24edo: Difference between revisions

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~~{{High priority}}~~

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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.

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. ~~Additionally~~, ~~like [[22edo]]~~, 24edo ~~tempers out~~ the ~~[[quartisma]]~~, ~~linking~~ the ~~otherwise sub-par [[~~7-limit]~~] harmonies with those of the [~~[~~11-limit]~~].

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.

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.

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 [[just intonation subgroup]], 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.

Its step, at 50 cents, is notable for ~~having some of the highest [[harmonic entropy]] possible, making it, in theory,~~ one of the most dissonant intervals possible (~~using the relatively common values of {{nowrap|''~~a~~'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}, the~~ peak ~~occurs at~~ around ~~46.4 cents~~). 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.

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.

=== Prime harmonics ===

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* [[24edo Chord Names]]

* [[Ups and ~~Downs Notation~~#Chords and Chord Progressions]].

* [[Ups and downs notation#Chords and Chord Progressions]].

== Approximation to JI ==

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=== Interval mappings ===

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 90~~

~~| steps = 24.0057421830853~~

~~| step size = 49.9880399800983~~

~~| tempered height = 5.721613~~

~~| pure height = 5.709993~~

~~| integral = 1.092055~~

~~| gap = 14.821136~~

~~| octave = 1199.71295952236~~

~~| consistent = 6~~

~~| distinct = 6~~

}}

== Regular temperament properties ==

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[[File:Eastwood-guitars-phase-4-mt-2307179.jpg|500px]]

=== Piano ===

=== Harp, Harpsichord, and Piano ===

Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface tuning]]. Iceface tuning is one type of scordatura piano (or other keyboard instrument) tuning. A more complex type of [[Wikipedia:scordatura|scordatura]] tuning was required for a performance of Charles Ives' 4th Symphony which calls for a quarter-tone piano, but for which no quarter-tone piano was available, as described by Thomas Broadhead in [https://www.youtube.com/watch?v=T1G2XFVtnXU this video]. For this composition the gamut of notes needed would not be met using a simple transformation such as Iceface. ~~{{todo|inline=1|provide example|text=Find a~~ legally freely available ~~recording using~~ this tuning.}}

==== Scordatura tuning of 12edo instruments ====

Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface tuning]]. Iceface tuning is one type of scordatura piano (or other keyboard instrument) tuning. A more complex type of [[Wikipedia:scordatura|scordatura]] tuning was required for a performance of Charles Ives' 4th Symphony which calls for a quarter-tone piano, but for which no quarter-tone piano was available, as described by Thomas Broadhead in [https://www.youtube.com/watch?v=T1G2XFVtnXU this video]. For this composition the gamut of notes needed would not be met using a simple transformation such as Iceface.

Although no recording using the above tuning is currently legally freely available, [[Paweł Mykietyn]] has used a similar idea with harp and harpsichord. A score video of this is available as [https://www.youtube.com/watch?v=_7o0uwPrYas ''Klave for Microtonal Harpsichord and Chamber Orchestra (Score-Video)''] (2004, performed by Elżbieta Chojnacka with Marek Moś conducting the AUKSO chamber orchestra of the city of Tychy, uploaded by Quinone Bob with permission from Paweł Mykietyn); the video starts with slides explaining the scordatura tuning of each manual of the Revival harpsichord (with each manual having a differrent scordatura tuning), followed by the scordatura tuning of the harp.

==== Quarter-tone instruments ====

A very small number of quarter-tone pianos have been built — here are a couple of videos of these instruments being tested/played experimentally (to demonstrate their capabilities rather than to play specific compositions that would qualify for the 24edo Music section):

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 | ja = 24平均律
 }}
-{{High priority}}
 {{Infobox ET}}
 {{ED intro}}
@@ Line 17: / Line 16: @@
 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.
-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. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].
+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.
+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.
 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 [[just intonation subgroup]], 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.
-Its step, at 50 cents, is notable for having some of the highest [[harmonic entropy]] possible, making it, in theory, one of the most dissonant intervals possible (using the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}, the peak occurs at around 46.4 cents). 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.
+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.
 === Prime harmonics ===
@@ Line 540: / Line 541: @@
 * [[24edo Chord Names]]
-* [[Ups and Downs Notation#Chords and Chord Progressions]].
+* [[Ups and downs notation#Chords and Chord Progressions]].
 == Approximation to JI ==
@@ Line 547: / Line 548: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|24}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 90
-| steps = 24.0057421830853
-| step size = 49.9880399800983
-| tempered height = 5.721613
-| pure height = 5.709993
-| integral = 1.092055
-| gap = 14.821136
-| octave = 1199.71295952236
-| consistent = 6
-| distinct = 6
-}}
 == Regular temperament properties ==
@@ Line 1,198: / Line 1,185: @@
 [[File:Eastwood-guitars-phase-4-mt-2307179.jpg|500px]]
-=== Piano ===
+=== Harp, Harpsichord, and Piano ===
-Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface tuning]]. Iceface tuning is one type of scordatura piano (or other keyboard instrument) tuning. A more complex type of [[Wikipedia:scordatura|scordatura]] tuning was required for a performance of Charles Ives' 4th Symphony which calls for a quarter-tone piano, but for which no quarter-tone piano was available, as described by Thomas Broadhead in [https://www.youtube.com/watch?v=T1G2XFVtnXU this video]. For this composition the gamut of notes needed would not be met using a simple transformation such as Iceface. {{todo|inline=1|provide example|text=Find a legally freely available recording using this tuning.}}
+==== Scordatura tuning of 12edo instruments ====
+Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface tuning]]. Iceface tuning is one type of scordatura piano (or other keyboard instrument) tuning. A more complex type of [[Wikipedia:scordatura|scordatura]] tuning was required for a performance of Charles Ives' 4th Symphony which calls for a quarter-tone piano, but for which no quarter-tone piano was available, as described by Thomas Broadhead in [https://www.youtube.com/watch?v=T1G2XFVtnXU this video]. For this composition the gamut of notes needed would not be met using a simple transformation such as Iceface.
+Although no recording using the above tuning is currently legally freely available, [[Paweł Mykietyn]] has used a similar idea with harp and harpsichord. A score video of this is available as [https://www.youtube.com/watch?v=_7o0uwPrYas ''Klave for Microtonal Harpsichord and Chamber Orchestra (Score-Video)''] (2004, performed by Elżbieta Chojnacka with Marek Moś conducting the AUKSO chamber orchestra of the city of Tychy, uploaded by Quinone Bob with permission from Paweł Mykietyn); the video starts with slides explaining the scordatura tuning of each manual of the Revival harpsichord (with each manual having a differrent scordatura tuning), followed by the scordatura tuning of the harp.
+==== Quarter-tone instruments ====
 A very small number of quarter-tone pianos have been built — here are a couple of videos of these instruments being tested/played experimentally (to demonstrate their capabilities rather than to play specific compositions that would qualify for the 24edo Music section):