22edo: Difference between revisions

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, with patterns including [[nicetone]].

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 (approximating [[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 [[36:45:54:64|1–5/4–3/2–16/9]] dominant seventh chord and an otonal tetrad are represented by the same chord.

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. 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]] harmonic sixth chord or minor tetrad. 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 comma), so that the [[marvel]] augmented triad is a chord of 22et, as it also is of any meantone tuning. 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|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]]. However, 22edo has a leg-up on the others melodically, as the large and small steps of [[Orwell[9]]] are easier to distinguish.