24edo: Difference between revisions

(24 intermediate revisions by 9 users not shown)

Line 6:

}}

24edo is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones, since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic, Turkish, Persian music|Arabic music]].

It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.

It is easy to jump into this tuning and make [[microtonal music]] right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.

== 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.

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 ===

Line 33:

Line 35:

! Degree

! Cents

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]]}}</ref>

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>

! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and d2)

! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}

Line 317:

Line 319:

== Notation ==

=== Ups and ~~Downs~~ notation ===

=== Ups and downs notation ===

~~Spoken~~ as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

Line 341:

Line 343:

'''Pros:''' familiar, intuitive, and fairly easy to learn.

'''Cons:''' can clutter a score easily (especially when used ~~to indicate~~ microtonal key signatures), can get confusing when sight read at faster paces.

'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.

=== Persian quartertone accidentals ===

Line 348:

Line 350:

{| class="wikitable"

|-

| width="40px" |[[File:Koron_sign.svg|39px|center]]

| width="40px" | [[File:Koron_sign.svg|39px|center]]

| '''Koron''' = quarter-tone flat

|-

| width="40px" |[[File:Sori_sign.svg|39px|center]]

| width="40px" | [[File:Sori_sign.svg|39px|center]]

| '''Sori''' = quarter-tone sharp

|}

Line 392:

Line 394:

</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#~~Stein–Zimmermann accidentals~~|Stein–Zimmerman notation]].

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein.E2.80.93Zimmermann_accidentals|Stein–Zimmerman notation]].

==== Pros and cons ====

Line 539:

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 546:

Line 548:

=== Interval mappings ===

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

~~{| class="wikitable center-all"~~

|-

~~! colspan="3" | Tuning~~

~~! colspan="3" | Strength~~

~~! colspan="2" | Closest edo~~

~~! colspan="2" | Integer limit~~

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[90zpi]]~~

~~| 24.0057421830853~~

~~| 49.9880399800983~~

~~| 5.721613~~

~~| 1.092055~~

~~| 14.821136~~

~~| 24edo~~

~~| 1199.71295952236~~

~~| 6~~

|}

== Regular temperament properties ==

Line 620:

Line 591:

=== Uniform maps ===

=== Commas ===

Line 843:

Line 814:

| Thulu

| Grossma

|-

| 13

| [[351/350]]

| {{Monzo| -1 3 -2 -1 0 1 }}

| 4.94

| Thorugugu

| Ratwolfsma

|-

| 13

| [[352/351]]

| {{monzo| 5 -3 0 0 1 -1 }}

| 4.93

| Thulo

| Minor minthma

|-

| 13

Line 1,067:

Line 1,052:

| {{dash|C, E{{sesquiflat2}}, G|hair}}

| Cvm Cm({{demiflat2}}3), Cmin({{demiflat2}}3)

| C ~~subminor~~ C minor semiflat-three

| C inframinor C minor semiflat-three

|-

| 10:12:15

Line 1,091:

Line 1,076:

| {{dash|C, E{{demisharp2}}, G|hair}}

| C^ C({{demisharp2}}3), Cmaj({{demisharp2}}3)

| C ~~supermajor~~ C major semisharp-three

| C ultramajor C major semisharp-three

|}

These chords tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads. For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: {{dash|0, 7, 14, 21|hair}}. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: {{dash|0, 14, 21, 35|hair}} [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.

24edo also is very good at 15 limit and does 13 quite well allowing ~~barbodos~~ 10:13:15 and barbodos minor ~~triad~~ 26:30:39 to be used as an entirely new harmonic system.

24edo also is very good at 15 limit and does 13 quite well allowing barbados major (10:13:15) and barbodos minor (26:30:39) triads to be used as an entirely new harmonic system.

More good chords in 24edo:

Line 1,122:

Line 1,107:

| {{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|hair}}

| smin7 min7({{demiflat2}}3, {{demiflat2}}7)

| ~~Subminor~~ seven Minor seven semiflat-three semiflat-seven

| Inframinor seven Minor seven semiflat-three semiflat-seven

|-

| {{dash|0, 6, 14, 20|hair}}

Line 1,147:

Line 1,132:

| {{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|hair}}

| smaj7 maj7({{demisharp2}}3, {{demisharp2}}7)

| ~~Supermajor~~ seven Major seven semisharp-three semisharp-seven

| Ultramajor seven Major seven semisharp-three semisharp-seven

|-

| {{dash|0, 8, 14, 20|hair}}

Line 1,200:

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):

