th, down 8ve | | major 7th, down 8ve | ||
| M7, v8 | | M7, v8 | ||
| Line 314: | Line 317: | ||
| M7 / k8 | | M7 / k8 | ||
| C#, kD | | C#, kD | ||
|- | |- | ||
| 22 | | 22 | ||
| 1200.0 | | 1200.0 | ||
| [[2/1]] | | [[2/1]] | ||
| [[File:0-1200.000c_P8.mp3]] | |||
| perfect octave | | perfect octave | ||
| P8 | | P8 | ||
| Line 325: | Line 328: | ||
| P8 | | P8 | ||
| D | | D | ||
|} | |} | ||
== Notation == | == Notation == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
Since a sharp raises by three steps, 22edo is a good candidate for [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to 29edo: | |||
{{Sharpness-sharp3-szg}} | |||
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with double arrows. | |||
=== Kite's ups and downs notation === | |||
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud. | Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud. | ||
{{sharpness-sharp3a}} | {{sharpness-sharp3a}} | ||
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 | Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 edostep (1\22). Note that E♭ and D♯ are different notes and that E♭ is significantly lower in pitch than D♯. | ||
{| class="wikitable right-1 right-2 center-3 center-4" | {| class="wikitable right-1 right-2 center-3 center-4" | ||
|+ style="font-size: 105%;" | Notation of 22edo | |+ style="font-size: 105%;" | Notation of 22edo | ||
|- | |- | ||
! rowspan="2" | [[Degree]] | ! rowspan="2" | [[Degree|#]] | ||
! rowspan="2" | [[Cent]]s | ! rowspan="2" | [[Cent]]s | ||
! colspan="2" | [[ | ! colspan="2" | [[Kite's ups and downs notation]] | ||
|- | |- | ||
! [[5L 2s|Diatonic | ! [[5L 2s|Diatonic interval names]] | ||
! Note | ! Note names | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 352: | Line 360: | ||
| 1 | | 1 | ||
| 54.5 | | 54.5 | ||
| Minor second (m2)<br | | Minor second (m2)<br>Up unison (^1) | ||
| Eb<br | | Eb<br>^D | ||
|- | |- | ||
| 2 | | 2 | ||
| 109.1 | | 109.1 | ||
| Upminor second (^m2)<br | | Upminor second (^m2)<br>Downaugmented unison (vA1)<br>Diminished third (d3) | ||
| ^Eb<br | | ^Eb<br>vD#<br>Fb | ||
|- | |- | ||
| 3 | | 3 | ||
| 163.6 | | 163.6 | ||
| Downmajor second (vM2)<br | | Downmajor second (vM2)<br>Augmented unison (A1) | ||
| vE<br | | vE<br>D# | ||
|- | |- | ||
| 4 | | 4 | ||
| 218.2 | | 218.2 | ||
| '''Major second (M2)'''<br | | '''Major second (M2)'''<br>Upaugmented unison (^A1)<br>Downminor third (vm3) | ||
| '''E'''<br | | '''E'''<br>^D#<br />vF | ||
|- | |- | ||
| 5 | | 5 | ||
| 272.7 | | 272.7 | ||
| Upmajor second (^M2)<br | | Upmajor second (^M2)<br>'''Minor third (m3)''' | ||
| ^E<br | | ^E<br>'''F''' | ||
|- | |- | ||
| 6 | | 6 | ||
| 327.3 | | 327.3 | ||
| '''Upminor third (^m3)'''<br | | '''Upminor third (^m3)'''<br>Diminished fourth (d4) | ||
| '''^F'''<br | | '''^F'''<br>Gb | ||
|- | |- | ||
| 7 | | 7 | ||
| 381.8 | | 381.8 | ||
| '''Downmajor third (vM3)'''<br | | '''Downmajor third (vM3)'''<br>Augmented second (A2)<br>Updiminished fourth (^d4) | ||
| '''vF#'''<br | | '''vF#'''<br>E#<br>^Gb | ||
|- | |- | ||
| 8 | | 8 | ||
| 436.4 | | 436.4 | ||
| '''Major third (M3)'''<br | | '''Major third (M3)'''<br>Upaugmented second (^A2)<br>Down fourth (v4) | ||
| '''F#'''<br | | '''F#'''<br>^E#<br>vG | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 397: | Line 405: | ||
| 10 | | 10 | ||
| 545.5 | | 545.5 | ||
| Up fourth (^4)<br | | Up fourth (^4)<br>Diminished fifth (d5) | ||
| ^G<br | | ^G<br>Ab | ||
|- | |- | ||
| 11 | | 11 | ||
| 600.0 | | 600.0 | ||
| Downaugmented fourth (vA4)<br | | Downaugmented fourth (vA4)<br>Updiminished fifth (^d5) | ||
| vG#<br | | vG#<br>^Ab | ||
|- | |- | ||
| 12 | | 12 | ||
| 654.5 | | 654.5 | ||
| Augmented fourth (A4)<br | | Augmented fourth (A4)<br>Down fifth (v5) | ||
| G#<br | | G#<br>vA | ||
|- | |- | ||
| 13 | | 13 | ||
| Line 417: | Line 425: | ||
| 14 | | 14 | ||
| 763.6 | | 763.6 | ||
| Up fifth (^5)<br | | Up fifth (^5)<br>Minor sixth (m6) | ||
| ^A<br | | ^A<br>Bb | ||
|- | |- | ||
| 15 | | 15 | ||
| 818.2 | | 818.2 | ||
| Downaugmented fifth (vA5)<br | | Downaugmented fifth (vA5)<br>Upminor sixth (^m6) | ||
| vA#<br | | vA#<br>^Bb | ||
|- | |- | ||
| 16 | | 16 | ||
| 872.7 | | 872.7 | ||
| Augmented fifth (A5)<br | | Augmented fifth (A5)<br>'''Downmajor sixth (vM6)''' | ||
| A#<br | | A#<br>'''vB''' | ||
|- | |- | ||
| 17 | | 17 | ||
| 927.3 | | 927.3 | ||
| '''Major sixth (M6)'''<br | | '''Major sixth (M6)'''<br>Upaugmented fifth (^A5)<br>Downminor seventh (vm7) | ||
| '''B'''<br | | '''B'''<br>^A#<br />vC | ||
|- | |- | ||
| 18 | | 18 | ||
| 981.8 | | 981.8 | ||
| '''Minor seventh (m7)'''<br | | '''Minor seventh (m7)'''<br>Upmajor sixth (^M6)<br>Downdiminished octave (vd8) | ||
| '''C'''<br | | '''C'''<br>^B<br>vDb | ||
|- | |- | ||
| 19 | | 19 | ||
| 1036.4 | | 1036.4 | ||
| '''Upminor seventh (^m7)'''<br | | '''Upminor seventh (^m7)'''<br>Diminished octave (d8) | ||
| '''^C'''<br | | '''^C'''<br>Db | ||
|- | |- | ||
| 20 | | 20 | ||
| 1090.9 | | 1090.9 | ||
| Downmajor seventh (vM7)<br | | Downmajor seventh (vM7)<br>Updiminished octave (^d8)<br>Augmented sixth (A6) | ||
| vC#<br | | vC#<br>^Db<br>B# | ||
|- | |- | ||
| 21 | | 21 | ||
| 1145.5 | | 1145.5 | ||
| Major seventh (M7)<br | | Major seventh (M7)<br>Down octave (v8) | ||
| C#<br | | C#<br>vD | ||
|- | |- | ||
| 22 | | 22 | ||
| Line 461: | Line 469: | ||
|} | |} | ||
Treating | Treating ups and downs as "fused" with sharps and flats, and never appearing separately: | ||
[[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]] | [[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]] | ||
| Line 472: | Line 480: | ||
[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]] | [[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]] | ||
Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs. | Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs. | ||
| Line 487: | Line 489: | ||
=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as | This notation uses the same sagittal sequence as edos [[15edo #Sagittal notation|15]] and [[29edo #Sagittal notation|29]], is a subset of the notations for edos [[44edo #Sagittal notation|44]] and [[66edo #Sagittal notation|66]], and is a superset of the notation for [[11edo #Sagittal notation|11edo]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
| Line 507: | Line 509: | ||
[[File:22edo Sagittal.png|800px]] | [[File:22edo Sagittal.png|800px]] | ||
=== Superpyth/ | === Superpyth/porcupine notation === | ||
Superpyth/ | 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 === | === Porcupine notation === | ||
| Line 534: | Line 536: | ||
=== Comparison of 22edo notation systems === | === Comparison of 22edo notation systems === | ||
{| class="wikitable center-all right-2" | {| class="wikitable center-all right-2 mw-collapsible mw-collapsed" | ||
|- | |- | ||
! [[Degree]] | ! [[Degree]] | ||
! [[Cent]]s | ! [[Cent]]s | ||
! colspan="2" | Superpyth/ | ! colspan="2" | Superpyth/porcupine | ||
! colspan="3" | Porcupine (Onyx) | ! colspan="3" | Porcupine (Onyx) | ||
! colspan="3" |Porcupine (Zarlino) | ! colspan="3" | Porcupine (Zarlino) | ||
! colspan="3" | Pentatonic | ! colspan="3" | Pentatonic | ||
! colspan="3" | Decatonic | ! colspan="3" | Decatonic | ||
! colspan="3" | [[Ups and downs notation|Ups and | ! colspan="3" | [[Ups and downs notation|Ups and downs]] | ||
! colspan="3" | [[SKULO interval names]] | ! colspan="3" | [[SKULO interval names]] | ||
|- | |- | ||
| 0 | | 0 | ||
| 0 | | 0 | ||
| Natural | | Natural unison | ||
| 1 | | 1 | ||
| perfect unison | | perfect unison | ||
| P1 | | P1 | ||
| D | | D | ||
|perfect unison | | perfect unison | ||
|P1 | | P1 | ||
|C | | C | ||
| perfect unison | | perfect unison | ||
| P1 | | P1 | ||
| Line 576: | Line 578: | ||
| A1 | | A1 | ||
| D# | | D# | ||
|augmented unison | | augmented unison | ||
|A1 | | A1 | ||
|C# | | C# | ||
| aug unison | | aug unison | ||
| A1 | | A1 | ||
| Line 599: | Line 601: | ||
| d2 | | d2 | ||
| Eb | | Eb | ||
|minor second | | minor second | ||
|m2 | | m2 | ||
|Db | | Db | ||
| double-aug unison,<br | | double-aug unison,<br>double-dim sub3rd | ||
| AA1,<br | | AA1,<br>dds3 | ||
| Dx,<br | | Dx,<br>Fb<span style="vertical-align: super;">3</span> | ||
| natural 2nd | | natural 2nd | ||
| N2 | | N2 | ||
| Line 622: | Line 624: | ||
| P2 | | P2 | ||
| E | | E | ||
|narrow major second | | narrow major second | ||
|nM2 | | nM2 | ||
|D | | D | ||
| dim sub3rd | | dim sub3rd | ||
| ds3 | | ds3 | ||
| Line 640: | Line 642: | ||
| 4 | | 4 | ||
| 218 | | 218 | ||
| (s/p) | | (s/p) major second | ||
| M2 | | M2 | ||
| aug 2nd | | aug 2nd | ||
| A2 | | A2 | ||
| E# | | E# | ||
|wide major second | | wide major second | ||
|WM2 | | WM2 | ||
|D# | | D# | ||
| minor sub3rd | | minor sub3rd | ||
| ms3 | | ms3 | ||
| Line 668: | Line 670: | ||
| d3 | | d3 | ||
| Fb | | Fb | ||
|wolf third | | wolf third | ||
|w3 | | w3 | ||
|Ebb | | Ebb | ||
| major sub3rd | | major sub3rd | ||
| Ms3 | | Ms3 | ||
| Line 691: | Line 693: | ||
| m3 | | m3 | ||
| F | | F | ||
|minor third | | minor third | ||
|m3 | | m3 | ||
|Eb | | Eb | ||
| aug sub3rd | | aug sub3rd | ||
| As3 | | As3 | ||
| Line 709: | Line 711: | ||
| 7 | | 7 | ||
| 382 | | 382 | ||
| p- | | p-major third | ||
| pM3 | | pM3 | ||
| major 3rd | | major 3rd | ||
| M3 | | M3 | ||
| F# | | F# | ||
|major third | | major third | ||
|M3 | | M3 | ||
|E | | E | ||
| double-aug sub3rd,<br | | double-aug sub3rd,<br>double-dim 4thoid | ||
| AAs3,<br | | AAs3,<br>dd4d | ||
| Fx,<br | | Fx,<br>Gbb | ||
| natural 4th | | natural 4th | ||
| N4 | | N4 | ||
| Line 732: | Line 734: | ||
| 8 | | 8 | ||
| 436 | | 436 | ||
| s- | | s-major third | ||
| sM3 | | sM3 | ||
| aug 3rd, dim 4th | | aug 3rd, dim 4th | ||
| A3, d4 | | A3, d4 | ||
| Fx, Gb | | Fx, Gb | ||
|augmented third | | augmented third | ||
|A3 | | A3 | ||
|E# | | E# | ||
| dim 4thoid | | dim 4thoid | ||
| d4d | | d4d | ||
| Line 755: | Line 757: | ||
| 9 | | 9 | ||
| 491 | | 491 | ||
| Natural | | Natural fourth | ||
| 4, N4 | | 4, N4 | ||
| minor 4th | | minor 4th | ||
| m4 | | m4 | ||
| G | | G | ||
|perfect fourth | | perfect fourth | ||
|P4 | | P4 | ||
|F | | F | ||
| perfect 4thoid | | perfect 4thoid | ||
| P4d | | P4d | ||
| Line 778: | Line 780: | ||
| 10 | | 10 | ||
| 545 | | 545 | ||
| p- | | p-major fourth, s-dim fifth | ||
| pM4, sd5 | | pM4, sd5 | ||
| major 4th | | major 4th | ||
| M4 | | M4 | ||
| G# | | G# | ||
|wolf fourth | | wolf fourth | ||
|w4 | | w4 | ||
|F# | | F# | ||
| aug 4thoid | | aug 4thoid | ||
| A4d | | A4d | ||
| Line 801: | Line 803: | ||
| 11 | | 11 | ||
| 600 | | 600 | ||
| p- | | p-augmented fourth,<br>p-diminished fifth,<br>half-octave | ||
| A4, HO | | A4, HO | ||
