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! Approximate Ratios<ref>based on treating 24edo as a 2.3.11.17.19 [[subgroup]]; other approaches are possible.</ref>
! Approximate Ratios<ref>based on treating 24edo as a 2.3.11.17.19 [[subgroup]]; other approaches are possible.</ref>
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="3" |[[SKULO interval names|SKULO notation]] (U or S = 1)
! colspan="3" | [[SKULO interval names|SKULO notation]] (U or S = 1)
! [[24edo solfege|Solfege]]
! [[24edo solfege|Solfege]]
|-
|-
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| unison
| unison
| D
| D
|unison
| unison
|P1
| P1
|D
| D
| Do
| Do
|-
|-
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| up-unison, downminor 2nd
| up-unison, downminor 2nd
| ^D, vEb
| ^D, vEb
|super unison, uber unison
| super unison, uber unison
|S1, U1
| S1, U1
|SD, UD
| SD, UD
| Da/Ru
| Da/Ru
|-
|-
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| aug unison, minor 2nd
| aug unison, minor 2nd
| D#, Eb
| D#, Eb
|aug unison,  
| aug unison, minor 2nd
minor 2nd
| A1, m2
|A1, m2
| D#, Eb
|D#, Eb
| Ro
| Ro
|-
|-
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| mid 2nd
| mid 2nd
| vE
| vE
|neutral 2nd
| neutral 2nd
|N2
| N2
|UEb, uE
| UEb, uE
| Ra
| Ra
|-
|-
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| major 2nd
| major 2nd
| E
| E
|major 2nd
| major 2nd
|M2
| M2
|E
| E
| Re
| Re
|-
|-
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| upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
| ^E, vF
| ^E, vF
|supermajor 2nd, subminor 3rd
| supermajor 2nd, subminor 3rd
|SM2, sm3
| SM2, sm3
|SE, sF
| SE, sF
| Ri/Mu
| Ri/Mu
|-
|-
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| minor 3rd
| minor 3rd
| F
| F
|minor 3rd
| minor 3rd
|m3
| m3
|F
| F
| Mo
| Mo
|-
|-
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| mid 3rd
| mid 3rd
| vF#
| vF#
|neutral 3rd
| neutral 3rd
|N3
| N3
|UF, uF#
| UF, uF#
| Ma
| Ma
|-
|-
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| major 3rd
| major 3rd
| F#
| F#
|major 3rd
| major 3rd
|M3
| M3
|F#
| F#
| Me
| Me
|-
|-
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| upmajor 3rd, down-4th
| upmajor 3rd, down-4th
| ^F#, vG
| ^F#, vG
|supermajor 3rd, sub 4th
| supermajor 3rd, sub 4th
|SM3, s4
| SM3, s4
|SF#, sG
| SF#, sG
| Mi/Fu
| Mi/Fu
|-
|-
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| fourth
| fourth
| G
| G
|perfect 4th
| perfect 4th
|P4
| P4
|G
| G
| Fo
| Fo
|-
|-
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| up-4th, mid-4th
| up-4th, mid-4th
| ^G
| ^G
|uber 4th/neutral 4th
| uber 4th/neutral 4th
|U4/N4
| U4/N4
|UG
| UG
| Fa/Su
| Fa/Su
|-
|-
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| aug 4th, dim 5th
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|aug 4th, dim 5th
| aug 4th, dim 5th
|A4, d5
| A4, d5
|G#/Ab
| G#/Ab
| Fe/So
| Fe/So
|-
|-
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| down-5th, mid-5th
| down-5th, mid-5th
| vA
| vA
|unter 5th/neutral 5th
| unter 5th/neutral 5th
|u5/N5
| u5/N5
|uA
| uA
| Fi/Sa
| Fi/Sa
|-
|-
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| fifth
| fifth
| A
| A
|perfect 5th
| perfect 5th
|P5
| P5
|A
| A
| Se
| Se
|-
|-
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| up-fifth, downminor 6th
| up-fifth, downminor 6th
| ^A, vBb
| ^A, vBb
|super 5th, subminor 6th
| super 5th, subminor 6th
|S5, sm6
| S5, sm6
|SA, sBb
| SA, sBb
| Si/Lu
| Si/Lu
|-
|-
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| minor 6th
| minor 6th
| Bb
| Bb
|minor 6th
| minor 6th
|m6
| m6
|Bb
| Bb
| Lo
| Lo
|-
|-
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| mid 6th
| mid 6th
| vB
| vB
|neutral 6th
| neutral 6th
|N6
| N6
|UBb, uB
| UBb, uB
| La
| La
|-
|-
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| major 6th
| major 6th
| B
| B
|major 6th
| major 6th
|M6
| M6
|B
| B
| Le
| Le
|-
|-
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| upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
| ^B, vC
| ^B, vC
|supermajor 6th, subminor 7th
| supermajor 6th, subminor 7th
|SM6, sm7
| SM6, sm7
|SB, sC
| SB, sC
| Li/Tu
| Li/Tu
|-
|-
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| minor 7th
| minor 7th
| C
| C
|minor 7th
| minor 7th
|m7
| m7
|C
| C
| To
| To
|-
|-
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| mid 7th
| mid 7th
| vC#
| vC#
|neutral 7th
| neutral 7th
|N7
| N7
|UC, uC#
| UC, uC#
| Ta
| Ta
|-
|-
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| major 7th
| major 7th
| C#
| C#
|major 7th
| major 7th
|M7
| M7
|C#
| C#
| Te
| Te
|-
|-
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| upmajor 7th, down-8ve
| upmajor 7th, down-8ve
| ^C#, vD
| ^C#, vD
|sub 8ve, unter 8ve
| sub 8ve, unter 8ve
|s8, u8
| s8, u8
|C#, uD
| C#, uD
| Ti/Du
| Ti/Du
|-
|-
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| perfect 8ve
| perfect 8ve
| D
| D
|perfect 8ve
| perfect 8ve
|P8
| P8
|D
| D
| Do
| Do
|}
|}
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[[Category:IMPORTDEBUG - Change External Images]]
[[Category:IMPORTDEBUG - Change External Images]]


