User:Overthink/Draft edits: Difference between revisions

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= 24edo =

== Theory ==

~~The~~ [[5-~~limit~~]] ~~approximations in 24edo are~~ the ~~same as those~~ in 12edo, ~~so 24edo offers nothing new~~ as ~~far as approximating the 5-limit~~ is ~~concerned~~.

24edo is the double of [[12edo|12edo/12-TET]], so it contains all of the notes of 12edo. It adds to 12edo another copy 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 accessible to new microtonalists who want new intervals while still having access to familiar ones.

The ~~7th harmonic~~ and ~~its intervals (~~[[7/4]], [[7/5]], [[7/6]], ~~and~~ [[9/7]]) are ~~almost as inaccurate in 24edo as~~ in 12edo. ~~To achieve a satisfactory level of approximation~~ to intervals of 7 while ~~maintaining~~ the 12 notes of 12edo ~~requires high-degree~~ tunings like [[36edo~~|36et~~]], [[~~72edo|72et~~]], [[~~84edo|84et]], [[156edo|156et~~]], or [[~~192edo|192et~~]]. However, 24edo ~~excels at~~ the 11th harmonic ~~and most~~ intervals ~~involving~~ 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). ~~The 24-tone interval~~ of ~~550~~ cents ~~is 1~~.3 cents ~~flatter than 11~~/8 and ~~is almost indistinguishable from it. In addition,~~ the ~~interval approximating 11/9 is 7 steps which is exactly half~~ the ~~perfect fifth~~. ~~24edo~~ is ~~also~~ good at the ~~13th harmonic, which makes it a good~~ 2.3.5.11.13 ~~system~~. ~~Specifically, intervals of~~ 13/5 are ~~particularly well approximated~~. ~~And of course~~, 24edo shares its 17 and 19 ~~tunings~~ with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.

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.

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 more ultramajor and inframinor.

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 = 24edo =
 == Theory ==
-The [[5-limit]] approximations in 24edo are the same as those in 12edo, so 24edo offers nothing new as far as approximating the 5-limit is concerned.
+edo is the double of [[12edo|12edo/12-TET]], so it contains all of the notes of 12edo. It adds to 12edo another copy 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 accessible to new microtonalists who want new intervals while still having access to familiar ones.
-The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth. 24edo is also good at the 13th harmonic, which makes it a good 2.3.5.11.13 system. Specifically, intervals of 13/5 are particularly well approximated. And of course, 24edo shares its 17 and 19 tunings with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.
+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.
 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 more ultramajor and inframinor.