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24edo
Theory
24edo/24-TET, also known as the quarter-tone system, is the double of 12edo/12-TET, so it contains all of the notes of 12edo. It adds to 12edo another circle of it spaced a quarter tone apart, which contains unfamiliar intervals not found in 12edo, such as neutral seconds and thirds. Since it contains 12edo, it is very desirable for microtonalists who want new intervals while still having access to familiar ones.
The 5-limit approximations in 24edo are the same as those in 12edo, tempering out 81/80, 128/125, 648/625, and 531441/524288, so 24edo offers nothing new as far as approximating the 5-limit is concerned. However, it maps the 7th harmonic differently from 12edo, with 7/4 mapped to 950 cents rather than 1000 cents in 12edo, being 18.8 cents flat of just rather than 31.2 cents sharp in 12edo. Most intervals of 7 are still approximated quite poorly for its size, though chords like 6:7:9 are nonetheless closer to just than in 12edo. Still, if one wishes to approximate intervals of 7 while still having access to the notes of 12edo, it is best to use finer divisions like 36edo, 48edo, 72edo, or 84edo.
However, 24edo approximates the 11th harmonic very accurately at 550 cents, only 1.3 cents flat of just. Most intervals of 11, such as 11/8, 11/6, 11/10, and 11/9, are approximated accurately as well. It is thus usable as an 2.3.11 or 2.3.5.11 subgroup system, notably tempering out 121/120, splitting 6/5 into two neutral seconds of 11/10~12/11, and 243/242, splitting 3/2 into two 11/9 neutral thirds. It also has a decent approximation of the 13th harmonic at 850 cents, being 9.5 cents sharp of just. Intervals of 13 are thus represented decently, with 13/10, 15/13, and their octave complements being especially close to just due to the cancellation of the sharpness of harmonics 5 and 13. It is thus a good tuning for the 2.3.5.11.13 and 2.3.11.13/5 subgroups, tempering out 144/143 in the former, so that 11/9 and 16/13 are equated, and 676/675 in both subgroups, so two 15/13's add up to 4/3. Finally, 24edo shares its tunings of harmonics 17 and 19 with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly. Nonetheless, it is not the best system for approximating full-prime-limit JI, with other equal temperaments like 22edo, 27edo, and 31edo being more accurate.
Aside from harmony, it also preserves the melodic resources of 12edo, containing minor and major seconds and thirds. However, it adds several new intervals, including neutral seconds and thirds, so new melodies can be written in 24edo that aren't possible in 12edo. This also means 24edo contains new scales, most notably the neutralized diatonic 3L 4s MOS with step pattern LssLsLs, where L is a major second and s is a neutral second. These scales also contain chords unfamiliar to 12edo, such as the neutral tetrad.
While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically, though it equates 7/6 with 8/7 due to vanishing of 49/48, leading to semaphore. Nonetheless, scales of semaphore are quite interesting, especially the 9-note 5L 4s MOS. A supermajor chord is available as [0 9 14], and a subminor chord as [0 5 14], though they're better described as ultramajor and inframinor, being interpreted much more accurately as 10:13:15 and 1/(10:13:15) respectively, the corresponding temperament being barbados, the 2.3.13/5 temperament tempering out 676/675. These chords are relatively simple and may serve as alternatives to the regular 4:5:6 and 1/(4:5:6) triads as bases for harmony; see Extraclassical tonality.
A notable superset of 24edo is 72edo, which has good approximations up to the 19-limit, and especially the 11-limit. The tunings supplied by 72edo cannot be used for all low-limit just intervals, but they can be used on the 17-limit 3*24 subgroup 2.3.125.35.11.325.17.19, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. One will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 subgroup is where it is the most accurate.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | +13.69 | -18.83 | -1.32 | +9.47 | -4.96 | +2.49 | +21.73 | +20.42 | +4.96 |
| Relative (%) | +0.0 | -3.9 | +27.4 | -37.7 | -2.6 | +18.9 | -9.9 | +5.0 | +43.5 | +40.8 | +9.9 | |
| Steps (reduced) |
24 (0) |
38 (14) |
56 (8) |
67 (19) |
83 (11) |
89 (17) |
98 (2) |
102 (6) |
109 (13) |
117 (21) |
119 (23) | |
Subsets and supersets
24edo is the 6th highly composite edo. Its nontrivial divisors are 2, 3, 4, 6, 8, and 12. Some of its supersets, most notably 72edo and 96edo, have been used by a variety of composers.
Miscellaneous properties
Its step, at 50 cents, is notable for being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant.