22 Tone Equal Temperament
Improvisations on a 22-ET guitar.
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. 22-ET is arguably the most xenharmonic tuning that matches all these intervals, and the matches to minor thirds and fourth/fifths are poor enough to sound out-of-tune to the trained ear.
22-ET matches the 5th, 11th, 15th, and 17th harmonics well, and the 3rd harmonic is somewhat closely matched. The 7th harmonic is poorly matched, meaning that the few good matches to intervals involving this harmonic are an accident of mismatches to various harmonics cancelling out.
Some factors that make the 22-ET system sound xenharmonic, and make it difficult to play conventional Western music and diatonic scales in include:
- 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.
- 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".
- 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.
- 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.
Because of the last two points, 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.
Summary of 22-ET Relative to 31-ET
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. But the selection of intervals involving the 7th and 11th harmonics, and their correspondence to the just intervals, is limited. 31-ET preserves the use of whole tones in scales, while providing a much more thorough array of harmonic possibilities involving the 7th and 11th harmonics. The only advantage that 22-ET offers is its larger interval size and lower complexity.
On a personal note, I experimented with, but abandoned writing music in 22 due to the poor intonation of fifths and minor thirds, incomplete matches to the intervals involving the 7th and 11th harmonic, and my inability to play diatonic scales in-tune enough for my tastes.
Intervals well-matched by 22-ET
- Major third - The major third (5:4) is very in-tune, only about 4.5 cents too narrow.
- Septimal major third - The septimal major third (9:7) is very in-tune, more so than in 31-ET.
- Greater undecimal neutral second - 22-ET provides a near-perfect match to the greater undecimal neutral second (11:10); by contrast, 31-ET better matches the lesser (12:11) undecimal neutral second.
- Undecimal tritones - 22-ET provides a good match (within 5.86 cents) to the undecimal tritones, the 11:8 and 16:11 ratios; this match is a little better than in 31-ET.
- Diatonic semitone - The diatonic semitone (16:15) is nearly perfectly matched in 22-ET; this match is a little closer than in 31-ET.