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__FORCETOC__
{{interwiki
| de = 24-EDO
| en = 24edo
| es = 24 EDO
| ja = 24平均律
}}
{{Infobox ET}}
{{ED intro}}
{{Wikipedia|Quarter tone}}


<span style="display: block; text-align: right;">[[:de:24edo|Deutsch]] - [[24平均律|日本語]]</span>
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]].


=Basics=
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]]''.
The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It 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 and 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|Arabic]] music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - [[DIY_Quartertone_Composition_with_12_equal_tools|see this page]].


=24edo as a temperament=
== Theory ==
The [[Harmonic_Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.
24edo/24-TET, also known as the quarter-tone system, is the double of [[12edo|12edo/12-TET]], so it contains all of the notes of 12edo. It adds to 12edo another circle 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 desirable for microtonalists who want new intervals while still having access to familiar ones.


The tunings supplied by 72 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|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 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.
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{{c}} rather than 1000{{c}} in 12edo, being 18.8{{c}} flat of just rather than 31.2{{c}} 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 divisions like [[36edo]], [[48edo]], [[72edo]], or [[84edo]].


{| class="wikitable"
However, 24edo approximates the [[11/1|11th harmonic]] very accurately at 550{{c}}, only 1.3{{c}} 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 {{nowrap|[[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{{c}}, being 9.5{{c}} sharp of just. Intervals of 13 are thus represented decently, with [[13/10]], [[15/13]], and their [[octave complement]]s 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 JI if one wishes to use prime 7, with other equal temperaments like [[22edo]], [[27edo]], and especially [[31edo]] being more 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 "neutral diatonic" [[3L&nbsp;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]].
 
While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically, though it equates [[7/6]] with [[8/7]] due to vanishing of [[49/48]], leading to [[semaphore]]. Nonetheless, scales of semaphore are quite interesting, especially the 9-note [[5L&nbsp;4s]] MOS. A supermajor chord is available as [0&nbsp;9&nbsp;14], and a subminor chord as [0&nbsp;5&nbsp;14]; however, they are better described as ultramajor and inframinor, being interpreted much more accurately as [[10:13:15]] and [[26:30:39|1/(10:13:15)]] respectively, the corresponding temperament being [[barbados]], the 2.3.13/5 temperament tempering out 676/675. These chords are relatively simple and 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. 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.
 
=== Prime harmonics ===
{{Harmonics in equal|24|prec=2}}
 
=== Subsets and supersets ===
24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}. Some of its supersets, most notably [[72edo]] and [[96edo]], have been used by a variety of composers.
 
=== Miscellaneous properties ===
Its step, at 50{{c}}, 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{{c}} tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60{{c}} are often perceived as having less "tension", and thus are also considered to be more consonant.
 
== Intervals ==
{| class="wikitable center-all left-3"
|-
|-
| style="text-align:center;" | Degree
! Degree
| style="text-align:center;" | Cents
! Cents
| style="text-align:center;" | Approximate Ratios*
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
! 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)}}
! [[24edo solfege|Solfege]]
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 1/1
| [[1/1]]
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | unison
| unison
| style="text-align:center;" | C
| D
| unison
| P1
| D
| Do
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 50
| 50
| style="text-align:center;" | 33/32, 34/33
| [[33/32]], [[34/33]]
| style="text-align:center;" | ^P1, vm2
| ^P1, vm2
| style="text-align:center;" | up-unison, downminor 2nd
| up-unison, downminor 2nd
| style="text-align:center;" | C^, Dbv
| ^D, vEb
| super unison, uber unison
| S1, U1
| SD, UD
| Da/Ru
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 100
| 100
| style="text-align:center;" | 17/16, 18/17
| [[16/15]], [[17/16]], [[18/17]], [[19/18]]
| style="text-align:center;" | A1, m2
| A1, m2
| style="text-align:center;" | aug unison, minor 2nd
| aug unison, minor 2nd
| style="text-align:center;" | C#, Db
| D#, Eb
| aug unison, minor 2nd
| A1, m2
| D#, Eb
| Ro
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 150
| 150
| style="text-align:center;" | 12/11
| [[13/12]], [[12/11]], [[11/10]]
| style="text-align:center;" | ~2
| ~2
| style="text-align:center;" | mid 2nd
| mid 2nd
| style="text-align:center;" | Dv
| vE
| neutral 2nd
| N2
| UEb, uE
| Ra
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 200
| 200
| style="text-align:center;" | 9/8
| [[9/8]], [[10/9]]
| style="text-align:center;" | M2
| M2
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | D
| E
| major 2nd
| M2
| E
| Re
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 250
| 250
| style="text-align:center;" | 22/19
| [[15/13]], [[22/19]]
| style="text-align:center;" | ^M2, vm3
| ^M2, vm3
| style="text-align:center;" | upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
| style="text-align:center;" | D^, Ebv
| ^E, vF
| supermajor 2nd, subminor 3rd
| SM2, sm3
| SE, sF
| Ri/Mu
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 300
| 300
| style="text-align:center;" | 19/16
| [[6/5]], [[13/11]], [[19/16]]
| style="text-align:center;" | m3
| m3
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | Eb
| F
| minor 3rd
| m3
| F
| Mo
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 350
| 350
| style="text-align:center;" | 11/9
| [[11/9]], [[16/13]], [[27/22]], [[39/32]]
| style="text-align:center;" | ~3
| ~3
| style="text-align:center;" | mid 3rd
| mid 3rd
| style="text-align:center;" | Ev
| vF#
| neutral 3rd
| N3
| UF, uF#
| Ma
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 400
| 400
| style="text-align:center;" | 24/19
| [[5/4]], [[24/19]]
| style="text-align:center;" | M3
| M3
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | E
| F#
| major 3rd
| M3
| F#
| Me
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 450
| 450
| style="text-align:center;" | 22/17
| [[13/10]], [[17/13]], [[22/17]]
| style="text-align:center;" | ^M3, vP4
| ^M3, v4
| style="text-align:center;" | upmajor 3rd, down-fourth
| upmajor 3rd, down-4th
| style="text-align:center;" | E^, Fv
| ^F#, vG
| supermajor 3rd, sub 4th
| SM3, s4
| SF#, sG
| Mi/Fu
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 500
| 500
| style="text-align:center;" | 4/3
| [[4/3]]
| style="text-align:center;" | P4
| P4
| style="text-align:center;" | fourth
| fourth
| style="text-align:center;" | F
| G
| perfect 4th
| P4
| G
| Fo
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 550
| 550
| style="text-align:center;" | 11/8
| [[11/8]], [[15/11]]
| style="text-align:center;" | ^P4
| ^4, ~4
| style="text-align:center;" | up-fourth
| up-4th, mid-4th
| style="text-align:center;" | F^
| ^G
| uber 4th/neutral 4th
| U4/N4
| UG
| Fa/Su
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 600
| 600
| style="text-align:center;" | 17/12
| [[17/12]], [[24/17]], [[45/32]], [[64/45]]
| style="text-align:center;" | A4, d5
| A4, d5
| style="text-align:center;" | aug 4th, dim 5th
| aug 4th, dim 5th
| style="text-align:center;" | F#, Gb
| G#, Ab
| aug 4th, dim 5th
| A4, d5
| G#/Ab
| Fe/So
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 650
| 650
| style="text-align:center;" | 16/11
| [[16/11]], [[22/15]]
| style="text-align:center;" | vP5
| v5, ~5
| style="text-align:center;" | down-fifth
| down-5th, mid-5th
| style="text-align:center;" | Gv
| vA
| unter 5th/neutral 5th
| u5/N5
| uA
| Fi/Sa
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 700
| 700
| style="text-align:center;" | 3/2
| [[3/2]]
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | fifth
| fifth
| style="text-align:center;" | G
| A
| perfect 5th
| P5
| A
| Se
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 750
| 750
| style="text-align:center;" | 17/11
| [[17/11]], [[20/13]]
| style="text-align:center;" | ^P5, vm6
| ^5, vm6
| style="text-align:center;" | up-fifth, downminor 6th
| up-fifth, downminor 6th
| style="text-align:center;" | G^, Abv
| ^A, vBb
| super 5th, subminor 6th
| S5, sm6
| SA, sBb
| Si/Lu
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 800
| 800
| style="text-align:center;" | 19/12
| [[8/5]], [[19/12]]
| style="text-align:center;" | m6
| m6
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | Ab
| Bb
| minor 6th
| m6
| Bb
| Lo
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 850
| 850
| style="text-align:center;" | 18/11
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| style="text-align:center;" | ~6
| ~6
| style="text-align:center;" | mid 6th
| mid 6th
| style="text-align:center;" | Av
| vB
| neutral 6th
| N6
| UBb, uB
| La
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 900
| 900
| style="text-align:center;" | 32/19
| [[5/3]], [[22/13]], [[32/19]]
| style="text-align:center;" | M6
| M6
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | A
| B
| major 6th
| M6
| B
| Le
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 950
| 950
| style="text-align:center;" | 19/11
| [[19/11]], [[26/15]]
| style="text-align:center;" | ^M6, vm7
| ^M6, vm7
| style="text-align:center;" | upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
| style="text-align:center;" | A^, Bbv
| ^B, vC
| supermajor 6th, subminor 7th
| SM6, sm7
| SB, sC
| Li/Tu
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | 1000
| 1000
| style="text-align:center;" | 16/9
| [[9/5]], [[16/9]]
| style="text-align:center;" | m7
| m7
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | Bb
| C
| minor 7th
| m7
| C
| To
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 1050
| 1050
| style="text-align:center;" | 11/6
| [[11/6]], [[20/11]]
| style="text-align:center;" | ~7
| ~7
| style="text-align:center;" | mid 7th
| mid 7th
| style="text-align:center;" | Bv
| vC#
| neutral 7th
| N7
| UC, uC#
| Ta
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:center;" | 1100
| 1100
| style="text-align:center;" | 17/9, 32/17
| [[15/8]], [[17/9]], [[32/17]]
| style="text-align:center;" | M7
| M7
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | B
| C#
| major 7th
| M7
| C#
| Te
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | 1150
| 1150
| style="text-align:center;" | 33/17, 64/33
| [[33/17]], [[64/33]]
| style="text-align:center;" | ^M7, vP8
| ^M7, vP8
| style="text-align:center;" | upmajor 7th, down-8ve
| upmajor 7th, down-8ve
| style="text-align:center;" | B^, Cv
| ^C#, vD
| sub 8ve, unter 8ve
| s8, u8
| C#, uD
| Ti/Du
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | 1200
| 1200
| style="text-align:center;" | 2/1
| [[2/1]]
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | perfect 8ve
| perfect 8ve
| style="text-align:center;" | C
| D
| perfect 8ve
| P8
| D
| Do
|}
|}
*based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.
<references group="note" />