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|24}}
+{{ED intro}}
 {{Wikipedia|Quarter tone}}
 edo is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones, since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic, Turkish, Persian music|Arabic music]].
-It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.
+It is easy to jump into this tuning and make [[microtonal music]] right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.
 == 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.
-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 33: / Line 35: @@
 ! Degree
 ! Cents
-! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]]}}</ref>
+! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>
 ! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and d2)
 ! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
@@ Line 317: / Line 319: @@
 == Notation ==
-=== Ups and Downs notation ===
+=== Ups and downs notation ===
-Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
+Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
 {{sharpness-sharp2a|24}}
@@ Line 341: / Line 343: @@
 '''Pros:''' familiar, intuitive, and fairly easy to learn.
-'''Cons:''' can clutter a score easily (especially when used to indicate microtonal key signatures), can get confusing when sight read at faster paces.
+'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.
 === Persian quartertone accidentals ===
@@ Line 348: / Line 350: @@
 {| class="wikitable"
 |-
-| width="40px" |[[File:Koron_sign.svg|39px|center]]
+| width="40px" | [[File:Koron_sign.svg|39px|center]]
 | '''Koron''' = quarter-tone flat
 |-
-| width="40px" |[[File:Sori_sign.svg|39px|center]]
+| width="40px" | [[File:Sori_sign.svg|39px|center]]
 | '''Sori''' = quarter-tone sharp
 |}
@@ Line 392: / Line 394: @@
 </imagemap>
-Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein–Zimmermann accidentals|Stein–Zimmerman notation]].
+Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein.E2.80.93Zimmermann_accidentals|Stein–Zimmerman notation]].
 ==== Pros and cons ====
@@ Line 539: / 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 546: / Line 548: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|24}}
-=== Zeta peak index ===
-{| class="wikitable center-all"
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest edo
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[90zpi]]
-| 24.0057421830853
-| 49.9880399800983
-| 5.721613
-| 1.092055
-| 14.821136
-| 24edo
-| 1199.71295952236
-| 6
-| 6
-|}
 == Regular temperament properties ==
@@ Line 620: / Line 591: @@
 === Uniform maps ===
-{{Uniform map|13|23.5|24.5}}
+{{Uniform map|edo=24}}
 === Commas ===
@@ Line 843: / Line 814: @@
 | Thulu
 | Grossma
+|-
+| 13
+| [[351/350]]
+| {{Monzo| -1 3 -2 -1 0 1 }}
+| 4.94
+| Thorugugu
+| Ratwolfsma
+|-
+| 13
+| [[352/351]]
+| {{monzo| 5 -3 0 0 1 -1 }}
+| 4.93
+| Thulo
+| Minor minthma
 |-
 | 13
@@ Line 1,067: / Line 1,052: @@
 | {{dash|C, E{{sesquiflat2}}, G|hair}}
 | Cvm<br>Cm({{demiflat2}}3), Cmin({{demiflat2}}3)
-| C subminor<br>C minor semiflat-three
+| C inframinor<br>C minor semiflat-three
 |-
 | 10:12:15
@@ Line 1,091: / Line 1,076: @@
 | {{dash|C, E{{demisharp2}}, G|hair}}
 | C^<br>C({{demisharp2}}3), Cmaj({{demisharp2}}3)
-| C supermajor<br>C major semisharp-three
+| C ultramajor<br>C major semisharp-three
 |}
 These chords tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads. For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: {{dash|0, 7, 14, 21|hair}}. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: {{dash|0, 14, 21, 35|hair}} [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.
-edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.
+edo also is very good at 15 limit and does 13 quite well allowing barbados major (10:13:15) and barbodos minor (26:30:39) triads to be used as an entirely new harmonic system.
 More good chords in 24edo:
@@ Line 1,122: / Line 1,107: @@
 | {{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|hair}}
 | smin7<br>min7({{demiflat2}}3, {{demiflat2}}7)
-| Subminor seven<br>Minor seven semiflat-three semiflat-seven
+| Inframinor seven<br>Minor seven semiflat-three semiflat-seven
 |-
 | {{dash|0, 6, 14, 20|hair}}
@@ Line 1,147: / Line 1,132: @@
 | {{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|hair}}
 | smaj7<br>maj7({{demisharp2}}3, {{demisharp2}}7)
-| Supermajor seven<br>Major seven semisharp-three semisharp-seven
+| Ultramajor seven<br>Major seven semisharp-three semisharp-seven
 |-
 | {{dash|0, 8, 14, 20|hair}}
@@ Line 1,200: / 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):