| aug 4th, <br | | aug 4th, <br>dim 5th | ||
| A4, d5 | |||
| Gx, <br>Abb | |||
| augmented fourth, diminished fifth | |||
| A4, d5 | | A4, d5 | ||
| F##, Gbb | |||
| double-aug 4thoid,<br>double-dim 5thoid | |||
| AA4d, <br>dd5d | |||
|F##, Gbb | | Gx, <br>Abb | ||
| double-aug 4thoid,<br | |||
| AA4d, <br | |||
| Gx, <br | |||
| natural 6th | | natural 6th | ||
| N6 | | N6 | ||
| Line 818: | Line 820: | ||
| vA4, ^d5 | | vA4, ^d5 | ||
| vG#, ^Ab | | vG#, ^Ab | ||
| comma-narrow augmented 4th<br | | comma-narrow augmented 4th<br>comma-wide diminished 5th | ||
| kA4<br | | kA4<br>Kd5 | ||
| kG#, KAb | | kG#, KAb | ||
|- | |- | ||
| 12 | | 12 | ||
| 655 | | 655 | ||
| p-minor | | p-minor fifth, s-aug fourth | ||
| pm5, sA4 | | pm5, sA4 | ||
| minor 5th | | minor 5th | ||
| m5 | | m5 | ||
| Ab | | Ab | ||
|wolf fifth | | wolf fifth | ||
|w5 | | w5 | ||
|Gb | | Gb | ||
| dim 5thoid | | dim 5thoid | ||
| d5d | | d5d | ||
| Line 847: | Line 849: | ||
| 13 | | 13 | ||
| 709 | | 709 | ||
| Natural | | Natural fifth | ||
| 5, N5 | | 5, N5 | ||
| major 5th | | major 5th | ||
| M5 | | M5 | ||
| A | | A | ||
|perfect fifth | | perfect fifth | ||
|P5 | | P5 | ||
|G | | G | ||
| perfect 5thoid | | perfect 5thoid | ||
| P5d | | P5d | ||
| Line 875: | Line 877: | ||
| A5, d6 | | A5, d6 | ||
| A#, Bbb | | A#, Bbb | ||
|diminished sixth | | diminished sixth | ||
|d6 | | d6 | ||
|Abb | | Abb | ||
| aug 5thoid | | aug 5thoid | ||
| A5d | | A5d | ||
| Line 898: | Line 900: | ||
| m6 | | m6 | ||
| Bb | | Bb | ||
|minor sixth | | minor sixth | ||
|m6 | | m6 | ||
|Ab | | Ab | ||
| double-aug 5thoid,<br | | double-aug 5thoid,<br>double-dim sub7th | ||
| AA5d,<br | | AA5d,<br>dds7 | ||
| Ax,<br | | Ax,<br>Cb<span style="vertical-align: super;">3</span> | ||
| flat 8th | | flat 8th | ||
| f8 | | f8 | ||
| Line 916: | Line 918: | ||
| 16 | | 16 | ||
| 873 | | 873 | ||
| p- | | p-major sixth | ||
| pM6 | | pM6 | ||
| major 6th | | major 6th | ||
| M6 | | M6 | ||
| B | | B | ||
|major sixth | | major sixth | ||
|M6 | | M6 | ||
|A | | A | ||
| dim sub7th | | dim sub7th | ||
| ds7 | | ds7 | ||
| Line 939: | Line 941: | ||
| 17 | | 17 | ||
| 927 | | 927 | ||
| s- | | s-major sixth | ||
| sM6 | | sM6 | ||
| aug 6th | | aug 6th | ||
| A6 | | A6 | ||
| B# | | B# | ||
|wolf sixth | | wolf sixth | ||
|w6 | | w6 | ||
|A# | | A# | ||
| minor sub7th | | minor sub7th | ||
| ms7 | | ms7 | ||
| Line 967: | Line 969: | ||
| d7 | | d7 | ||
| Cb | | Cb | ||
|narrow minor seventh | | narrow minor seventh | ||
|nm7 | | nm7 | ||
|Bbb | | Bbb | ||
| major sub7th | | major sub7th | ||
| Ms7 | | Ms7 | ||
| Line 985: | Line 987: | ||
| 19 | | 19 | ||
| 1036 | | 1036 | ||
| p- | | p-major seventh | ||
| pM7 | | pM7 | ||
| perfect 7th | | perfect 7th | ||
| P7 | | P7 | ||
| C | | C | ||
|wide minor seventh | | wide minor seventh | ||
|Wm7 | | Wm7 | ||
|Bb | | Bb | ||
| aug sub7th | | aug sub7th | ||
| As7 | | As7 | ||
| Line 1,008: | Line 1,010: | ||
| 20 | | 20 | ||
| 1091 | | 1091 | ||
| p- | | p-augmented seventh | ||
| pA7 | | pA7 | ||
| aug 7th | | aug 7th | ||
| A7 | | A7 | ||
| C# | | C# | ||
|major seventh | | major seventh | ||
|M7 | | M7 | ||
|B | | B | ||
| double-aug sub7th,<br | | double-aug sub7th,<br>double-dim octave | ||
| AAs7,<br | | AAs7,<br>dd8 | ||
| Cx,<br | | Cx,<br>Dbb | ||
| natural 10th | | natural 10th | ||
| N10 | | N10 | ||
| Line 1,031: | Line 1,033: | ||
| 21 | | 21 | ||
| 1145 | | 1145 | ||
| s- | | s-major seventh | ||
| sM7 | | sM7 | ||
| dim 8ve | | dim 8ve | ||
| d8 | | d8 | ||
| Db | | Db | ||
|diminished octave | | diminished octave | ||
|d8 | | d8 | ||
|Cb | | Cb | ||
| dim octave | | dim octave | ||
| d8 | | d8 | ||
| Line 1,059: | Line 1,061: | ||
| P8 | | P8 | ||
| D | | D | ||
|perfect octave | | perfect octave | ||
|P8 | | P8 | ||
|C | | C | ||
| perfect octave | | perfect octave | ||
| P8 | | P8 | ||
| Line 1,081: | Line 1,083: | ||
=== Interval mappings === | === Interval mappings === | ||
{{Q-odd-limit intervals|22}} | {{Q-odd-limit intervals|22}} | ||
{{Q-odd-limit intervals|22.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 22f val mapping}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 1,149: | Line 1,152: | ||
| 3 | | 3 | ||
| <abbr title="34359738368/31381059609">(22 digits)</abbr> | | <abbr title="34359738368/31381059609">(22 digits)</abbr> | ||
| {{ | | {{Monzo| 35 -22 }} | ||
| 156.98 | | 156.98 | ||
| Trisawa | | Trisawa | ||
| Line 1,156: | Line 1,159: | ||
| 5 | | 5 | ||
| [[20480/19683]] | | [[20480/19683]] | ||
| {{ | | {{Monzo| 12 -9 1 }} | ||
| 68.72 | | 68.72 | ||
| Sayo | | Sayo | ||
| Line 1,163: | Line 1,166: | ||
| 5 | | 5 | ||
| [[250/243]] | | [[250/243]] | ||
| {{ | | {{Monzo| 1 -5 3 }} | ||
| 49.17 | | 49.17 | ||
| Triyo | | Triyo | ||
| Line 1,170: | Line 1,173: | ||
| 5 | | 5 | ||
| [[3125/3072]] | | [[3125/3072]] | ||
| {{ | | {{Monzo| -10 -1 5 }} | ||
| 29.61 | | 29.61 | ||
| Laquinyo | | Laquinyo | ||
| Line 1,177: | Line 1,180: | ||
| 5 | | 5 | ||
| [[2048/2025]] | | [[2048/2025]] | ||
| {{ | | {{Monzo| 11 -4 -2 }} | ||
| 19.55 | | 19.55 | ||
| Sagugu | | Sagugu | ||
| Line 1,184: | Line 1,187: | ||
| 5 | | 5 | ||
| [[2109375/2097152| (14 digits)]] | | [[2109375/2097152| (14 digits)]] | ||
| {{ | | {{Monzo| -21 3 7 }} | ||
| 10.06 | | 10.06 | ||
| Lasepyo | | Lasepyo | ||
| Line 1,191: | Line 1,194: | ||
