22edo: Difference between revisions

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The 22edo system is the third equal division, 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{{c}} per octave. Moreover, there is more to it than just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly.

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] (S9, syntonic), instead mapping it to one step. 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, like 22-tone guitars.

22edo's approximation to the 7th harmonic is about 12 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. In that manner, 22edo can be thought of as widening the gap of [[49/48]] (S7, semaphore) between septimal intervals to a full quarter-tone. However, the opposite effect is consequently had on the 5-limit: 5/4 is narrow, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] (S5, dicot) narrowed to a quarter tone. Reasonably, [[36/35]] (S6, mint) is tempered to the same 1-step interval.  

22edo's 11-limit status is somewhat contentious: while it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit an inaccurate one), it lacks a [[neutral third]] (unless the sharp 6/5 is considered to be one), meaning that 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 cents sharp.  

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]].

=== Temperaments ===

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 their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale

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 (S8, archytas) is tempered out, rather than the syntonic comma of 81/80, which one of 22et's core features.  

Superpyth as a temperament equates the Pythagorean sevenths (such as A-G, C–Bb in chain-of-fifths notation) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]). In addition to the more uneven diatonic scale as compared to meantone systems and 12edo, 22edo has a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size). The step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively. In superpyth (and thus in 22edo and technically 12edo) the dominant seventh chord and an otonal tetrad are represented by the same chord.

22edo additionally tempers out the porcupine comma or maximal diesis of [[250/243]] (S10² × S11, porcupine), 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 4-3-2-4-3-4-2 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 scale]]s of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3|med}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3|med}} (and their respective modes).

A third important temperament that 22edo supports is pajara. In the 5-limit, [[2048/2025]] (S16² × S17, [[diaschismic]]) 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]] (S5 / S7, [[jubilismic]]) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are also merged to the semioctave, and consequently 64/63 is tempered out as in superpyth - 5/4 is a semioctave away from 7/4 (which is why in 22edo, their tunings are inaccurate in the opposite directions, and why both semitwelfths and thirds have one particularly accurate interval (12/7, 5/4) and the opposite being rather sharp (7/4, 6/5)). Pajara temperament is also supported by [[12edo]], as that system also tempers out 50/49 and 64/63.

Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} (S15, [[marvel]]), 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 [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third as a generator (5 degrees) and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2|med}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2|med}}. Harmonically, orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]]. But 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish.

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 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]].  

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.

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. Let us also mind 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 2.3.5.7.11.17.29.31.  

 

 

=== Ups and downs notation ===

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♯.

! rowspan="2" | [[Degree]]

! colspan="2" | [[Ups and downs notation|Ups and downs notation]]

! [[5L 2s|Diatonic Interval Names]]

! Note Names

| Minor second (m2)<br />Up unison (^1)

| Eb<br />^D

| Upminor second (^m2)<br />Downaugmented unison (vA1)<br />Diminished third (d3)

| ^Eb<br />vD#<br />Fb

| Downmajor second (vM2)<br />Augmented unison (A1)

| vE<br />D#

| '''Major second (M2)'''<br />Upaugmented unison (^A1)<br />Downminor third (vm3)

| '''E'''<br />^D#<br />vF

| Upmajor second (^M2)<br />'''Minor third (m3)'''

| ^E<br />'''F'''

| '''Upminor third (^m3)'''<br />Diminished fourth (d4)

| '''^F'''<br />Gb

| '''Downmajor third (vM3)'''<br />Augmented second (A2)<br />Updiminished fourth (^d4)

| '''vF#'''<br />E#<br />^Gb

| '''Major third (M3)'''<br />Upaugmented second (^A2)<br />Down fourth (v4)

| '''F#'''<br />^E#<br />vG

| Up fourth (^4)<br />Diminished fifth (d5)

| ^G<br />Ab

| Downaugmented fourth (vA4)<br />Updiminished fifth (^d5)

| vG#<br />^Ab

| Augmented fourth (A4)<br />Down fifth (v5)

| G#<br />vA

| Up fifth (^5)<br />Minor sixth (m6)

| ^A<br />Bb

| Downaugmented fifth (vA5)<br />Upminor sixth (^m6)

| vA#<br />^Bb

| Augmented fifth (A5)<br />'''Downmajor sixth (vM6)'''

| A#<br />'''vB'''

| '''Major sixth (M6)'''<br />Upaugmented fifth (^A5)<br />Downminor seventh (vm7)

| '''B'''<br />^A#<br />vC

| '''Minor seventh (m7)'''<br />Upmajor sixth (^M6)<br />Downdiminished octave (vd8)

| '''C'''<br />^B<br />vDb

| '''Upminor seventh (^m7)'''<br />Diminished octave (d8)

| '''^C'''<br />Db

| Downmajor seventh (vM7)<br />Updiminished octave (^d8)<br />Augmented sixth (A6)

| vC#<br />^Db<br />B#

| Major seventh (M7)<br />Down octave (v8)

| C#<br />vD

Treating [[Ups and downs notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

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|11-EDO]].