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
In many other edos, 5/4 is downmajor and 11/9 is mid. To agree with this, the term mid is generally preferred over down or downmajor.


{| class="wikitable"
== Notation ==
|-
=== Ups and downs notation ===
! | quality
Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
! | [[Kite's color notation|color]]
{{Ups and downs sharpness}}
! | monzo format
 
! | examples
=== Stein–Zimmermann accidentals ===
|-
{{Sharpness-sharp2|24}}
| style="text-align:center;" | downminor
 
| style="text-align:center;" | zo
{| class="wikitable center-1"
| style="text-align:center;" | {a, b, 0, 1}
| style="text-align:center;" | 7/6, 7/4
|-
| style="text-align:center;" | minor
| style="text-align:center;" | fourthward wa
| style="text-align:center;" | {a, b}, b &lt; -1
| style="text-align:center;" | 32/27, 16/9
|-
| style="text-align:center;" | "
| style="text-align:center;" | gu
| style="text-align:center;" | {a, b, -1}
| style="text-align:center;" | 6/5, 9/5
|-
| style="text-align:center;" | mid
| style="text-align:center;" | lova
| style="text-align:center;" | {a, b, 0, 0, 1}
| style="text-align:center;" | 11/9, 11/6
|-
|-
| style="text-align:center;" | "
| style="width: 40px;" | [[File:HeQu1.svg|21px|center]]
| style="text-align:center;" | lu
| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.
| style="text-align:center;" | {a, b, 0, 0, -1}
| style="text-align:center;" | 12/11, 18/11
|-
|-
| style="text-align:center;" | major
| style="width: 40px;" | [[File:HeQu3.svg|32px|center]]
| style="text-align:center;" | yo
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
| style="text-align:center;" | {a, b, 1}
| style="text-align:center;" | 5/4, 5/3
|-
|-
| style="text-align:center;" | "
| style="width: 40px;" | [[File:HeQd1.svg|22px|center]]
| style="text-align:center;" | fifthward wa
| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
| style="text-align:center;" | {a, b}, b &gt; 1
| style="text-align:center;" | 9/8, 27/16
|-
|-
| style="text-align:center;" | upmajor
| style="width: 40px;" | [[File:HeQd3.svg|40px|center]]
| style="text-align:center;" | ru
| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.
| style="text-align:center;" | {a, b, 0, -1}
| style="text-align:center;" | 9/7, 12/7
|}
|}


The 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. 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. Some good chords in 24-tET are (the numbers are edosteps, e.g. 4 is a major second, 8 is a major third):
'''Pros:''' familiar, intuitive, and fairly easy to learn.  
 
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")
 
0-7-14 ("neutral")


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|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|mohajira]]), a heptatonic scale close to several Arabic scales.)
'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.


===Commas===
=== Persian quartertone accidentals ===
24 EDO tempers out the following commas. (Note: This assumes val &lt; 24 38 56 67 83 89 |.)
{{Wikipedia|Koron (music)|Sori (music)}}


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Comma
| style="width: 40px;" | [[File:Koron_sign.svg|39px|center]]
! | Monzo
| '''Koron''' = quarter-tone flat
! | Value (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| style="text-align:center;" |<nowiki> | -19 12 </nowiki>&gt;
| style="text-align:center;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 648/625
| style="text-align:center;" |<nowiki> | 3 4 -4 </nowiki>&gt;
| style="text-align:center;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 128/125
| style="text-align:center;" |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:center;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 81/80
| style="width: 40px;" | [[File:Sori_sign.svg|39px|center]]
| style="text-align:center;" |<nowiki> | -4 4 -1 </nowiki>&gt;
| '''Sori''' = quarter-tone sharp
| style="text-align:center;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| style="text-align:center;" |<nowiki> | 11 -4 -2 </nowiki>&gt;
| style="text-align:center;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 5201701/5149091
| style="text-align:center;" |<nowiki> | 26 -12 -3 </nowiki>&gt;
| style="text-align:center;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 32805/32768
| style="text-align:center;" |<nowiki> | -15 8 1 </nowiki>&gt;
| style="text-align:center;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1465155/1465142
| style="text-align:center;" |<nowiki> | 161 -84 -12 </nowiki>&gt;
| style="text-align:center;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 49/48
| style="text-align:center;" |<nowiki> | -4 -1 0 2 </nowiki>&gt;
| style="text-align:center;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 245/243
| style="text-align:center;" |<nowiki> | 0 -5 1 2 </nowiki>&gt;
| style="text-align:center;" | 14.19
| style="text-align:center;" | Sensamagic
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 19683/19600
| style="text-align:center;" |<nowiki> | -4 9 -2 -2 </nowiki>&gt;
| style="text-align:center;" | 7.32
| style="text-align:center;" | Cataharry
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 6144/6125
| style="text-align:center;" |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
| style="text-align:center;" | 5.36
| style="text-align:center;" | Porwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 121/120
| style="text-align:center;" |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
| style="text-align:center;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 176/175
| style="text-align:center;" |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| style="text-align:center;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 896/891
| style="text-align:center;" |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
| style="text-align:center;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 243/242
| style="text-align:center;" |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| style="text-align:center;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 385/384
| style="text-align:center;" |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:center;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9801/9800
| style="text-align:center;" |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| style="text-align:center;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 91/90
| style="text-align:center;" |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| style="text-align:center;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 676/675
| style="text-align:center;" |<nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
| style="text-align:center;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| style="text-align:center;" |
|}
|}


=Intervals=
'''Pros:''' easy to read.


24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.
'''Cons:''' hard to write on a computer, does not fit with standard notation well.