| 5 | | 5 | ||
| <abbr title="4294967296/4271484375">(20 digits)</abbr> | | <abbr title="4294967296/4271484375">(20 digits)</abbr> | ||
| {{ | | {{Monzo| 32 -7 -9 }} | ||
| 9.49 | | 9.49 | ||
| Sasa-tritrigu | | Sasa-tritrigu | ||
| Line 1,198: | Line 1,201: | ||
| 5 | | 5 | ||
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr> | | <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr> | ||
| {{ | | {{Monzo| -53 10 16 }} | ||
| 0.57 | | 0.57 | ||
| Quadla-quadquadyo | | Quadla-quadquadyo | ||
| Line 1,205: | Line 1,208: | ||
| 7 | | 7 | ||
| [[50/49]] | | [[50/49]] | ||
| {{ | | {{Monzo| 1 0 2 -2 }} | ||
| 34.98 | | 34.98 | ||
| Biruyo | | Biruyo | ||
| Line 1,212: | Line 1,215: | ||
| 7 | | 7 | ||
| [[64/63]] | | [[64/63]] | ||
| {{ | | {{Monzo| 6 -2 0 -1 }} | ||
| 27.26 | | 27.26 | ||
| Ru | | Ru | ||
| Line 1,219: | Line 1,222: | ||
| 7 | | 7 | ||
| [[875/864]] | | [[875/864]] | ||
| {{ | | {{Monzo|-5 -3 3 1 }} | ||
| 21.90 | | 21.90 | ||
| Zotriyo | | Zotriyo | ||
| Line 1,226: | Line 1,229: | ||
| 7 | | 7 | ||
| [[2430/2401]] | | [[2430/2401]] | ||
| {{ | | {{Monzo| 1 5 1 -4 }} | ||
| 20.79 | | 20.79 | ||
| Quadru-ayo | | Quadru-ayo | ||
| Line 1,233: | Line 1,236: | ||
| 7 | | 7 | ||
| [[245/243]] | | [[245/243]] | ||
| {{ | | {{Monzo| 0 -5 1 2 }} | ||
| 14.19 | | 14.19 | ||
| Zozoyo | | Zozoyo | ||
| Line 1,240: | Line 1,243: | ||
| 7 | | 7 | ||
| [[1728/1715]] | | [[1728/1715]] | ||
| {{ | | {{Monzo| 6 3 -1 -3 }} | ||
| 13.07 | | 13.07 | ||
| Triru-agu | | Triru-agu | ||
| Line 1,247: | Line 1,250: | ||
| 7 | | 7 | ||
| [[225/224]] | | [[225/224]] | ||
| {{ | | {{Monzo| -5 2 2 -1 }} | ||
| 7.71 | | 7.71 | ||
| Ruyoyo | | Ruyoyo | ||
| Line 1,254: | Line 1,257: | ||
| 7 | | 7 | ||
| [[10976/10935]] | | [[10976/10935]] | ||
| {{ | | {{Monzo| 5 -7 -1 3 }} | ||
| 6.48 | | 6.48 | ||
| Trizo-agu | | Trizo-agu | ||
| Line 1,261: | Line 1,264: | ||
| 7 | | 7 | ||
| [[6144/6125]] | | [[6144/6125]] | ||
| {{ | | {{Monzo| 11 1 -3 -2 }} | ||
| 5.36 | | 5.36 | ||
| Saruru-atrigu | | Saruru-atrigu | ||
| Line 1,268: | Line 1,271: | ||
| 7 | | 7 | ||
| [[65625/65536]] | | [[65625/65536]] | ||
| {{ | | {{Monzo| -16 1 5 1 }} | ||
| 2.35 | | 2.35 | ||
| Lazoquinyo | | Lazoquinyo | ||
| Line 1,275: | Line 1,278: | ||
| 7 | | 7 | ||
| <abbr title="420175/419904">(12 digits)</abbr> | | <abbr title="420175/419904">(12 digits)</abbr> | ||
| {{ | | {{Monzo| -6 -8 2 5 }} | ||
| 1.12 | | 1.12 | ||
| Quinzo-ayoyo | | Quinzo-ayoyo | ||
| Line 1,282: | Line 1,285: | ||
| 11 | | 11 | ||
| [[99/98]] | | [[99/98]] | ||
| {{ | | {{Monzo| -1 2 0 -2 1 }} | ||
| 17.58 | | 17.58 | ||
| Loruru | | Loruru | ||
| Line 1,289: | Line 1,292: | ||
| 11 | | 11 | ||
| [[100/99]] | | [[100/99]] | ||
| {{ | | {{Monzo| 2 -2 2 0 -1 }} | ||
| 17.40 | | 17.40 | ||
| Luyoyo | | Luyoyo | ||
| Line 1,296: | Line 1,299: | ||
| 11 | | 11 | ||
| [[121/120]] | | [[121/120]] | ||
| {{ | | {{Monzo| -3 -1 -1 0 2 }} | ||
| 14.37 | | 14.37 | ||
| Lologu | | Lologu | ||
| Line 1,303: | Line 1,306: | ||
| 11 | | 11 | ||
| [[176/175]] | | [[176/175]] | ||
| {{ | | {{Monzo| 4 0 -2 -1 1 }} | ||
| 9.86 | | 9.86 | ||
| Lorugugu | | Lorugugu | ||
| Line 1,310: | Line 1,313: | ||
| 11 | | 11 | ||
| [[896/891]] | | [[896/891]] | ||
| {{ | | {{Monzo| 7 -4 0 1 -1 }} | ||
| 9.69 | | 9.69 | ||
| Saluzo | | Saluzo | ||
| Line 1,317: | Line 1,320: | ||
| 11 | | 11 | ||
| [[65536/65219]] | | [[65536/65219]] | ||
| {{ | | {{Monzo| 16 0 0 -2 -3 }} | ||
| 8.39 | | 8.39 | ||
| Satrilu-aruru | | Satrilu-aruru | ||
| Line 1,324: | Line 1,327: | ||
| 11 | | 11 | ||
| [[385/384]] | | [[385/384]] | ||
| {{ | | {{Monzo|-7 -1 1 1 1 }} | ||
| 4.50 | | 4.50 | ||
| Lozoyo | | Lozoyo | ||
| Line 1,331: | Line 1,334: | ||
| 11 | | 11 | ||
| [[540/539]] | | [[540/539]] | ||
| {{ | | {{Monzo| 2 3 1 -2 -1 }} | ||
| 3.21 | | 3.21 | ||
| Lururuyo | | Lururuyo | ||
| Line 1,338: | Line 1,341: | ||
| 11 | | 11 | ||
| [[4000/3993]] | | [[4000/3993]] | ||
| {{ | | {{Monzo| 5 -1 3 0 -3 }} | ||
| 3.03 | | 3.03 | ||
| Triluyo | | Triluyo | ||
| Line 1,345: | Line 1,348: | ||
| 11 | | 11 | ||
| [[9801/9800]] | | [[9801/9800]] | ||
| {{ | | {{Monzo| -3 4 -2 -2 2 }} | ||
| 0.18 | | 0.18 | ||
| Bilorugu | | Bilorugu | ||
| Line 1,352: | Line 1,355: | ||
| 13 | | 13 | ||
| [[65/64]] | | [[65/64]] | ||
| {{ | | {{Monzo| -6 0 1 0 0 1 }} | ||
| 26.84 | | 26.84 | ||
| Thoyo | | Thoyo | ||
| Line 1,359: | Line 1,362: | ||
| 13 | | 13 | ||
| [[78/77]] | | [[78/77]] | ||
| {{ | | {{Monzo| 1 1 0 -1 -1 1 }} | ||
| 22.34 | | 22.34 | ||
| Tholuru | | Tholuru | ||
| Line 1,366: | Line 1,369: | ||
| 13 | | 13 | ||
| [[91/90]] | | [[91/90]] | ||
| {{ | | {{Monzo| -1 -2 -1 1 0 1 }} | ||
| 19.13 | | 19.13 | ||
| Thozogu | | Thozogu | ||
| Line 1,373: | Line 1,376: | ||
| 13 | | 13 | ||
| [[31213/31104]] | | [[31213/31104]] | ||
| {{ | | {{Monzo| -7 -5 0 4 0 1 }} | ||
| 6.06 | | 6.06 | ||
| Thoquadzo | | Thoquadzo | ||
| Line 1,380: | Line 1,383: | ||
| 31 | | 31 | ||
| [[125/124]] | | [[125/124]] | ||
| {{ | | {{Monzo| -2 0 3 0 0 0 0 0 0 0 -1 }} | ||
| 13.91 | | 13.91 | ||
| Thiwutriyo | | Thiwutriyo | ||
| Line 1,439: | Line 1,442: | ||
| 11 | | 11 | ||
| 1\22 | | 1\22 | ||
| [[Undeka]]<br>[[Hendecatonic]] | | [[Undeka]]<br>[[Hendecatonic (temperament)|Hendecatonic]] | ||
|} | |} | ||
== Octave stretch or compression == | == Octave stretch or compression == | ||
