User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)

Quotations in this essay are not CC-BY; they are copyright Merit Exchange LLC and used for the purposes of commentary under fair use.

This section serves as a response to an unusually negative view on 22edo, and may serve to resolve common complaints about the system.

22-Tone Equal Temperament is the next smallest tuning after 31-Tone Equal Temperament (31-ET) that provides a usable match to both perfect fourths and fifths and major and minor thirds.

Incorrect. Even assuming "major third" is 5/4, 27edo provides a reasonable functional counterpart with its 400c major third, and 29edo has a schismic 5/4. 26edo and 27edo have especially good tunings of 6/5, at least more accurate than 22edo.

22-ET is arguably the most xenharmonic tuning that matches all these intervals

While "most xenharmonic" is subjective, 15edo could reasonably be concluded to be more xenharmonic than 22edo, because its structural properties are essentially completely different from any other 5-limit (and up to 11-limit) system there is.

22-ET matches the 5th, 11th, 15th, and 17th harmonics well, and the 3rd harmonic is somewhat closely matched.

The 3rd harmonic is not particularly less accurate than the other ones mentioned. Additionally, it could be seen as less sensitive to detuning due to being simpler.

The intonation of a variety of fundamental intervals in 22-ET is poor enough to be noticeable, and in many cases, off in a different direction from in 12-ET, making this tuning sound unfamiliar. In 22-ET, fifths are noticeable sharp, the minor third is sharp rather than flat, and the major third is flat rather than sharp. These differences contribute to the intervals sounding unfamiliar and out-of-tune, even if some of them do match their just intervals more than in 12-ET.

5/4 is not the only "major third"; the presence of two distinct varieties of major (one sharper than 12edo, and one flatter) is a major part of 22edo's structure. Additionally, 22edo's fifth is about as inaccurate as 19edo's and only slightly moreso than that of 31edo, which the website promotes. I will agree that the 6/5 is primarily justified by structure rather than tuning accuracy; it is almost as inaccurate as 12edo's for an edo with twice the resolution.

22-ET distinguishes between the major and minor whole tones, the 9:8 and 10:9 ratios in the harmonic series, respectively, yet does not provide a particularly close or in-tune match to either of these intervals. As a result, this tuning does not effectively have an interval corresponding to what most people will know and hear as a "whole tone" or "whole step".

There is a definitive structural answer to what constitutes a "whole tone" in 22edo. Hint: it's not 10/9. The detuning of 9/8 is a structural feature of 22edo, which allows it to function as a 7-limit system despite its small size. The whole tone serves its function as a stack of two fifths and is more in-tune than the perfect fifth of 15edo. This provides notable structural differences from meantone tunings and is thus useful in xenharmonic composition, or as an exaggeration of systems which already make the distinction to begin with.

22-ET does not distinguish between the greater and lesser septimal tritones, instead matching these both with a half-division of the octave. This interval is a poor match to both just intervals. This, combined with the fact that the septimal whole tone is also poorly matched, makes the use of harmony involving the 7th harmonic more difficult in this system.

The main place the tritone appears in 22edo harmony is the chord 5:6:7 (and other chords that include it, such as 4:5:6:7). The structure of the chord in 22edo does a good enough job implying that the tritone is 7/5 even with a somewhat inaccurate tuning. Additionally, the tempering together of 7/5 and 10/7 allows for the structural parallels that make harmonic major and minor chords work the way they do. It is also an interval (and interpretation) shared with 12edo.

Although 22-ET matches the 11th harmonic well, in terms of the 11:8 and 16:11 ratios, the undecimal tritones, and also contains a good match to the greater undecimal neutral second (11:10), it does not contain a neutral third, nor a good match to the undecimal neutral third. This makes harmony involving the 11th harmonic more difficult in this system.

Porcupine presents an entirely alternative organization scheme for the 11-limit compared to the standard neutral ("rastmic") functional system, focusing more on 2.3.5.11 than 2.3.11 itself. It is just as valid and just as usable.

Compared to 12-ET, 22-ET offers a glimpse of the harmonic possibilities utilizing the 7th and 11th overtones, at the expense of the loss of the familiar whole tone and possibility of constructing consonant scales and harmonies by stacking whole tones.

22edo's whole tone scale (machine temperament, which closes at 11edo) actually has MORE harmonic content than 12edo's.

[If] the primary motivation for exploring microtonal music is to explore harmonies involving the 7th or 11th harmonics, 31-ET would be a better choice than 22-ET. And because of the first point, if one desires to still play diatonic music in a microtonal tuning, 31-ET is also superior to 22-ET. [...] The only advantage that 22-ET offers is its larger interval size and lower complexity.

In general, tuning systems shouldn't really be thought of in terms of "advantages" or "disadvantages", but rather just different structures. Overall, 22-ET is close to the lower edge of the "medium" edos, where JI structural interpretations tend to phase out in favor of the more RTT-agnostic edo-centric schemes of systems like 12edo and 15edo. This governs a lot of how 22edo is to be interpreted and used, and it sort of acts like a boundary between both approaches. In short, the answer to most of these problems is "try it. it's not what you're used to, but that's kinda the point."