These are the intervals of 24 EDO that do not exist in 12 EDO: 2
=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[17edo #Sagittal notation|17]], [[31edo #Sagittal notation|31]], and [[38edo #Sagittal notation|38]], is a subset of the notations for edos [[48edo #Sagittal notation|48]] and [[72edo #Sagittal notation|72]], and is a superset of the notations for edos [[12edo #Sagittal notation|12]], [[8edo #Sagittal notation|8]], and [[6edo #Sagittal notation|6]].


[[24_EDO_Interval_names_and_Harmonies|See full article on 24 Edo intervals.]]
==== Evo flavor ====
<imagemap>
File:24-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:24-EDO_Evo_Sagittal.svg]]
</imagemap>


{| class="wikitable"
==== Revo flavor ====
|-
<imagemap>
! colspan="2" | The twelve new intervals in 24edo
File:24-EDO_Revo_Sagittal.svg
! colspan="3" | some nearby JI intervals
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
! | cents
rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
! | common names
rect 20 80 130 106 [[33/32]]
! | frequency ratio
default [[File:24-EDO_Revo_Sagittal.svg]]
! | cents
</imagemap>
! | common name
|-
| style="text-align:center;" | 50
| style="text-align:center;" | quartertone


infra second, wide unison
==== Evo-SZ flavor ====
| style="text-align:center;" | [[36/35|36/35]]
<imagemap>
File:24-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 407 0 567 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:24-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


[[35/34|35/34]]
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein.E2.80.93Zimmermann_accidentals|Stein–Zimmerman notation]].


[[34/33|34/33]]
==== Pros and cons ====
Revo [[Sagittal notation]] works extremely well for 24edo notation as well as other systems. It is easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat. A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them with Sagittal signs for sharp and flat.


[[33/32|33/32]]
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
| style="text-align:center;" | 48.770


50.184
'''Pros:''' easy to read, and less likely to clutter the score.  


51.682
'''Cons:''' not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality.  


53.273
We also have, from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 24edo in the Revo flavor of Sagittal:
| style="text-align:center;" | large septimal quarter-tone (Archytas)


large 17-limit quartertone
[[File:24edo Sagittal.png|800px]]


small 17-limit quartertone
== Interval and chord naming ==
==== Combining ups and downs with color notation ====
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


33rd harmonic
{| class="wikitable center-all"
|-
|-
| style="text-align:center;" | 150
! Quality
| style="text-align:center;" | neutral second
! [[Color name]]
| style="text-align:center;" | [[12/11|12/11]]
! Monzo format
| style="text-align:center;" | 150.637
! Examples
| style="text-align:center;" | large undecimal neutral second
|-
|-
| style="text-align:center;" | 250
| downminor
| style="text-align:center;" | ultra second
| zo
 
| {{nowrap|(a, b, 0, 1)}}
<span style="line-height: 1.5;">infra third</span>
| 7/6, 7/4
| style="text-align:center;" | [[144/125|144/125]]
|-
 
| rowspan="2" | minor
[[15/13|15/13]]
| fourthward wa
| {{nowrap|(a, b)}}; {{nowrap|b < −1}}
| 32/27, 16/9
|-
| gu
| {{nowrap|(a, b, −1)}}
| 6/5, 9/5
|-
| rowspan="2" | mid
| ilo
| {{nowrap|(a, b, 0, 0, 1)}}
| 11/9, 11/6
|-
| lu
| {{nowrap|(a, b, 0, 0, −1)}}
| 12/11, 18/11
|-
| rowspan="2" | major
| yo
| {{nowrap|(a, b, 1)}}
| 5/4, 5/3
|-
| fifthward wa
| {{nowrap|(a, b)}}; {{nowrap|b > 1}}
| 9/8, 27/16
|-
| upmajor
| ru
| {{nowrap|(a, b, 0, −1)}}
| 9/7, 12/7
|}


[[52/45|52/45]]
Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].
| style="text-align:center;" | 244.969


247.741
=== William Lynch's interval and chord names ===
24edo breaks intervals into two sets of five categories. {{dash|Infra, Minor, Neutral, Major, Ultra|space|med}} for seconds, thirds, sixths, and sevenths; and {{dash|diminished, narrow, perfect, wide, augmented|space|med}} for fourths, fifths, unison, and octave.  


250.304
For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.
| style="text-align:center;" | diminished third (6/5 x 24/25)


..
These are the intervals of 24edo that do not exist in 12edo:


..
{| class="wikitable center-all right-6"
|-
|-
| style="text-align:center;" | 350
! [[Cent]]s
| style="text-align:center;" | neutral third
! Names
| style="text-align:center;" | [[11/9|11/9]]
 
[[27/22|27/22]]
 
[[16/13|16/13]]
| style="text-align:center;" | 347.408
 
354.547
 
359.472
| style="text-align:center;" | undecimal neutral third
 
..
 
tridecimal neutral third
|-
|-
| style="text-align:center;" | 450
| 50
| style="text-align:center;" | minor fourth, ultra third, narrow fourth
| Quarter tone, infra second, wide unison
| style="text-align:center;" | [[22/17|22/17]]
 
[[35/27|35/27]]
 
[[13/10|13/10]]
| style="text-align:center;" | 446.363
 
449.275
 
454.214
| style="text-align:center;" | 17-limit supermajor third
 
..
 
tridecimal subfourth
|-
|-
| style="text-align:center;" | 550
| 150
| style="text-align:center;" | wide fourth
| Neutral second
| style="text-align:center;" | [[11/8|11/8]]
| style="text-align:center;" | 551.318
| style="text-align:center;" | undecimal superfourth, harmonic 11th
|-
|-
| style="text-align:center;" | 650
| 250
| style="text-align:center;" | narrow fifth
| Ultra second, infra third
| style="text-align:center;" | [[16/11|16/11]]
| style="text-align:center;" | 648.682
| style="text-align:center;" | undecimal subfifth, 11th subharmonic
|-
|-
| style="text-align:center;" | 750
| 350
| style="text-align:center;" | wide fifth, infra sixth
| Neutral third
| style="text-align:center;" | [[20/13|20/13]]
 
[[54/35|54/35]]
 
[[17/11|17/11]]
| style="text-align:center;" | 745.786
 
750.725
 
753.637
| style="text-align:center;" | tridecimal superfifth
 
..
 
17-limit subminor sixth
|-
|-
| style="text-align:center;" | 850
| 450
| style="text-align:center;" | neutral sixth
| Minor fourth, ultra third, narrow fourth
| style="text-align:center;" | [[13/8|13/8]]
 
[[44/27|44/27]]
 
[[18/11|18/11]]
| style="text-align:center;" | 840.528
 
845.453
 
852.592
| style="text-align:center;" | overtone sixth, 13th harmonic
 
..
 
undecimal neutral sixth
|-
|-
| style="text-align:center;" | 950
| 550
| style="text-align:center;" | ultra sixth , infra seventh
| Wide fourth
| style="text-align:center;" | [[45/26|45/26]]
 
[[26/15|26/15]]
 
[[125/72|125/72]]
| style="text-align:center;" | 949.696
 
952.259
 
955.031
| style="text-align:center;" | ..
 
..
 
..
|-
|-
| style="text-align:center;" | 1050
| 650
| style="text-align:center;" | neutral seventh
| Narrow fifth
| style="text-align:center;" | [[11/6|11/6]]
| style="text-align:center;" | 1049.363
| style="text-align:center;" | undecimal neutral seventh
|-
|-
| style="text-align:center;" | 1150
| 750
| style="text-align:center;" | ultra seventh, narrow octave
| Wide fifth, infra sixth
| style="text-align:center;" | [[31/16|31/16]]
 
[[33/17|33/17]]
 
[[35/18|35/18]]
| style="text-align:center;" | 1145.036
 
1148.318
 
1151.230
| style="text-align:center;" | 31st harmonic
 
..
 
..
|}
 
=Notation=
There have been disputes about disadvantages of various systems for notating quarter tones. Here are some of the few systems along with pros and cons.
 