[[File:Accidental semisharp.png|thumb|alt=A "semisharp" accidental comprising one half of a regular musical sharp symbol.|left]]
{| class="wikitable center-all"
 
|-
or ^ = quarter-tone sharp or "Jump" or "up"
| width="40px" |[[File:HeQu1.svg|21px|center]]
 
| A "semisharp" accidental comprising one half of a regular musical sharp symbol.
 
|}
[[File:Accidental_sesquisharp.png|thumb|alt=A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.|left]]
 
 
or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
 
 
[[File:Accidental semiflat.png|thumb|alt=A "semiflat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.|left]]
 
 
or v = quarter-tone flat or "Drop" or "down"
 
 
[[File:Accidental_sesquiflat.png|thumb|alt=A "flat and a half" or "sesquiflat" accidental, comprising the above half-flat symbol and a regular flat symbol placed back to back.|left]]
 
 
or bv or vb = three-quarter-tone sharp or "Drop-Flat" or "downflat"


{| class="wikitable center-all"
|-
| width="40px" |[[File:HeQu3.svg|30px|center]]
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
|}


{| class="wikitable center-all"
|-
| width="40px" |[[File:HeQd1.svg|22px|center]]
| A "semiflat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
|}


{| class="wikitable center-all"
|-
| width="40px" |[[File:HeQd3.svg|36px|center]]
| A "flat and a half" or "sesquiflat" accidental, comprising the above half-flat symbol and a regular flat symbol placed back to back.
|}


'''Pros:''' Familiar, fairly easy to learn
'''Pros:''' Familiar, fairly easy to learn
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'''Cons:''' Clutters a score easily, can get confusing when sight read at faster paces
'''Cons:''' Clutters a score easily, can get confusing when sight read at faster paces


==== Persian accidentals ====
====Persian accidentals ====


{| class="wikitable"
{| class="wikitable"
|-
|-
|[[File:200px-Koron_sign.svg.png|40px|left]]
| width="40px" | [[File:200px-Koron_sign.svg.png|39px|center]]
|'''Koron''' ([https://en.wikipedia.org/wiki/Koron_%28music%29 en] <nowiki>|</nowiki> [https://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86 fa]) = quarter-tone flat
| '''Koron''' ([[wikipedia:Koron_(music)|en]] <nowiki>|</nowiki> [https://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86 fa]) = quarter-tone flat
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|-
|[[File:200px-Sori_sign.svg.png|40px|left]]
| width="40px" | [[File:200px-Sori_sign.svg.png|39px|center]]
|'''Sori''' ([https://fa.wikipedia.org/wiki/%D8%B3%D8%B1%DB%8C_%28%D9%85%D9%88%D8%B3%DB%8C%D9%82%DB%8C%29 fa]) = quarter-tone sharp
| '''Sori''' ([https://fa.wikipedia.org/wiki/%D8%B3%D8%B1%DB%8C_%28%D9%85%D9%88%D8%B3%DB%8C%D9%82%DB%8C%29 fa]) = quarter-tone sharp
|}
|}


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'''Cons:''' Hard to write on a computer, doesn't fit with standard notation well
'''Cons:''' Hard to write on a computer, doesn't fit with standard notation well


==== Sagittal notation ====
====Sagittal notation====
[[Sagittal notation]] works extremely well for 24edo notation as well as other systems.
[[Sagittal notation]] works extremely well for 24edo notation as well as other systems.


