22edo: Difference between revisions

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The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian music|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 {{w|James Murray Barbour|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''.

The 22edo system is in fact 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. Furthermore, 22edo, unlike 12 and 19, is not a [[meantone]] system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

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.

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]].

{{Harmonics in equal|22|columns=11}}

=== Subsets and supersets ===

As 22 is divisible by 11, a 22edo instrument can play any music in 11edo, in the same way that [[12edo]] can play [[6edo]] (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.

== Defining features ==

=== Septimal vs. syntonic comma ===

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80. This means that 22 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 and [[27edo]] 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.

The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A–G♮ and C–B♭ are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.

=== Porcupine comma ===

It additionally tempers out the porcupine comma or maximal diesis of 250/243, 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. 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. It 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).

Porcupine temperament also allows 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.

Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

=== 7-limit commas ===

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. [[50/49]] (the jubilisma), and 64/63 (the septimal comma) are tempered out in both systems, so they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 the dominant seventh chord and an otonal tetrad are represented by the same chord. Hence both also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}}, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

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 also supports the [[orwell]] temperament, which 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.

 

22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.

=== 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

|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

Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.

{{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}

 

* Step size: 54.494{{c}}, octave size: 1198.859{{c}}

Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.

{{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}

 

Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.

{{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}

* Step size: 54.420{{c}}, octave size: 1197.246{{c}}

Compressing the octave of 22edo by around 2.8{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7 subgroup|2.3.7-subgroup]] tuning, e.g. for [[archy]] (2.3.7-subgroup superpyth) temperament. The tuning 57ed6 does this.

{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}

 

; [[35edt]]  

Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.