==Mainstream Quartertone Notation==
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://www.wfg.woodwind.org/n/s14_blue.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
or ^ = quarter-tone sharp or "Jump" or "up"
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://www.wfg.woodwind.org/n/s34_blue.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://www.wfg.woodwind.org/n/f14_blue.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
or v = quarter-tone flat or "Drop" or "down"
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://www.wfg.woodwind.org/n/f34_blue.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
or bv or vb = three-quarter-tone flat or "Drop-Flat" or "downflat"
 
Pros: Familiar, fairly easy to learn
 
Cons: Clutters a score easily, can get confusing when sight read at faster paces
 
==Alternate Quartertone Accidentals==
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
= quarter-tone sharp or Jump
 
††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://www.dolmetsch.com/onequarterflat.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
= quarter-tone flat or Drop
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
= three quarter-tones flat or drop-flat
 
For example, the scale 0-5-10-15-20 is written as C-D
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
(or E
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
) F G
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
(or A
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
) Bb.
 
Pros: Very easy to distinguish accidentals from one another
 
Cons: Not practical, tends to clutter a score
 
==<span style="line-height: 1.5;">And in Persian music</span>==
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
Koron ([http://en.wikipedia.org/wiki/Koron_%28music%29 en] | [http://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86 fa]) = quarter-tone flat or Jump
 
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
External image: http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Sori_sign.svg/200px-Sori_sign.svg.png<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
 
Sori ([http://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 or Drop
 
Pros: Easy to read
 
Cons: Hard to write on a computer, doesn't fit with standard notation well
 
==Sagittal Notation==
[[Sagittal_notation|Sagittal notation]] works extremely well for 24 Edo notation as well as other systems.
 
It's easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.
 
A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them
 
with Sagittal signs for sharp and flat.
 
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
 
=Interval Alterations=
The special alterations of the intervals and chords of 12 equal can be notated like this:
 
Supermajor or "'''Tendo'''" is a major interval raised a quarter tone
 
Subminor or "'''Arto'''" is a minor interval lowered a quarter tone
 
Neutral are intervals that exist between the major and minor version of an interval
 
The prefix '''under''' indicates a perfect interval lowered by one quarter tone
 
The prefix '''over''' indicates a perfect interval raised by a quarter tone
 
The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.
 
=Chord Structures=
24edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo 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, 24edo features many possibilities for 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) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.
 
These chords though 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|William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 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.
 
William Lynch considers these as some possible good tetrads:
 
[[File:Three_chords.PNG|alt=Three chords.PNG|Three chords.PNG]]
 
{| class="wikitable"
|-
|-
! | Chord name
| 850
! | Degrees of 24edo
| Neutral sixth
! | Chord spelling
! | Audio example
|-
|-
| | neutral
| 950
| | 0 7 14 21
| Ultra sixth, infra seventh
| | 1 v3 5 v7
| | [[File:Neutral_Tetrad_on_C.mp3]]
|-
|-
| | arto
| 1050
| | 0 5 14 20
| Neutral seventh
| | 1 bv3 5 b7
| | [[File:arto_tetrad_on_C.mp3]]
|-
|-
| | tendo
| 1150
| | 0 9 14 19
| Ultra seventh, narrow octave
| | 1 ^3 5 bv7
| | ...
|}
|}


Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.
==== Interval alterations ====
The special alterations of the intervals and chords of 12edo can be notated like this:
 
* Supermajor or "'''Tendo'''" is a major interval raised a quarter tone
* Subminor or "'''Arto'''" is a minor interval lowered a quarter tone
* Neutral are intervals that exist between the major and minor version of an interval
* The prefix '''under''' indicates a perfect interval lowered by one quarter tone
* The prefix '''over''' indicates a perfect interval raised by a quarter tone
* The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.


=Naming Chords in 24edo=
==== Chord names ====
Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.
Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.


They are:
They are:


Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone
* Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone
 
* Sub + perfect interval means to lower a quarter tone
Sub + perfect interval means to lower a quarter tone
* Sharp is to raise by one half tone
 
* Flat is to raise by a half tone
Sharp is to raise by one half tone
* Neutral, arto and tendo refer to triads or tetrads
 
* Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively
Flat is to raise by a half tone
 
Neutral, arto and tendo refer to triads or tetrads
 
Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively


Examples:
Examples:


Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
* Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
* Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11


Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11
=== Further discussion of interval and chord naming ===
{{main|{{PAGENAME}}/Interval names and harmonies }}


Alternatively, [[Ups_and_Downs_Notation|ups and downs notation]] can be used. Here are the zo, gu, lova, yo and ru triads:
* [[24edo Chord Names]]
* [[Ups and downs notation#Chords and Chord Progressions]].


{| class="wikitable"
== Approximation to JI ==
|-
[[File:24ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]
! | [[Kite's color notation|color of the 3rd]]
! | JI chord
! | notes as edosteps
! | notes of C chord
! | written name
! | spoken name
|-
| style="text-align:center;" | zo
| style="text-align:center;" | 6:7:9
| style="text-align:center;" | 0-5-14
| style="text-align:center;" | C Ebv G
| style="text-align:center;" | C.vm
| style="text-align:center;" | C downminor
|-
| style="text-align:center;" | gu
| style="text-align:center;" | 10:12:15
| style="text-align:center;" | 0-6-14
| style="text-align:center;" | C Eb G
| style="text-align:center;" | Cm
| style="text-align:center;" | C minor
|-
| style="text-align:center;" | lova
| style="text-align:center;" | 18:22:27
| style="text-align:center;" | 0-7-14
| style="text-align:center;" | C Ev G
| style="text-align:center;" | C~
| style="text-align:center;" | C mid
|-
| style="text-align:center;" | yo
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 0-8-14
| style="text-align:center;" | C E G
| style="text-align:center;" | C
| style="text-align:center;" | C major or C
|-
| style="text-align:center;" | ru
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 0-9-14
| style="text-align:center;" | C E^ G
| style="text-align:center;" | C.^
| style="text-align:center;" | C upmajor or C dot up
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].


=Rank two temperaments=
=== Interval mappings ===
[[List_of_24et_rank_two_temperaments_by_badness|List of 24et rank two temperaments by badness]]
{{Q-odd-limit intervals|24}}


[[List_of_edo-distinct_24et_rank_two_temperaments|List of edo-distinct 24et rank two temperaments]]
== Regular temperament properties ==
 
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable"
|-
! | Periods per octave
! | Generator
! | Name
|-
| | 1
| | 1\24
| |
|-
| | 1
| | 5\24
| | [[Semaphore_and_Godzilla|Semaphore/godzilla]] / [[Chromatic_pairs#Bridgetown|Bridgetown]]
|-
| | 1
| | 7\24
| | [[Mohajira|Mohajira]] (or [[maqamic|maqamic]] with 24d val)
|-
| | 1
| | 11\24
| |
|-
|-
| | 2
! rowspan="2" | [[Subgroup]]
| | 1\24
! rowspan="2" | [[Comma list]]
| | [[Shrutar|Shrutar]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 2
! [[TE error|Absolute]] (¢)
| | 5\24
! [[TE simple badness|Relative]] (%)
| | Similar to [[Decimal|decimal]]
|-
|-
| | 3
| 2.3.5.11
| | 1\24
| 81/80, 121/120, 128/125
| | [[Semiaug|Semiaug]]
| {{mapping| 24 38 56 83 }}
| −1.08
| 2.82
| 5.63
|-
|-
| | 3
| 2.3.5.11.13
| | 3\24
| 66/65, 81/80, 128/125, 144/143
| | [[Triforce|Triforce]]
| {{mapping| 24 38 56 83 89 }}
| −1.37
| 2.59
| 5.19
|-
|-
| | 4
| 2.3.5.11.13.17
| | 1\24
| 51/50, 66/65, 81/80, 128/125, 144/143
| |  
| {{mapping| 24 38 56 83 89 98 }}
| −0.94
| 2.55
| 5.11
|-
|-
| | 6
| 2.3.5.11.13.17.19
| | 1\24
| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143
| |  
| {{mapping| 24 38 56 83 89 98 102 }}
|-
| −0.89
| | 8
| 2.37
| | 1\24
| 4.74
| |
|-
| | 12
| | 1\24
| | [[Catler|Catler]]
|}
|}


=Scales / Modes=
=== Uniform maps ===
 
{{Uniform map|edo=24}}
{| class="wikitable"
|-
| colspan="2" |
====Pentatonic:====
|-
| | <tt>'''2 8 3 6 5'''</tt>
| | Anchihoye: Ethiopia
|-
| | <tt>'''5 5 4 5 5'''</tt>
| | Quasi-equal Pentatonic - [[MOSScales|MOS]] of type [[4L_1s|4L 1s (bug)]]
|-
| | <tt>'''5 5 5 5 4'''</tt>
| | Hába's Pentatonic - [[MOSScales|MOS]] of type [[4L_1s|4L 1s (bug)]]
|}


{| class="wikitable"
=== Commas ===
|-
This is a partial list of the [[commas]] that 24edo [[tempering out|tempers out]] with its patent [[val]], {{val| 24 38 56 67 83 89 }}.
| colspan="2" |
====Hexatonic:====
|-
| | <tt>'''1 1 8 4 2 8'''</tt>
| | Spondeiakos
|}


{| class="wikitable"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
| colspan="2" |
! [[Harmonic limit|Prime<br>limit]]
====Heptatonic:====
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
|-
| | <tt>'''1 1 8 1 1 8 4'''</tt>
| 3
| | Enharmonic Mixolydian
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
|-
| | <tt>'''1 8 1 1 8 4 1'''</tt>
| 5
| | Enharmonic Lydian
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
|-
| | <tt>'''8 1 1 8 4 1 1'''</tt>
| 5
| | Enharmonic Phrygian
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
|-
| | <tt>'''1 1 8 4 1 1 8'''</tt>
| 5
| | Enharmonic Dorian
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
|-
| | <tt>'''1 8 4 1 1 8 1'''</tt>
| 5
| | Enharmonic Hypolydian
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| | <tt>'''8 4 1 1 8 1 1'''</tt>
| 5
| | Enharmonic Hypophrygian
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
|-
| | <tt>'''4 1 1 8 1 1 8'''</tt>
| 5
| | Enharmonic Hypodorian
| [[67108864/66430125| (16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
|-
| | <tt>'''2 3 5 2 3 5 4'''</tt>
| 5
| | Soft Diatonic Mixolydian
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
|-
| | <tt>'''3 5 2 3 5 4 2'''</tt>
| 5
| | Soft Diatonic Lydian
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadbigu
| [[Kirnberger's atom]]
|-
|-
| | <tt>'''5 2 3 5 4 2 3'''</tt>
| 7
| | Soft Diatonic Phrygian
| [[1323/1280]]
| {{monzo| -8 3 -1 2 }}
| 57.20
| Lazozogu
| Septimal two-seventh tone
|-
|-
| | <tt>'''2 3 5 4 2 3 5'''</tt>
| 7
| | Soft Diatonic Dorian
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
|-
| | <tt>'''3 5 4 2 3 5 2'''</tt>
| 7
| | Soft Diatonic Hypolydian
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
|-
| | <tt>'''5 4 2 3 5 2 3'''</tt>
| 7
| | Soft Diatonic Hypophrygian
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
|-
| | <tt>'''4 2 3 5 2 3 5'''</tt>
| 7
| | Soft Diatonic Hypodorian
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Sarurutrigu
| Porwell comma
|-
|-
| | <tt>'''3 3 4 3 3 4 4'''</tt>
| 11
| | Maqam Ouchairan-Hussaini, Bayatan, Neutral Diatonic Mixolydian - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
|-
| | <tt>'''3 4 3 3 4 4 3'''</tt>
| 11
| | Dastgah-e Sehgah, Neutral Diatonic Lydian - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
|-
| | <tt>'''4 3 3 4 4 3 3'''</tt>
| 11
| | Arabic Diatonic, Maqam Rast, Quasi-equal Heptatonic, Neutral Diatonic Phrygian - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
|-
| | <tt>'''3 3 4 4 3 3 4'''</tt>
| 11
| | Maqam Hussaini, Ushaq, Neutral Diatonic Dorian - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
|-
| | <tt>'''3 4 4 3 3 4 3'''</tt>
| 11
| | Maqam Sikah (Segah), Neutral Diatonic Hypolydian - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
|-
| | <tt>'''4 4 3 3 4 3 3'''</tt>
| 11
| | Neutral Diatonic Hypophrygian - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[243/242]]
| {{monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
|-
| | <tt>'''4 3 3 4 3 3 4'''</tt>
| 11
| | Miha'il Musaqa's mode: Egypt, Neutral Diatonic Hypodorian, Dastgah-e Sehgah, Maqam Nairuz - [[MODMOS_Scales|MODMOS]] of type [[3L_4s|3L 4s (mosh)]]
| <abbr title="214990848/214358881">(18 digits)</abbr>
| {{monzo| 15 8 0 0 -8 }}
| 5.10
| Quadbilu
| [[Octatonic comma]]
|-
|-
| | <tt>'''1 5 4 1 5 4 4'''</tt>
| 11
| | Diatonic + Enharmonic Diesis Mixolydian
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
|-
| | <tt>'''5 4 1 5 4 4 1'''</tt>
| 11
| | Diatonic + Enharmonic Diesis Lydian
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{monzo| 24 -6 0 1 -5 }}
| 0.51
| Saquinlu-azo
| [[Quartisma]]
|-
|-
| | <tt>'''4 1 5 4 4 1 5'''</tt>
| 11
| | Diatonic + Enharmonic Diesis Phrygian
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
|-
| | <tt>'''1 5 4 4 1 5 4'''</tt>
| 11
| | Diatonic + Enharmonic Diesis Dorian
| <abbr title="1771561/1771470">(14 digits)</abbr>
| {{monzo| -1 -11 -1 0 6 }}
| 0.089
| Satribilo-agu
| [[Parimo]]
|-
|-
| | <tt>'''5 4 4 1 5 4 1'''</tt>
| 13
| | Diatonic + Enharmonic Diesis Hypolydian
| [[66/65]]
| {{monzo| 1 1 -1 0 1 -1 }}
| 26.43
| Thulogu
| Winmeanma
|-
|-
| | <tt>'''4 4 1 5 4 1 5'''</tt>
| 13
| | Diatonic + Enharmonic Diesis Hypophrygian
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|-
| | <tt>'''4 1 5 4 1 5 4'''</tt>
| 13
| | Diatonic + Enharmonic Diesis Hypodorian
| [[512/507]]
| {{monzo| 9 -1 0 0 0 -2 }}
| 16.99
| Thuthu
| Tridecimal neutral thirds comma
|-
|-
| | <tt>'''1 3 6 1 3 6 4'''</tt>
| 13
| | Chromatic/Enharmonic Mixolydian
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| Thuzoyo
| Animist comma
|-
|-
| | <tt>'''3 6 1 3 6 4 1'''</tt>
| 13
| | Chromatic/Enharmonic Lydian
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
|-
| | <tt>'''6 1 3 6 4 1 3'''</tt>
| 13
| | Chromatic/Enharmonic Phrygian
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Thorugugu
| Ratwolfsma
|-
|-
| | <tt>'''1 3 6 4 1 3 6'''</tt>
| 13
| | Chromatic/Enharmonic Dorian
| [[352/351]]
| {{monzo| 5 -3 0 0 1 -1 }}
| 4.93
| Thulo
| Minor minthma
|-
|-
| | <tt>'''3 6 4 1 3 6 1'''</tt>
| 13
| | Chromatic/Enharmonic Hypolydian
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|-
|-
| | <tt>'''6 4 1 3 6 1 3'''</tt>
| 13
| | Chromatic/Enharmonic Hypophrygian
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
|-
| | <tt>'''4 1 3 6 1 3 6'''</tt>
| 17
| | Chromatic/Enharmonic Hypodorian
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
|-
| | <tt>'''3 4 3 3 4 3 4'''</tt>
| 17
| | Neutral Mixolydian - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
|-
| | <tt>'''4 3 3 4 3 4 3'''</tt>
| 17
| | Neutral Lydian - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[170/169]]
| {{monzo| 1 0 1 0 0 -2 1 }}
| 10.21
| Sothuthuyo
| Major naiadma
|-
|-
| | <tt>'''3 3 4 3 4 3 4'''</tt>
| 17
| | Neutral Phrygian - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| Sotholugu
| Minor naiadma
|-
|-
| | <tt>'''3 4 3 4 3 4 3'''</tt>
| 17
| | Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
|-
| | <tt>'''4 3 4 3 4 3 3'''</tt>
| 17
| | Neutral Hypolydian - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
|-
| | <tt>'''3 4 3 4 3 3 4'''</tt>
| 17
| | Neutral Hypophrygian - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[1225/1224]]
| {{monzo| -3 -2 2 2 0 0 -1 }}
| 1.41
| Subizoyo
| Noellisma
|-
|-
| | <tt>'''4 3 4 3 3 4 3'''</tt>
| 19
| | Neutral Hypodorian - [[MOSScales|MOS]] of type [[3L_4s|3L 4s (mosh)]]
| [[76/75]]
| {{monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.93
| Nogugu
| Large undevicesimal ninth tone
|-
|-
| | <tt>'''3 5 2 4 3 5 2'''</tt>
| 19
| | Athanasopoulos' Byzantine Liturgical Chromatic, Dastgah-e Chahargah
| [[77/76]]
| {{monzo| -2 0 0 1 1 0 0 -1 }}
| 22.63
| Nulozo
| Small undevicesimal ninth tone
|-
|-
| | <tt>'''4 2 4 4 3 3 4'''</tt>
| 19
| | Dastgah-e Nava, Maqam Ushaq Masri
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
|-
| | <tt>'''2 7 1 4 2 7 1'''</tt>
| 19
| | Second plagal Byzantine Liturgical mode
| [[133/132]]
| {{monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.07
| Noluzo
| Minithirdma
|-
|-
| | <tt>'''3 3 4 4 2 4 4'''</tt>
| 19
| | Maqam 'Ushshaq Turki, Urfa, Isfahan, Dastgah-e Shur
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
|-
| | <tt>'''3 3 4 4 4 2 4'''</tt>
| 19
| | Maqam Nahfat
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
|-
| | <tt>'''3 3 2 6 2 4 4'''</tt>
| 19
| | Maqam Saba
| [[209/208]]
| {{monzo| -4 0 0 0 1 -1 0 1 }}
| 8.30
| Nothulo
| Yama comma
|-
|-
| | <tt>'''3 3 2 6 2 6 2'''</tt>
| 19
| | Maqam Sabr Jadid
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
|-
| | <tt>'''4 3 3 4 2 6 2'''</tt>
| 19
| | Maqam Suznak (Soznak)
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|-
|-
| | <tt>'''4 3 3 4 4 4 2'''</tt>
| 19
| | Maqam Mahur
| [[5776/5775]]
| {{monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.30
| Nonolurugugu
| Neovish comma
|}
<references group="note" />
 
=== Rank-2 temperaments ===
* [[List of 24et rank two temperaments by badness]]
* [[List of edo-distinct 24et rank two temperaments]]
 
{| class="wikitable center-1 center-2"
|-
|-
| | <tt>'''3 3 4 2 6 2 4'''</tt>
! Periods<br>per 8ve
| | Maqam Qarjighar, Bayati Shuri
! Generator
! Name
|-
|-
| | <tt>'''3 4 2 6 2 4 3'''</tt>
| 1
| | Maqam Hizam (Huzam, El Houzam), Rahat al Arouah
| 1\24
| [[Hemiripple]] (24)
|-
|-
| | <tt>'''2 4 4 4 3 3 4'''</tt>
| 1
| | Maqam Nawa
| 5\24
| [[Godzilla]] (24) / [[baragon]] (24) / [[semaphoresmic clan #Varan|varan]] (24)
|-
|-
| | <tt>'''2 5 3 4 2 5 3'''</tt>
| 1
| | Maqam Higaz-kar
| 7\24
| [[Mohajira]] (24) / [[neutrominant]] (24d) / [[migration]] (24d)
|-
|-
| | <tt>'''3 4 4 2 4 4 3'''</tt>
| 1
| | Maqam Su'ar, Naghmeh Abuata, Naghmeh Afshari
| 11\24
| [[Cohemiripple]] (24), [[freivald]] (24)
|-
|-
| | <tt>'''4 4 2 4 4 3 3'''</tt>
| 2
| | Maqam Jahargah (Jiharkah), Naghmeh Bayat-e Tork, Naghmeh Dashti
| 1\24
| [[Shrutar]] (24)
|-
|-
| | <tt>'''3 5 2 4 2 4 4'''</tt>
| 2
| | Dastgah-e Homayun
| 5\24
| [[Sruti]] (24), [[anguirus]] (24), [[decimal]] (24c)
|-
|-
| | <tt>'''4 2 4 4 3 5 2'''</tt>
| 3
| | Naghmeh Esfahan
| 1\24
| [[Hemiaug]] (24)
|-
|-
| | <tt>'''3 6 1 5 2 6 1'''</tt>
| 3
| | Maqam 'Awg 'ara (Aug-ara)
| 3\24
| [[Triforce]] (24)
|-
|-
| | <tt>'''4 1 5 4 2 6 2'''</tt>
| 4
| | Maqam Buselik
| 1\24
| [[Hemidim]] (24)
|-
|-
| | <tt>'''4 2 6 2 2 5 3'''</tt>
| 6
| | Maqam Neuter
| 1\24
| [[Hemisemiaug]] (24)
|-
|-
| | <tt>'''4 3 3 4 4 2 4'''</tt>
| 8
| | Dance scale of Yi people: China
| 1\24
| [[Semidim]] (24)
|-
|-
| | <tt>'''4 4 2 3 1 4 6'''</tt>
| 12
| | Daniel-mode of Spanish-Arab Jews
| 1\24
| [[Catler]]
|}
|}


{| class="wikitable"
Important MOSes include:
* Semaphore 4L&nbsp;1s 55455 (generator: 5\24)
* Semaphore 5L&nbsp;4s 414141414 (generator: 5\24)
* Mohajira 3L&nbsp;4s 3434343 (generator: 7\24)
* Mohajira 7L&nbsp;3s 3313313313 (generator: 7\24)
 
== Octave stretch or compression ==
If one wishes to use 24edo as a full 19-or-lower-limit tuning, then it benefits from slight [[octave stretching]], mostly to improve its [[prime]] 7. If one wishes to use 24edo as a no-7s 19-or-lower-limit tuning, then it benefits from slight [[octave shrinking]], mostly to improve its primes 5 and 13.
* Stretched-octave tunings (least to most stretch): [[ed12|86ed12]], [[ed6|62ed6]], [[38edt]]
* Compressed-octave tunings (least to most compression): [[zpi|90zpi]], [[equal tuning|80ed10]], [[ed5|56ed5]]
 
== Scales and modes ==
''See: [[24edo scales]] and [[List of MOS scales in 24edo]].''
 
== Tetrachords ==
''See [[24edo tetrachords]]''.
 
== 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 4:5:6:9:11:13, a sort of super-extended major chord.
 
As for entirely new chords, there are three new fundamental options, giving five basic triads over 12edo's two:
 
{| class="wikitable center-all"
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental triads of 24edo
|-
|-
| colspan="2" |
! JI Chord
====Octatonic:====
! Edosteps
! Notes of C Chord
! Written name
! Spoken name
|-
|-
| | <tt>'''3 3 3 3 3 3 3 3'''</tt>
| 6:7:9, 26:30:39
| | 8-equal, Wyschnegradsky's octatonic
| {{dash|0, 5, 14|hair}}
| {{dash|C, E{{sesquiflat2}}, G|hair}}
| Cvm<br>Cm({{demiflat2}}3), Cmin({{demiflat2}}3)
| C inframinor<br>C minor semiflat-three
|-
|-
| | <tt>'''3 3 4 4 2 1 3 4'''</tt>
| 10:12:15
| | Maqam Bayati
| {{dash|0, 6, 14|hair}}
| {{dash|C, E♭, G|hair}}
| Cm, Cmin
| C minor
|-
|-
| | <tt>'''3 3 2 6 2 4 2 2'''</tt>
| 18:22:27, 22:27:33
| | Maqam Saba
| 0-7-14
| {{dash|C, E{{demiflat2}}, G|hair}}
| C~, Cneu
| C neutral
|-
|-
| | <tt>'''4 3 1 2 4 4 2 4'''</tt>
| 4:5:6
| | Maqam Suzidil 'ara
| {{dash|0, 8, 14|hair}}
| {{dash|C, E, G|hair}}
| C, Cmaj
| C, C major
|-
|-
| | <tt>'''3 3 2 2 4 3 3 4'''</tt>
| 14:18:21, 10:13:15
| | Maqam Mansuri
| {{dash|0, 9, 14|hair}}
| {{dash|C, E{{demisharp2}}, G|hair}}
| C^<br>C({{demisharp2}}3), Cmaj({{demisharp2}}3)
| 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.
 