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[[File:24edo Sagittal.png|800px]]
[[File:24edo Sagittal.png|800px]]


=== Further discussion of 24edo notation ===
* [[24edo interval names and harmonies]]
* [[24edo Chord Names]]
* [[Ups and Downs Notation#Chords and Chord Progressions]].


== Regular temperament properties ==
===Further discussion of 24edo notation===
*[[24edo interval names and harmonies]]
*[[24edo Chord Names]]
*[[Ups and Downs Notation#Chords and Chord Progressions]].
 
==Regular temperament properties==
[[File:24ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]
[[File:24ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]
=== Uniform maps ===
===Uniform maps===
{{Uniform map|13|23.5|24.5}}
{{Uniform map|13|23.5|24.5}}


=== Commas ===
=== Commas===
This is a partial list of the [[commas]] that 24edo [[tempers out]] with its patent [[val]], {{val| 24 38 56 67 83 89 }}.  
This is a partial list of the [[commas]] that 24edo [[tempers out]] with its patent [[val]], {{val| 24 38 56 67 83 89 }}.  


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit| Prime<br>Limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
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| 3
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo|-19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawa
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| 5
| 5
| [[81/80]]
| [[81/80]]
| {{monzo|-4 4 -1 }}
| {{monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
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| 5
| 5
| [[32805/32768]]
| [[32805/32768]]
| {{monzo|-15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layo
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| 7
| 7
| [[1323/1280]]
| [[1323/1280]]
| {{monzo|-8 3 -1 2 }}
| {{monzo| -8 3 -1 2 }}
| 57.20
| 57.20
| Lazozogu
| Lazozogu
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| 11
| 11
| [[121/120]]
| [[121/120]]
| {{monzo|-3 -1 -1 0 2 }}
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
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| 11
| 11
| [[243/242]]
| [[243/242]]
| {{monzo|-1 5 0 0 -2 }}
| {{monzo| -1 5 0 0 -2 }}
| 7.14
| 7.14
| Lulu
| Lulu
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| 11
| 11
| [[385/384]]
| [[385/384]]
| {{monzo|-7 -1 1 1 1 }}
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
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| 11
| 11
| [[9801/9800]]
| [[9801/9800]]
| {{monzo|-3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Bilorugu
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| 13
| 13
| [[91/90]]
| [[91/90]]
| {{monzo|-1 -2 -1 1 0 1 }}
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
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| 13
| 13
| [[105/104]]
| [[105/104]]
| {{monzo|-3 1 1 1 0 -1 }}
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| 16.57
| Thuzoyo
| Thuzoyo
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<references />
<references />


=== Rank-2 temperaments ===
===Rank-2 temperaments ===
*[[List of 24et rank two temperaments by badness]]
*[[List of 24et rank two temperaments by badness]]
*[[List of edo-distinct 24et rank two temperaments]]
*[[List of edo-distinct 24et rank two temperaments]]
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|}
|}


== Scales / modes ==
==Scales / modes==
''See [[24edo scales]].''
''See [[24edo scales]].''


== Tetrachords ==
==Tetrachords==
''See [[24edo tetrachords]]''.
''See [[24edo tetrachords]]''.


== Chord types ==
==Chord types==
24edo features a rich variety of not only new chords, but also alterations that can be used with regular 12edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.
24edo features a rich variety of not only new chords, but also alterations that can be used with regular 12edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.


As for entirely new chords, the most obvious is the neutral or mid triad 0-7-14. However there are other options such as:
As for entirely new chords, the most obvious is the neutral or mid triad 0-7-14. However there are other options such as:


* 0-9-14 (Ultra Triad or upmajor triad)
*0-9-14 (Ultra Triad or upmajor triad)
* 0-5-14 (Infra Triad or downminor triad)
*0-5-14 (Infra Triad or downminor triad)


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: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.  
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: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.  
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More good chords in 24edo:
More good chords in 24edo:


* 0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
*0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
* Its inversion, 0-3-6-10-14 ("minor")
*Its inversion, 0-3-6-10-14 ("minor")
* 0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 ([[godzilla]]), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[mohajira]]), a heptatonic scale close to several Arabic scales.)
* 0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 ([[godzilla]]), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[mohajira]]), a heptatonic scale close to several Arabic scales.)


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{| class="wikitable"
{| class="wikitable"
|-
|-
!Chord name
! Chord name
!Degrees of 24edo
! Degrees of 24edo
!Chord spelling
! Chord spelling
!Audio example
! Audio example
|-
|-
|neutral
| neutral
|0 7 14 21
| 0 7 14 21
| 1 v3 5 v7
| 1 v3 5 v7
|[[File:Neutral Tetrad on C.mp3]]
| [[File:Neutral Tetrad on C.mp3]]
|-
|-
|arto
| arto
|0 5 14 20
| 0 5 14 20
|1 vb3 5 b7
| 1 vb3 5 b7
|[[File:arto tetrad on C.mp3]]
| [[File:arto tetrad on C.mp3]]
|-
|-
|tendo
| tendo
| 0 9 14 19
| 0 9 14 19
|1 ^3 5 vb7
| 1 ^3 5 vb7
|...
| ...
|}
|}


The tendo chord can also be spelled 1 ^3 5 ^6. Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.
The tendo chord can also be spelled 1 ^3 5 ^6. Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.


== Counterpoint ==
==Counterpoint==
24edo is the first edo to have both a [[sqrt(25/24)]] distinct from [[25/24]] and a correct [[5-odd-limit]]. It is thus the first edo which allows to lead the two voices of a major third to a minor third by [[strict contrary motion]]. And vice versa.
24edo is the first edo to have both a [[sqrt(25/24)]] distinct from [[25/24]] and a correct [[5-odd-limit]]. It is thus the first edo which allows to lead the two voices of a major third to a minor third by [[strict contrary motion]]. And vice versa.


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|}
|}


== Instruments ==
== Instruments==
The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This [[Microtonal_Keyboards#twelvenoteoctavescales|"12 note octave scales"]] approach is used in a wide part of the existing literature - see below.
The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This [[Microtonal_Keyboards#twelvenoteoctavescales|"12 note octave scales"]] approach is used in a wide part of the existing literature - see below.


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24edo can also be played on the Lumatone: see [[Lumatone mapping for 24edo]]
24edo can also be played on the Lumatone: see [[Lumatone mapping for 24edo]]


== Music ==
==Music==
{{Wikipedia|List of quarter tone pieces}}
{{Wikipedia|List of quarter tone pieces}}
{{Main|Music in 24edo}}
{{Main|Music in 24edo}}


== Further reading ==
== Further reading==
* Ellis, Don. ''[https://archive.org/details/don-ellis-quarter-tones/ Quarter Tones: A Text with Musical Examples, Exercises and Etudes]''. 1975.
*Ellis, Don. ''[https://archive.org/details/don-ellis-quarter-tones/ Quarter Tones: A Text with Musical Examples, Exercises and Etudes]''. 1975.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitetraphonic Scales for Guitar: Theory and Scales for Twenty-four Equal Divisions of the Octave]''. 2009. (Features a practical approach to understanding the tuning, and over 550 scale examples on nine-string finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the scales/modes from the list above.)
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitetraphonic Scales for Guitar: Theory and Scales for Twenty-four Equal Divisions of the Octave]''. 2009. (Features a practical approach to understanding the tuning, and over 550 scale examples on nine-string finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the scales/modes from the list above.)


== See also ==
==See also==
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[48edo]] (2mu tuning)
**[[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
**[[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
**[[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
**[[384edo]] (5mu tuning)
** [[768edo]] (6mu tuning)
**[[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
**[[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
**[[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
**[[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
**[[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
**[[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
**[[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
**[[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)
**[[196608edo]] (14mu tuning)




== External links ==
==External links==
* [http://tonalsoft.com/enc/q/quarter-tone.aspx Tonalsoft Encyclopedia | ''quarter-tone / 24-edo / 24-ed2''] [https://www.webcitation.org/5xeFMH6cd Permalink]
*[http://tonalsoft.com/enc/q/quarter-tone.aspx Tonalsoft Encyclopedia | ''quarter-tone / 24-edo / 24-ed2''] [https://www.webcitation.org/5xeFMH6cd Permalink]
* [http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [https://www.webcitation.org/5xeFBNdQW Permalink]
*[http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [https://www.webcitation.org/5xeFBNdQW Permalink]
* [https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
*[https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
* [http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]     
*[http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]     


[[Category:Quartertone]]
[[Category:Quartertone]]
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