22edo can benefit from slightly compressing the octave, especially when using it as an | 22edo can benefit from slightly compressing the octave, especially when using it as an 7-limit equal temperament. With the right amount of compression we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11. | ||
Good compressed-22 options include: [[ZPI|80zpi]] or [[57ed6]]. | Good compressed-22 options include: [[ZPI|80zpi]] or [[57ed6]]. | ||
| Line 1,544: | Line 1,547: | ||
== Instruments == | == Instruments == | ||
== Scordatura piano == | |||
Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by [[Juhani Nuorvala]]'s [https://www.youtube.com/watch?v=raRiTvogBBA ''Improvisations on a piano tuned to 22edo''] (2026). | |||
=== Keyboards === | === Keyboards === | ||
[[File:22-tone halberstadt layout.png|alt=|frameless]] | [[File:22-tone halberstadt layout.png|alt=|frameless]] | ||
| Line 1,577: | Line 1,583: | ||
[[Category:Alpharabian]] | [[Category:Alpharabian]] | ||
[[Category:Superpyth]] | [[Category:Superpyth]] | ||
[[Category:Pajara]] | |||
[[Category:Orwell]] | [[Category:Orwell]] | ||
[[Category:Porcupine]] | [[Category:Porcupine]] | ||
Latest revision as of 01:15, 1 June 2026
| ← 21edo | 22edo | 23edo → |
22 equal divisions of the octave (abbreviated 22edo or 22ed2), also called 22-tone equal temperament (22tet) or 22 equal temperament (22et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 22 equal parts of about 54.5 ¢ each. Each step represents a frequency ratio of 21/22, or the 22nd root of 2. Because it distinguishes 10/9 and 9/8, it is not a meantone system.
History
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 supposed 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 19edo, and J. Murray Barbour in his classic survey of tuning history, Tuning and Temperament.
Theory
22edo is the third edo, after 12 and 19, which is capable of approximating the 5-limit to within a Tenney–Euclidean error of 4 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the 7- and 11-limit to within 3 cents of error, and in fact 22 is the smallest edo to represent the 11-odd-limit consistently, though 31edo is considerably more accurate.
Possibly the most striking characteristic of 22edo to those not used to it is that it does not temper out 81/80 (the syntonic comma), and instead maps it to one step. Additionally, it is a superset of 11edo and is close to 24edo, having only 2 fewer steps than it, and thus behaves like 11edo and 13edo in that melodic movements similar to 12edo can quickly arrive at an unfamiliar place. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.
22edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out 64/63, equating the pythagorean minor seventh with 7/4, and supporting superpyth. In that manner, 22edo can be thought of as widening the gap of 49/48 between septimal intervals like 7/6 and 8/7 to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in 25/24 being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out 50/49, so the 7/5 and 10/7 are equated to the 600 ¢ half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and 12/7. Reasonably, 36/35 is also tempered to 1 step just like 25/24 and 49/48.
22edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5–6 ¢ flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a neutral third dividing the perfect fifth in two, which means 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 ¢ sharp, but also because 22edo's step is just short of being small enough to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 24edo, 27edo, and 31edo all include fully). Because 22edo does not contain "neutral" intervals, 11/9 is mapped to the same interval as 6/5 and 12/11 is mapped to the submajor second, inflating 243/242 to a full step.
Since 22edo's fifth is sharp of just by approximately one quarter of the septimal comma (64/63), and since it tunes the septimal supermajor third (9/7) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended quarter-comma meantone.
22edo is also the third-smallest edo (after 10edo and 15edo) that maintains 25% or lower relative error on all of the first eight harmonics of the harmonic series.
Prime harmonics
| 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 |
| Relative (%) | +0.0 | +13.1 | -8.2 | +23.8 | -10.7 | -41.0 | +7.6 | -45.4 | +48.2 | +12.4 | +0.8 | |
| Steps (reduced) |
22 (0) |
35 (13) |
51 (7) |
62 (18) |
76 (10) |
81 (15) |
90 (2) |
93 (5) |
100 (12) |
107 (19) |
109 (21) | |
As a tuning of other temperaments
Observance of 81/80
22edo, unlike 12 and 19, is not a system of meantone temperament, and as such it distinguishes a number of 3-limit and 5-limit intervals that meantone tunings (most notably 12edo, 19edo, 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as 34edo, 41edo, and 53edo, allowing many opportunities for alternate interpretations of these intervals. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to the 5L 2s mos as in meantone systems. Instead, it is a ternary scale, having the nicetone pattern.
Superpyth temperament
The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from superpyth temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic mos has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 is tempered out rather than the syntonic comma of 81/80, which one of 22et's core features.
Superpyth temperament equates the Pythagorean sevenths (such as A–G and C–B♭ in chain-of-fifths notation) to harmonic sevenths instead of 5-limit minor sevenths (approximating 7/4 instead of 9/5). Due to the sharper fifths, the diatonic scale is more uneven than in meantone systems and 12edo. In addition to the more uneven diatonic scale, 22edo has a quasi-equal pentatonic scale (the major whole tone and subminor third are rather close in size). The step patterns of the pentatonic and diatonic scales in 22et are 4 – 4 – 5 – 4 – 5 and 4 – 4 – 1 – 4 – 4 – 4 – 1 respectively. In superpyth (and thus in 22edo and technically 12edo), the 1–5/4–3/2–16/9 dominant seventh chord and an otonal tetrad are represented by the same chord.
Porcupine temperament
22edo additionally tempers out the porcupine comma or maximal diesis of 250/243 (S102⋅S11), which means that 22edo supports porcupine temperament. The generator for porcupine is a very flat minor whole tone of ~10/9 (usually tuned slightly flat of 11/10), two of which is a 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 temperament allows the 5-limit diatonic scale (the zarlino scale), present as 4 – 3 – 2 – 4 – 3 – 4 – 2 and tuned particularly accurately in 22edo, to be notated with only 1 set of accidentals (conventionally sharps and flats) representing both the syntonic comma and the classical chromatic semitone, as the difference between them (250/243) is tempered out.
It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. Porcupine's generator forms mos scales of 7 and 8, which in 22edo are tuned respectively as 4 – 3 – 3 – 3 – 3 – 3 – 3 and 1 – 3 – 3 – 3 – 3 – 3 – 3 – 3 (and their respective modes).
Pajara temperament
A third important temperament that 22edo supports is pajara. In the 5-limit, 2048/2025 (diaschisma) is tempered out, meaning that the 5-limit tritones are equated to one another and to the semioctave. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, 50/49 (jubilisma) is tempered out, meaning that the tritones 7/5 and 10/7 are equated to the semioctave, and consequently 64/63 is tempered out as in superpyth—5/4 is a semioctave away from 7/4. Since 50/49 is tempered out, the 25/24 and 49/48 intervals are equated to a single interval, and it functions as a chroma in the 2L 8s mos. This suggests the use of a decatonic notation system, where 7/6 and 8/7 are the same number of scale degrees, and 7/4 is a major interval. Thus the 1–5/4–3/2–7/4 major tetrad has 5/4 and 7/4 as major intervals, and replacing them with the corresponding minor intervals gives us the 1–6/5–3/2–12/7 subharmonic sixth chord or minor tetrad. Pajara temperament is also supported by 12edo, as it also tempers out 50/49 and 64/63.
The decatonic scales of pajara have been considered by many to be a system in the 7-limit analogous to the diatonic scale of meantone temperament in the 5-limit, as described in Paul Erlich's paper Tuning, Tonality and 22-Tone Temperament.
Additional commas
Both 22edo and 12edo also temper out (50/49)/(64/63) = 225/224 (S15, marvel comma), so that the marvel augmented triad is a chord of 22et. A 7-limit comma not tempered out by 12et which 22et does temper out is 1728/1715, the orwell comma; therefore, the orwell tetrad is also a chord of 22et. The orwell temperament uses the septimal subminor third (5 degrees) as a generator, and forms mos scales with step patterns 2 – 3 – 2 – 3 – 2 – 3 – 2 – 3 – 2 and 2 – 1 – 2 – 2 – 1 – 2 – 2 – 1 – 2 – 2 – 1 – 2 – 2. While orwell can be tuned more accurately in other temperaments, such as 31edo, 53edo, and 84edo, 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish.
Subsets, supersets, and inheritances
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 quartertones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In particular, 22edo can be roughly conceptualized as 24 but with only two types of thirds rather than three. In Sagittal notation, 11 can be notated as every other note of 22.
22 inherits 11edo's 11/8 and 7/4, and inherits 2edo's tritone, which is mapped in both systems to 7/5 and 10/7.
Other features
The 163.6 ¢ "flat minor whole tone" or "submajor second" 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.
Higher-limit interpretations
22edo can also be treated as adding harmonics 3 and 5 to 11edo's 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 subgroup temperament. Also note that its approximation of the 31st harmonic is within half a cent, which is very 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 the 2.3.5.7.11.17.29.31 subgroup.
Intervals
| Degree | Cents | Approximate Ratios[note 1] | Audio | Ups and downs notation (EUs: v3A1 and ^^d2) |
SKULO notation (K = 1) | ||||
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0 | 1/1 | perfect unison | P1 | D | perfect unison | P1 | D | |
| 1 | 54.5 | 36/35, 34/33, 33/32, 32/31 | up-unison, minor 2nd | ^1, m2 | ^D, Eb | comma-wide unison, minor 2nd | K1, m2 | KD, Eb | |
| 2 | 109.1 | 18/17, 17/16, 16/15, 15/14 | downaug 1sn, upminor 2nd | vA1, ^m2 | vD#, ^Eb | classic minor 2nd | Km2 | KEb | |
| 3 | 163.6 | 12/11, 11/10, 10/9 | aug 1sn, downmajor 2nd | A1, vM2 | D#, vE | classic/comma-narrow major 2nd | kM2 | kE | |
| 4 | 218.2 | 9/8, 17/15, 8/7 | major 2nd | M2 | E | major 2nd | M2 | E | |
| 5 | 272.7 | 20/17, 7/6 | minor 3rd | m3 | F | minor 3rd | m3 | F | |
| 6 | 327.3 | 6/5, 17/14, 11/9 | upminor 3rd | ^m3 | ^F | classic minor 3rd | Km3 | KF | |
| 7 | 381.8 | 5/4, 96/77 | downmajor 3rd | vM3 | vF# | classic major 3rd | kM3 | kF# | |
| 8 | 436.4 | 14/11, 9/7, 22/17 | major 3rd | M3 | F# | major 3rd | M3 | F# | |
| 9 | 490.9 | 4/3 | perfect 4th | P4 | G | perfect 4th | P4 | G | |
| 10 | 545.5 | 15/11, 11/8 | up-4th, dim 5th | ^4, d5 | ^G, Ab | comma-wide 4th | K4 | KG | |
| 11 | 600.0 | 7/5, 24/17, 17/12, 10/7 | downaug 4th, updim 5th | vA4, ^d5 | vG#, ^Ab | comma-narrow augmented 4th comma-wide diminished 5th |
kA4 Kd5 |
kG#, KAb | |
| 12 | 654.5 | 16/11, 22/15 | aug 4th, down-5th | A4, v5 | G#, vA | comma-narrow 5th | k5 | kA | |
| 13 | 709.1 | 3/2 | perfect 5th | P5 | A | perfect 5th | P5 | A | |
| 14 | 763.6 | 17/11, 14/9, 11/7 | minor 6th | m6 | Bb | minor 6th | m6 | Bb | |
| 15 | 818.2 | 8/5, 77/48 | upminor 6th | ^m6 | ^Bb | classic minor 6th | Km6 | KBb | |
| 16 | 872.7 | 18/11, 28/17, 5/3 | downmajor 6th | vM6 | vB | classic major 6th | kM6 | kB | |
| 17 | 927.3 | 17/10, 12/7 | major 6th | M6 | B | major 6th | M6 | B | |
| 18 | 981.8 | 7/4, 30/17, 16/9 | minor 7th | m7 | C | minor 7th | m7 | C | |
| 19 | 1036.4 | 9/5, 11/6, 20/11 | upminor 7th, dim 8ve | ^m7, d8 | ^C, Db | classic minor 7th | Km7 | kC | |
| 20 | 1090.9 | 28/15, 15/8, 32/17, 17/9 | downmajor 7th, updim 8ve | vM7, ^d8 | vC#, ^Db | classic major 7th | kM7 | kC# | |
| 21 | 1145.5 | 31/16, 64/33, 33/17, 35/18 | major 7th, down 8ve | M7, v8 | C#, vD | major 7th / comma-narrow 8ve | M7 / k8 | C#, kD | |
| 22 | 1200.0 | 2/1 | perfect octave | P8 | D | perfect 8ve | P8 | D | |
Notation
Stein–Zimmermann–Gould notation
Since a sharp raises by three steps, 22edo is a good candidate for Stein–Zimmermann–Gould notation, using sharps and flats with arrows similar to 29edo:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Sharp symbol | | | | | | | | |
| Flat symbol | | | | | | | |
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with double arrows.