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:
 
* {{dash|0, 4, 8, 11, 14|hair}} ("major" chord with a 9:8 and a 11:8 above the root)
* Its inversion, {{dash|0, 3, 6, 10, 14|hair}} ("minor")
* 0-5-10 (another kind of "neutral", splitting the fourth in two. The {{dash|0, 5, 10|hair}} can be extended into a ([[Godzilla]]) pentatonic scale ({{dash|0, 5, 10, 14, 19, 24|hair}}), that is close to equi-pentatonic and also close to several Indonesian slendro scales. In a similar way {{dash|0, 7, 14|hair}} extends to {{dash|0, 4, 7, 11, 14, 18, 21, 24|hair}} ([[mohajira]]), a heptatonic scale close to several Arabic scales.)
 
William Lynch considers these as some possible good tetrads:
 
[[File:Three_chords.PNG|alt=Three chords.PNG|Three chords.PNG]]
 
{| class="wikitable center-all"
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental tetrads of 24edo
|-
|-
| | <tt>'''4 3 3 4 4 2 1 3'''</tt>
! Degrees of 24edo
| | Maqam Rast, Dilkashidah, Dilnishin
! Chord spelling
! Notes of C chord
! Written name
! Spoken name
! Audio example
|-
|-
| | <tt>'''3 4 2 6 2 4 2 1'''</tt>
| {{dash|0, 5, 14, 19|hair}}
| | Maqam Rahat al-Arwah
| {{dash|1, vb3, 5, vb7|hair}}
| {{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|hair}}
| smin7<br>min7({{demiflat2}}3, {{demiflat2}}7)
| Inframinor seven<br>Minor seven semiflat-three semiflat-seven
|-
|-
| | <tt>'''3 4 3 3 4 4 2 1'''</tt>
| {{dash|0, 6, 14, 20|hair}}
| | Maqam Iraq
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E♭, G, B♭|hair}}
| m7, min7
| Minor seven
|-
|-
| | <tt>'''2 6 2 4 2 1 3 4'''</tt>
| {{dash|0, 7, 14, 21|hair}}
| | Maqam Hijaz
| {{dash|1, v3, 5, v7|hair}}
| {{dash|C, E{{demiflat2}}, G, B{{demiflat2}}|hair}}
| n7, neu7
| Neutral seven
| [[File:Neutral Tetrad on C.mp3]]
|-
|-
| | <tt>'''3 4 4 2 1 3 4 3'''</tt>
| {{dash|0, 8, 14, 22|hair}}
| | Maqam Musta'ar
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E, G, B|hair}}
| maj7
| Major seven
|-
|-
| | <tt>'''3 4 4 4 2 4 2 1'''</tt>
| {{dash|0, 8, 14, 22|hair}}
| | Maqam Farahnak
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|hair}}
| smaj7<br>maj7({{demisharp2}}3, {{demisharp2}}7)
| Ultramajor seven<br>Major seven semisharp-three semisharp-seven
|-
|-
| | <tt>'''3 4 3 3 2 6 2 1'''</tt>
| {{dash|0, 8, 14, 20|hair}}
| | Maqam Bastanikar, Tarz Nuin
| {{dash|1, 3, 5, b7|hair}}
| {{dash|C, E, G, B♭|hair}}
| 7, dom7
| Dominant seven
|-
|-
| | <tt>'''4 2 6 2 4 2 1 3'''</tt>
| {{dash|0, 8, 14, 19|hair}}
| | Maqam Farah Faza, Maqam Nakriz
| {{dash|1, 3, 5, vb7|hair}}
| {{dash|C, E, G, B{{sesquiflat2}}|hair}}
| h7<br>7({{demiflat2}}7)
| Harmonic seven<br>Dominant 7 semiflat-seven
|-
|-
| | <tt>'''3 1 2 4 4 2 4 4'''</tt>
| {{dash|0, 5, 14, 20|hair}}
| | Maqam Jabburi
| {{dash|1, vb3, 5, b7|hair}}
| {{dash|C, E{{sesquiflat2}}, G, B♭|hair}}
| min7({{demiflat2}}3)
| Arto<br>Minor seven semiflat-three
| [[File:arto tetrad on C.mp3]]
|-
|-
| | <tt>'''1 4 4 2 4 4 4 1'''</tt>
| {{dash|0, 9, 14, 19|hair}}
| | Giancarlo Dalmonte's new quarter-tone scale (see [http://www.ottavanota.info/ http://www.ottavanota.info])
| {{dash|1, ^3, 5, vb7|hair}}
| {{dash|C, E{{demisharp2}}, G, B{{sesquiflat2}}|hair}}
| h7({{demisharp2}}3)<br>7({{demisharp2}}3, {{demiflat2}}7)
| Tendo<br>Harmonic seven semisharp-three<br>Dominant seven semisharp-three semiflat-seven
|  
|}
|}


{| class="wikitable"
The tendo chord can also be spelled {{nowrap|1 ^3 5 ^6}}. Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.
|-
 
| colspan="2" |
== Counterpoint ==
====Enneatonic:====
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.
|-
| | <tt>'''1 2 3 4 4 1 2 3 4'''</tt>
| | Progressive Enneatonic
|-
| | <tt>'''4 1 4 1 4 1 4 1 4'''</tt>
| | de Vries 9-tone - [[MOSScales|MOS]] of type [[5L_4s|5L 4s (unfair bug)]]
|-
| | <tt>'''3 4 2 2 2 2 2 4 3'''</tt>
| | Maqam Huzam
|-
| | <tt>'''4 4 2 1 3 2 2 4 2'''</tt>
| | Maqam Shawq Afza
|}


{| class="wikitable"
Furthermore, in the same fashion, every sequence of intervals available in 12edo are reachable by equal contrary motion in 24edo.
|-
[[File:Strict-contrary-motion-24edo.png|left|frame|Every sequence of 12edo intervals are reachable by strict contrary motion in 24edo. [[File:24-EDO_Contrary_Motion.flac]]]] {{clear}}
| colspan="2" |
====Decatonic:====
|-
| | <tt>'''3 1 3 3 3 1 3 3 1 3'''</tt>
| | Breed Decatonic - [[MOSScales|MOS]] of type [[7L_3s|7L 3s (unfair mosh)]]
|-
| | <tt>'''2 3 2 2 3 2 3 2 2 3'''</tt>
| | Oljare Decatonic - [[MOSScales|MOS]] of type [[4L_6s|4L 6s (fair bicycle)]]
|-
| | <tt>'''2 1 3 2 2 4 2 4 2 2'''</tt>
| | Maqam Shawq Tarab
|-
| | <tt>'''4 2 1 3 2 2 4 2 1 3'''</tt>
| | Maqam Basandida
|-
| | <tt>'''4 3 1 2 1 3 4 2 1 3'''</tt>
| | Maqam Yakah
|}


{| class="wikitable"
== 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.
| colspan="2" |
====Hendecatonic:====
|-
| | <tt>'''4 2 1 3 2 2 2 2 3 1 2'''</tt>
| | Maqam Hayyan
|}


{| class="wikitable"
=== Guitar ===
|-
Adam Hoey Xen ([https://www.youtube.com/@adamhoeyxen2199/videos on YouTube]) has used a "neutral thirds tuning" of F#-At-C#-Et-G#-Bt on a standard guitar to play in quartertones.
| colspan="2" |
====Tridecatonic:====
|-
| | <tt>'''1 2 2 2 2 2 2 1 2 2 2 2 2'''</tt>
| | de Vries 13-tone - [[MOSScales|MOS]] of type [[11L_2s|11L 2s]]
|-
| | <tt>'''2 2 2 2 2 1 2 2 2 2 2 2 1'''</tt>
| | Agmon Diatonic DS5, Ivan Wyschnegradsky's diatonicized chromatic scale - [[MOSScales|MOS]] of type [[11L_2s|11L 2s]]
|}


{| class="wikitable"
Guitars with 24 frets per octave are also an option, although only [https://eastwoodguitars.com/products/hi-flier-edo-24-electric-microtonal-guitar Eastwood] offer this as a standard production model at the time of writing. Other luthiers you can commission custom microtonal instruments from, including 24edo ones, include [https://www.etsy.com/uk/listing/1154683769 JLJ instruments] and [https://meantoneguitar.com Meantone Guitar].
|-
| colspan="2" |
====Tetradecatonic:====
|-
| | '''<span style="font-family: monospace;">2 2 1 2 2 2 1 2 2 1 2 2 2 1</span>'''
| | Young Half-Octave Diatonic - [[MOSScales|MOS]] of type [[10L_4s|10L 4s]]
|}


=Instruments=
[[File:24edo_guitar.jpg|500px]]
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.


