User:Overthink/Draft edits: Difference between revisions

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=== ~~Approximation to JI~~ ===

=== 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.

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.

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]].

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. ~~Expanding this, 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.

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 ===

@@ Line 8: / Line 8: @@
 {{Harmonics in equal|24|prec=2}}
-=== Approximation to JI ===
+=== 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.
+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&nbsp;9&nbsp;14], and a subminor chord at [0&nbsp;5&nbsp;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.
+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&nbsp;9&nbsp;14], and a subminor chord at [0&nbsp;5&nbsp;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]].
-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&nbsp;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. Expanding this, 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.
+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&nbsp;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 ===