=== 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.

{| class="wikitable center-all right-2"

! colspan="2" | Superpyth/Porcupine

! colspan="3" |Porcupine (Zarlino)

! colspan="3" | [[Ups and downs notation|Ups and Downs]]

| Natural Unison

|perfect unison

|P1

|C

|augmented unison

|A1

|C#

|minor second

|m2

|Db

| double-aug unison,<br />double-dim sub3rd

| AA1,<br />dds3

| Dx,<br />Fb<span style="vertical-align: super;">3</span>

|narrow major second

|nM2

|D

| (s/p) Major second

|wide major second

|WM2

|D#

|wolf third

|w3

|Ebb

|minor third

|m3

|Eb

| p-Major third

|major third

|M3

|E

| double-aug sub3rd,<br />double-dim 4thoid

| AAs3,<br />dd4d

| Fx,<br />Gbb

| s-Major third

|augmented third

|A3

|E#

| Natural Fourth

|perfect fourth

|P4

|F

| p-Major fourth, s-dim fifth

|wolf fourth

|w4

|F#

| p-Augmented Fourth,<br />p-diminished Fifth,<br />Half-Octave

| aug 4th, <br />dim 5th

| Gx, <br />Abb

|augmented fourth, diminished fifth

|A4, d5

|F##, Gbb

| double-aug 4thoid,<br />double-dim 5thoid

| AA4d, <br />dd5d

| Gx, <br />Abb

| comma-narrow augmented 4th<br />comma-wide diminished 5th

| kA4<br />Kd5

| p-minor Fifth, s-aug Fourth

|wolf fifth

|w5

|Gb

| Natural Fifth

|perfect fifth

|P5

|G

|diminished sixth

|d6

|Abb

|minor sixth

|m6

|Ab

| double-aug 5thoid,<br />double-dim sub7th

| AA5d,<br />dds7

| Ax,<br />Cb<span style="vertical-align: super;">3</span>

| p-Major sixth

|major sixth

|M6

|A

| s-Major sixth

|wolf sixth

|w6

|A#

|narrow minor seventh

|nm7

|Bbb

| p-Major seventh

|wide minor seventh

|Wm7

|Bb

| p-Augmented seventh

|major seventh

|M7

|B

| double-aug sub7th,<br />double-dim octave

| AAs7,<br />dd8

| Cx,<br />Dbb

| s-Major seventh

|diminished octave

|d8

|Cb

|perfect octave

|P8

|C

| {{monzo| 35 -22 }}

| {{monzo| 12 -9 1 }}

| {{monzo| 1 -5 3 }}

| {{monzo|-10 -1 5 }}

| {{monzo| 11 -4 -2 }}

| {{monzo|-21 3 7 }}

| {{monzo| 32 -7 -9 }}

| {{monzo|-53 10 16 }}

| {{monzo| 1 0 2 -2 }}

| {{monzo| 6 -2 0 -1 }}

| {{monzo|-5 -3 3 1 }}

| {{monzo| 1 5 1 -4 }}

| {{monzo| 0 -5 1 2 }}

| {{monzo| 6 3 -1 -3 }}

| {{monzo|-5 2 2 -1 }}

| {{monzo| 5 -7 -1 3 }}

| {{monzo| 11 1 -3 -2 }}

| {{monzo|-16 1 5 1 }}

| {{monzo|-6 -8 2 5 }}

| {{monzo|-1 2 0 -2 1 }}

| {{monzo| 2 -2 2 0 -1 }}

| {{monzo|-3 -1 -1 0 2 }}

| {{monzo| 4 0 -2 -1 1 }}

| {{monzo| 7 -4 0 1 -1 }}

| {{monzo| 16 0 0 -2 -3 }}

| {{monzo|-7 -1 1 1 1 }}

| {{monzo| 2 3 1 -2 -1 }}

| {{monzo| 5 -1 3 0 -3 }}

| {{monzo|-3 4 -2 -2 2 }}

| {{monzo|-6 0 1 0 0 1 }}

| {{monzo| 1 1 0 -1 -1 1 }}

| {{monzo|-1 -2 -1 1 0 1 }}

| {{monzo|-7 -5 0 4 0 1 }}

| {{monzo|-2 0 3 0 0 0 0 0 0 0 -1 }}

| [[Undeka]]<br>[[Hendecatonic]]

22edo can benefit from slightly compressing the octave, especially when using it as an 11-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.