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
While these are playable, the extra frets can make playing chords and navigating the fretboard significantly more challenging for [[12edo]] chords and scales.
External image: http://www.swordguitars.com/Sword_quartertone_stratsm.jpg<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]


24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars
More common is the "Sazocaster" tuning popularised by Australian band [[King Gizzard & the Lizard Wizard]], which adds quarter tones between approximately half the regular frets. Multiple guitar makers, including [https://eastwoodguitars.com/products/sg2c-flying-banana-mt Eastwood] and [https://salamuzik.com/collections/guitar/products/professional-microtonal-electric-classical-guitar-with-equalizer-kg-5 Sala Muzik] have produced Sazocaster variations.


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


=Compositions=
It is also possible to create a guitar that has quartertones on all the lower frets, then switches to regular 12edo at some point on the neck to keep the upper notes easily acessible, as demonstrated by the band [[Angine de Poitrine]]. Guitars using this layout are available at [https://www.microtonalguitar.org/product-page/angine-de-poitrine-style-fixed-microtonal-electric-guitar-ap24 microtonalguitar.org] and doubleneck guitar/basses are available from [https://eastwoodguitars.com/products/eastwood-microtonal-doubleneck-electric-guitar-bass Eastwood].


==Music==
=== Harp, Harpsichord, and Piano ===
<span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex3.mp3 Microhex3]</span> <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex4.mp3 Microhex4]</span> and <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex5.mp3 Microhex5]</span> by [http://www.96edo.com/About_me.html Shaahin Mohajeri]


<span style="">[http://micro.soonlabel.com/24et/quarterpicnic.mp3 Quarterpicnic]</span> by [[Chris_Vaisvil|Chris Vaisvil]]
==== Scordatura tuning of 12edo instruments ====


[http://micro.soonlabel.com/gene_ward_smith/Others/BeDell/Quarter%20Tone%20Prelude%20for%20two%20Harps.mp3 Quarter Tone Prelude For Two Harps] by [http://soundcloud.com/cerbeus-1/quarter-tone-prelude-for-two Nathan BeDell]
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 simpler version of this (fewer notes retuned) is demonstrated in [https://www.youtube.com/watch?v=KS-mmj5kuxw ''when it blooms (24edo)''] (2021). 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.


<span style="">[http://micro.soonlabel.com/24et/daily20111021-fretless-1.mp3 Fretless Chrome 1]</span> and <span style="">[http://micro.soonlabel.com/24et/daily20111021-fretless-2.mp3 Fretless Chrome 2]</span> by [http://chrisvaisvil.com/?p=1477 Chris Vaisvil]
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 different scordatura tuning), followed by the scordatura tuning of the harp.


[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lament.mp3 Lament] by [http://soundcloud.com/jdfreivald/lament Jake Freivald]. In the [[freivaldneutral24|freivaldneutral24]] scale.
==== Quarter-tone instruments ====


[https://soundcloud.com/jdfreivald/mo-happy-happy Mo - Happy - Happy] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/mo-happy-happy.mp3 play] by Jake Freivald in Neuter[7] (2.3.11 mohajira), 24et tuning
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):


[http://www.youtube.com/watch?v=q1Lp8AtKK9o Autumn Winds], [http://www.youtube.com/watch?v=Igxe3DwbFJ4&feature=c4-overview&list=UUvq5bg-LvOS6adpB5efPTyQ Easter Time at Nine], [http://www.youtube.com/watch?v=z_3uhA9Cq08&list=UUvq5bg-LvOS6adpB5efPTyQ Waters of Persia] by William Lynch in mohajira, 24et tuning.
; Quarter-tone grand piano, Czech Museum of Music (this piano is essentially two stacked grand pianos, and as such is massive, in order to avoid sacrificing strings per note)
* [https://www.youtube.com/shorts/Ieqi54XE2lI Demonstration short video by Nahre Sol] (2024)


[https://soundcloud.com/mason-l-green/serena Serena], by Mason Green (intro and coda in 24edo, the rest is in 12edo)
; Quarter-tone upright piano, Academy of Music in Prague (Czech Republic) (this piano apparently sacrificed number of strings per note in order to be able to fit into a reasonable amount of space)
* [https://www.youtube.com/watch?v=PdP4epQIUrU Demonstration video by Steve Cohn] (2011)


[https://www.youtube.com/watch?v=yzvXEMYgHCY Autumn Girl], by Mason Green
=== Electronic Keyboards ===


==About==
24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see [[Lumatone mapping for 24edo]]
"Prométhée enchaîné" by [http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy Fromental Halévy] (considered the first mainstream western orchestral composition to use quarter tones.)


"3 Hommages" by [http://en.wikipedia.org/wiki/Georg_Friedrich_Haas Georg Friedrich Haas]
=== Flute ===
Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:


[http://en.wikipedia.org/wiki/List_of_quarter_tone_pieces List of quartertone pieces on Wikipedia]
; Quarter-tone flute, made by Eva Kingma
* [https://www.youtube.com/watch?v=F3GD0Omr4Z0 Visit to the workshop of Eva Kingma, followed by test by Manuel Luis Cochofel] (2010) (demonstration of fingering starts at 06:56)


=Practical Theory / Books=
=== Brass ===
Since the trombone is a free-pitched instrument, playing quartertones, or any other edo simply requires increased precision in moving the slide. If you want a brass instrument with fixed steps, [https://www.a-courtois.com/en/instruments/trompettes-2/t-o-m-a Courtois] and [https://www.vanlaartrumpets.nl/en/trumpets/quartertone Van Laar] both produce trumpets with an additional valve that enable you to easily play quartertones. In addition, {{W|Renold Schilke|Schilke Music Products Incorporated}} built quartertone trumpets (model B5), as shown in All Things Brass And Technology's [https://www.youtube.com/watch?v=1ip0lOlQ2Xo&list=WL&index=257 ''Schilke B5 Quartertone Trumpet from 1971''] (2023).


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== Music ==
External image: http://ronsword.com/images/24_tet_Coversm.jpg<br>
{{Wikipedia|List of quarter tone pieces}}
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'''"Icosakaiteraphonic Scales for Guitar"''' - A Book for Twenty-Four Equal Divisions of the Octave on guitar, or 'Quarter-tones'. 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.
== Further reading ==
* 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.)


=External links=
== See also ==
[http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory] [http://www.webcitation.org/5xeFMH6cd Permalink]
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)


[http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [http://www.webcitation.org/5xeFBNdQW Permalink]
== External links ==
* [http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo / 24-ed2] [https://www.webcitation.org/5xeFMH6cd Permalink] on [[Tonalsoft Encyclopedia]]
* [http://www.96edo.com/24_EDO.html About 24-EDO] [https://www.webcitation.org/5xeFBNdQW Permalink] by Shaahin Mohajeri
* [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://tonalsoft.com/enc/q/quarter-tone.aspx Tonalsoft Encyclopedia | ''quarter-tone / 24-edo / 24-ed2'']  


[https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
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[http://sonic-arts.org/darreg/contents.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor_Darreg|Ivor Darreg]] [http://sonic-arts.org/darreg/dar8.htm Permalink]     
[[Category:Semaphore]]
[[Category:edo]]
[[Category:Godzilla]]
[[Category:intervals]]
[[Category:Meantone]]
[[Category:listen]]
[[Category:Quartertone]]
[[Category:quartertone]]
[[Category:Quartismic]]
[[Category:subgroup]]
[[Category:Subgroup temperaments]]
[[Category:table]]
[[Category:Twentuning]]
[[Category:theory]]
[[Category:todo:unify_precision]]
[[Category:twentuning]]
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