22edo: Difference between revisions

=== Overview to JI approximation quality ===

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.

=== Prime harmonics ===

{{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.

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.

=== 5-limit commas ===

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

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. [[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.

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.