User:Overthink/Draft edits: Difference between revisions

This is a page where I will draft edits before making them on the actual page. If you have something to add to any of them, or any concerns, please suggest them on the talk page.

= 24edo =

24edo/24-TET, also known as the quarter-tone system, is the double of [[12edo|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.

=== Prime harmonics ===

{{Harmonics in equal|24|prec=2}}

=== Resources ===

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 [[7/1|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 tunings like [[36edo]], [[48edo]], [[72edo]], or [[84edo]].

However, 24edo approximates the [[11/1|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 subgroup|2.3.11]] or [[2.3.5.11 subgroup|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 [[13/1|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 [[17/1|17]] and [[19/1|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.

While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically. A supermajor chord is available at [0 9 14], and a subminor chord at [0 5 14], though they're better described as ultramajor and inframinor, being interpreted as [[10:13:15]] and [[26:30:39|1/(10:13:15)]] respectively. These chords may serve as alternatives to the regular [[4:5:6]] and [[10:12:15|1/(4:5:6)]] triads as bases for harmony; see [[Extraclassical tonality]].

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]].

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 [[k*N subgroups|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.

=== Subsets and supersets ===

24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}. Its multiples [[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.