Kite's ups and downs notation
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
|
 |
 
|
|
|
|
|
|
| Flat symbol | |
|
|
|
|
|
|
Standard Pythagorean chain-of-fifths notation can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 edostep (1\22). Note that E♭ and D♯ are different notes and that E♭ is significantly lower in pitch than D♯.
| # | Cents | Kite's ups and downs notation | |
|---|---|---|---|
| Diatonic interval names | Note names | ||
| 0 | 0.0 | Perfect unison (P1) | D |
| 1 | 54.5 | Minor second (m2) Up unison (^1) |
Eb ^D |
| 2 | 109.1 | Upminor second (^m2) Downaugmented unison (vA1) Diminished third (d3) |
^Eb vD# Fb |
| 3 | 163.6 | Downmajor second (vM2) Augmented unison (A1) |
vE D# |
| 4 | 218.2 | Major second (M2) Upaugmented unison (^A1) Downminor third (vm3) |
E ^D# vF |
| 5 | 272.7 | Upmajor second (^M2) Minor third (m3) |
^E F |
| 6 | 327.3 | Upminor third (^m3) Diminished fourth (d4) |
^F Gb |
| 7 | 381.8 | Downmajor third (vM3) Augmented second (A2) Updiminished fourth (^d4) |
vF# E# ^Gb |
| 8 | 436.4 | Major third (M3) Upaugmented second (^A2) Down fourth (v4) |
F# ^E# vG |
| 9 | 490.9 | Perfect fourth (P4) | G |
| 10 | 545.5 | Up fourth (^4) Diminished fifth (d5) |
^G Ab |
| 11 | 600.0 | Downaugmented fourth (vA4) Updiminished fifth (^d5) |
vG# ^Ab |
| 12 | 654.5 | Augmented fourth (A4) Down fifth (v5) |
G# vA |
| 13 | 709.1 | Perfect fifth (P5) | A |
| 14 | 763.6 | Up fifth (^5) Minor sixth (m6) |
^A Bb |
| 15 | 818.2 | Downaugmented fifth (vA5) Upminor sixth (^m6) |
vA# ^Bb |
| 16 | 872.7 | Augmented fifth (A5) Downmajor sixth (vM6) |
A# vB |
| 17 | 927.3 | Major sixth (M6) Upaugmented fifth (^A5) Downminor seventh (vm7) |
B ^A# vC |
| 18 | 981.8 | Minor seventh (m7) Upmajor sixth (^M6) Downdiminished octave (vd8) |
C ^B vDb |
| 19 | 1036.4 | Upminor seventh (^m7) Diminished octave (d8) |
^C Db |
| 20 | 1090.9 | Downmajor seventh (vM7) Updiminished octave (^d8) Augmented sixth (A6) |
vC# ^Db B# |
| 21 | 1145.5 | Major seventh (M7) Down octave (v8) |
C# vD |
| 22 | 1200.0 | Perfect octave (P8) | D |
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)
Sagittal notation
This notation uses the same sagittal sequence as edos 15 and 29, is a subset of the notations for edos 44 and 66, and is a superset of the notation for 11edo.
Evo flavor
Revo flavor
When 22edo is treated as generated by a cycle of its fifths, the natural notes 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:
Superpyth/porcupine 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.
Porcupine notation
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.
A score video demonstrating this type of notation using redefined sharp and flat symbols is available: Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic) (2020) by John Moriarty. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.
Pentatonic notation
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.
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.
Comparison of 22edo notation systems
| Degree | Cents | Superpyth/porcupine | Porcupine (Onyx) | Porcupine (Zarlino) | Pentatonic | Decatonic | Ups and downs | SKULO interval names | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | Natural unison | 1 | perfect unison | P1 | D | perfect unison | P1 | C | perfect unison | P1 | D | natural 1st | N1 | C | perfect unison | P1 | D | perfect unison | P1 | D |
| 1 | 55 | s-minor second | sm2 | aug unison | A1 | D# | augmented unison | A1 | C# | aug unison | A1 | D# | flat 2nd | f2 | C#, δb | up-unison, minor 2nd | ^1, m2 | ^D, Eb | comma-wide unison, minor 2nd | K1, m2 | KD, Eb |
| 2 | 109 | p-diminished second | pd2 | dim 2nd | d2 | Eb | minor second | m2 | Db | double-aug unison, double-dim sub3rd |
AA1, dds3 |
Dx, Fb3 |
natural 2nd | N2 | δ | downaug 1sn, upminor 2nd | vA1, ^m2 | vD#, ^Eb | classic minor 2nd | Km2 | KEb |
| 3 | 164 | p-minor second | pm2 | perfect 2nd | P2 | E | narrow major second | nM2 | D | dim sub3rd | ds3 | Fbb | sharp 2nd, flat 3rd | s2, f3 | δ#, Db | aug 1sn, downmajor 2nd | A1, vM2 | D#, vE | classic/comma-narrow major 2nd | kM2 | kE |
| 4 | 218 | (s/p) major second | M2 | aug 2nd | A2 | E# | wide major second | WM2 | D# | minor sub3rd | ms3 | Fb | natural 3rd | N3 | D | major 2nd | M2 | E | major 2nd | M2 | E |
| 5 | 273 | s-minor third | sm3 | dim 3rd | d3 | Fb | wolf third | w3 | Ebb | major sub3rd | Ms3 | F | sharp 3rd | s3 | D# | minor 3rd | m3 | F | minor 3rd | m3 | F |
| 6 | 327 | p-minor third | pm3 | minor 3rd | m3 | F | minor third | m3 | Eb | aug sub3rd | As3 | F# | flat 4th | f4 | εb | upminor 3rd | ^m3 | ^F | classic minor 3rd | Km3 | KF |
| 7 | 382 | p-major third | pM3 | major 3rd | M3 | F# | major third | M3 | E | double-aug sub3rd, double-dim 4thoid |
AAs3, dd4d |
Fx, Gbb |
natural 4th | N4 | ε | downmajor 3rd | vM3 | vF# | classic major 3rd | kM3 | kF# |
| 8 | 436 | s-major third | sM3 | aug 3rd, dim 4th | A3, d4 | Fx, Gb | augmented third | A3 | E# | dim 4thoid | d4d | Gb | sharp 4th, flat 5th | s4, f5 | ε#, Eb | major 3rd | M3 | F# | major 3rd | M3 | F# |
| 9 | 491 | Natural fourth | 4, N4 | minor 4th | m4 | G | perfect fourth | P4 | F | perfect 4thoid | P4d | G | natural 5th | N5 | E | perfect 4th | P4 | G | perfect 4th | P4 | G |
| 10 | 545 | p-major fourth, s-dim fifth | pM4, sd5 | major 4th | M4 | G# | wolf fourth | w4 | F# | aug 4thoid | A4d | G# | sharp 5th, flat 6th | s5, f6 | E#, γb | up-4th, dim 5th | ^4, d5 | ^G, Ab | comma-wide 4th | K4 | KG |
| 11 | 600 | p-augmented fourth, p-diminished fifth, half-octave |
A4, HO | aug 4th, dim 5th |
A4, d5 | Gx, Abb |
augmented fourth, diminished fifth | A4, d5 | F##, Gbb | double-aug 4thoid, double-dim 5thoid |
AA4d, dd5d |
Gx, Abb |
natural 6th | N6 | γ | downaug 4th, updim 5th | vA4, ^d5 | vG#, ^Ab | comma-narrow augmented 4th comma-wide diminished 5th |
kA4 Kd5 |
kG#, KAb |
| 12 | 655 | p-minor fifth, s-aug fourth | pm5, sA4 | minor 5th | m5 | Ab | wolf fifth | w5 | Gb | dim 5thoid | d5d | Ab | sharp 6th, flat 7th | s6, f7 | γ#, Gb | aug 4th, down-5th | A4, v5 | G#, vA | comma-narrow 5th | k5 | kA |
| 13 | 709 | Natural fifth | 5, N5 | major 5th | M5 | A | perfect fifth | P5 | G | perfect 5thoid | P5d | A | natural 7th | N7 | G | perfect 5th | P5 | A | perfect 5th | P5 | A |
| 14 | 764 | s-minor sixth | sm6 | aug 5th, dim 6th | A5, d6 | A#, Bbb | diminished sixth | d6 | Abb | aug 5thoid | A5d | A# | sharp 7th | s7 | G# | minor 6th | m6 | Bb | minor 6th | m6 | Bb |
| 15 | 818 | p-minor sixth | pm6 | minor 6th | m6 | Bb | minor sixth | m6 | Ab | double-aug 5thoid, double-dim sub7th |
AA5d, dds7 |
Ax, Cb3 |
flat 8th | f8 | αb | upminor 6th | ^m6 | ^Bb | classic minor 6th | Km6 | KBb |
| 16 | 873 | p-major sixth | pM6 | major 6th | M6 | B | major sixth | M6 | A | dim sub7th | ds7 | Cbb | natural 8th | N8 | α | downmajor 6th | vM6 | vB | classic major 6th | kM6 | kB |
| 17 | 927 | s-major sixth | sM6 | aug 6th | A6 | B# | wolf sixth | w6 | A# | minor sub7th | ms7 | Cb | sharp 8th, flat 9th | s8, f9 | α#, Ab | major 6th | M6 | B | major 6th | M6 | B |
| 18 | 982 | (s/p) minor seventh | m7 | dim 7th | d7 | Cb | narrow minor seventh | nm7 | Bbb | major sub7th | Ms7 | C | natural 9th | N9 | A | minor 7th | m7 | C | minor 7th | m7 | C |
| 19 | 1036 | p-major seventh | pM7 | perfect 7th | P7 | C | wide minor seventh | Wm7 | Bb | aug sub7th | As7 | C# | sharp 9th, flat 10th | s9, f10 | A#, βb | upminor 7th, dim 8ve | ^m7, d8 | ^C, Db | classic minor 7th | Km7 | kC |
| 20 | 1091 | p-augmented seventh | pA7 | aug 7th | A7 | C# | major seventh | M7 | B | double-aug sub7th, double-dim octave |
AAs7, dd8 |
Cx, Dbb |
natural 10th | N10 | β | downmajor 7th, updim 8ve | vM7, ^d8 | vC#, ^Db | classic major 7th | kM7 | kC# |
| 21 | 1145 | s-major seventh | sM7 | dim 8ve | d8 | Db | diminished octave | d8 | Cb | dim octave | d8 | Db | sharp 10th | s10 | β#, Cb | major 7th, down 8ve | M7, v8 | C#, vD | major 7th / comma-narrow 8ve | M7 / k8 | C#, kD |
| 22 | 1200 | Octave | 8 | perfect octave | P8 | D | perfect octave | P8 | C | perfect octave | P8 | D | natural 11th | N11 | C | perfect octave | P8 | D | perfect 8ve | P8 | D |
Approximation to JI
Interval mappings
The following tables show how 15-odd-limit intervals are represented in 22edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 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 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 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 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 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 |
| 7/5, 10/7 | 17.488 | 32.1 |
| 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 |
| 13/12, 24/13 | 25.064 | 46.0 |
| 15/13, 26/15 | 29.559 | 54.2 |
| 13/8, 16/13 | 32.200 | 59.0 |
| 13/10, 20/13 | 36.695 | 67.3 |
| 13/11, 22/13 | 38.063 | 69.8 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| 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 does best in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is 46.
Uniform maps
| Min. size | Max. size | Wart notation | Map |
|---|---|---|---|
| 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] |
Commas
22et tempers out the following commas. This assumes the val ⟨22 35 51 62 76 81].
| Prime limit |
Ratio[note 2] | Monzo | Cents | Color name | Name |
|---|---|---|---|---|---|
| 3 | (22 digits) | [35 -22⟩ | 156.98 | Trisawa | 22-comma |
| 5 | 20480/19683 | [12 -9 1⟩ | 68.72 | Sayo | Superpyth comma |
| 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 comma |
| 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 comma |
| 7 | 245/243 | [0 -5 1 2⟩ | 14.19 | Zozoyo | Sensamagic comma |
| 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 comma |
| 7 | 6144/6125 | [11 1 -3 -2⟩ | 5.36 | Saruru-atrigu | Porwell comma |
| 7 | 65625/65536 | [-16 1 5 1⟩ | 2.35 | Lazoquinyo | Horwell comma |
| 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 comma |
| 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 comma |
| 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 comma, biome comma |
| 13 | 31213/31104 | [-7 -5 0 4 0 1⟩ | 6.06 | Thoquadzo | Praveensma |
| 31 | 125/124 | [-2 0 3 0 0 0 0 0 0 0 -1⟩ | 13.91 | Thiwutriyo | Twizzler comma |
Rank-2 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 8ve |
Generator | Temperaments |
|---|---|---|
| 1 | 1\22 | Escapade / escaped 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 |
Octave stretch or compression
22edo can benefit from slightly compressing the octave, especially when using it as an 7-limit equal temperament. With the right amount of compression we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11.
Good compressed-22 options include: 80zpi or 57ed6.
Scales
Tetrachords
Chords
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 |
Examples:
- 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)
Instruments
Scordatura piano
Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by Juhani Nuorvala's Improvisations on a piano tuned to 22edo (2026).
Keyboards
A potential layout for a 22edo keyboard with both split black and white keys.
Lumatone mappings for 22edo are available.
Music
- See also: Category:22edo tracks
See also
Further reading
- 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
Notes
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