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


=Theory=
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]].  
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 as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.


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|7:4]], [[7/5|7:5]] and [[7/6|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 [[36edo|36-tET]], [[72edo|72-tET]], [[84edo|84-tET]] or [[156edo|156-tET]].  However, it should be noted that 24edo, like [[22edo]], ''does'' temper out [[Quartisma|117440512/117406179]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit, and speaking of 11-limit representation in 24edo, 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 this EDO. 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.  
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 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.
== Theory ==
The [[5-limit]] approximations in 24edo are the same as those in 12edo, so 24edo offers nothing new as far as approximating the 5-limit is concerned.


= Notation =
The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth. 24edo is also good at the 13th harmonic, which makes it a good 2.3.5.11.13 system. Specifically, intervals of 13/5 are particularly well approximated. And of course, 24edo shares its 17 and 19 tunings with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.


== Ups and down notation ==
While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically: a supermajor chord is available at [0 9 14] and a subminor chord at [0 5 14], though they're more ultramajor and inframinor.
{| class="wikitable"
|-
| style="text-align:center;" | Degree
| style="text-align:center;" | Cents
| style="text-align:center;" | Approximate Ratios*
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| colspan="2" style="text-align:center;" | 0
| style="text-align:center;" | 1/1
| style="text-align:center;" | P1
| style="text-align:center;" | unison
| style="text-align:center;" | C
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 50
| style="text-align:center;" | 33/32, 34/33
| style="text-align:center;" | ^P1, vm2
| style="text-align:center;" | up-unison, downminor 2nd
| style="text-align:center;" | ^C, vDb
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 100
| style="text-align:center;" | 17/16, 18/17
| style="text-align:center;" | A1, m2
| style="text-align:center;" | aug unison, minor 2nd
| style="text-align:center;" | C#, Db
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 150
| style="text-align:center;" | 12/11
| style="text-align:center;" | ~2
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | vD
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 200
| style="text-align:center;" | 9/8
| style="text-align:center;" | M2
| style="text-align:center;" | major 2nd
| style="text-align:center;" | D
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 250
| style="text-align:center;" | 22/19
| style="text-align:center;" | ^M2, vm3
| style="text-align:center;" | upmajor 2nd, downminor 3rd
| style="text-align:center;" | ^D, vEb
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 300
| style="text-align:center;" | 19/16
| style="text-align:center;" | m3
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | Eb
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 350
| style="text-align:center;" | 11/9, 27/22
| style="text-align:center;" | ~3
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | vE
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 400
| style="text-align:center;" | 24/19
| style="text-align:center;" | M3
| style="text-align:center;" | major 3rd
| style="text-align:center;" | E
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 450
| style="text-align:center;" | 22/17
| style="text-align:center;" | ^M3, v4
| style="text-align:center;" | upmajor 3rd, down-4th
| style="text-align:center;" | ^E, vF
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 500
| style="text-align:center;" | 4/3
| style="text-align:center;" | P4
| style="text-align:center;" | fourth
| style="text-align:center;" | F
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 550
| style="text-align:center;" | 11/8
| style="text-align:center;" | ^4, ~4
| style="text-align:center;" | up-4th, mid-4th
| style="text-align:center;" | ^F
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 600
| style="text-align:center;" | 17/12
| style="text-align:center;" | A4, d5
| style="text-align:center;" | aug 4th, dim 5th
| style="text-align:center;" | F#, Gb
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 650
| style="text-align:center;" | 16/11
| style="text-align:center;" | v5, ~5
| style="text-align:center;" | down-5th, mid-5th
| style="text-align:center;" | vG
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 700
| style="text-align:center;" | 3/2
| style="text-align:center;" | P5
| style="text-align:center;" | fifth
| style="text-align:center;" | G
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 750
| style="text-align:center;" | 17/11
| style="text-align:center;" | ^5, vm6
| style="text-align:center;" | up-fifth, downminor 6th
| style="text-align:center;" | ^G, vAb
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 800
| style="text-align:center;" | 19/12
| style="text-align:center;" | m6
| style="text-align:center;" | minor 6th
| style="text-align:center;" | Ab
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 850
| style="text-align:center;" | 18/11, 44/27
| style="text-align:center;" | ~6
| style="text-align:center;" | mid 6th
| style="text-align:center;" | vA
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 900
| style="text-align:center;" | 32/19
| style="text-align:center;" | M6
| style="text-align:center;" | major 6th
| style="text-align:center;" | A
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 950
| style="text-align:center;" | 19/11
| style="text-align:center;" | ^M6, vm7
| style="text-align:center;" | upmajor 6th, downminor 7th
| style="text-align:center;" | ^A, vBb
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 1000
| style="text-align:center;" | 16/9
| style="text-align:center;" | m7
| style="text-align:center;" | minor 7th
| style="text-align:center;" | Bb
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 1050
| style="text-align:center;" | 11/6
| style="text-align:center;" | ~7
| style="text-align:center;" | mid 7th
| style="text-align:center;" | vB
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 1100
| style="text-align:center;" | 17/9, 32/17
| style="text-align:center;" | M7
| style="text-align:center;" | major 7th
| style="text-align:center;" | B
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 1150
| style="text-align:center;" | 33/17, 64/33
| style="text-align:center;" | ^M7, vP8
| style="text-align:center;" | upmajor 7th, down-8ve
| style="text-align:center;" | ^B, vC
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 1200
| style="text-align:center;" | 2/1
| style="text-align:center;" | P8
| style="text-align:center;" | perfect 8ve
| style="text-align:center;" | C
|}
*based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.


{| class="wikitable"
Its step, at 50 cents, is notable for being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant.
|-
! | quality
! | [[Kite's color notation|color]]
! | monzo format
! | examples
|-
| style="text-align:center;" | downminor
| style="text-align:center;" | zo
| 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 < -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;" | ilo
| style="text-align:center;" | {a, b, 0, 0, 1}
| style="text-align:center;" | 11/9, 11/6
|-
| style="text-align:center;" | "
| style="text-align:center;" | lu
| style="text-align:center;" | {a, b, 0, 0, -1}
| style="text-align:center;" | 12/11, 18/11
|-
| style="text-align:center;" | major
| style="text-align:center;" | yo
| style="text-align:center;" | {a, b, 1}
| style="text-align:center;" | 5/4, 5/3
|-
| style="text-align:center;" | "
| style="text-align:center;" | fifthward wa
| style="text-align:center;" | {a, b}, b > 1
| style="text-align:center;" | 9/8, 27/16
|-
| style="text-align:center;" | upmajor
| style="text-align:center;" | ru
| style="text-align:center;" | {a, b, 0, -1}
| style="text-align:center;" | 9/7, 12/7
|}


Ups and downs notation can be used to name chords. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).
=== Prime harmonics ===
{{Harmonics in equal|24|prec=2}}


In many other edos, especially larger ones, 5/4 is downmajor and 11/9 is mid. To agree with this, the term mid is generally preferred over down or downmajor.
=== Subsets and supersets ===
24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}.


{| class="wikitable"
== Intervals ==
{| class="wikitable center-all left-3"
|-
|-
!edosteps
! Degree
!notes
! Cents
!written name
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>
!spoken name
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and d2)
!JI chord, if any
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
! [[24edo solfege|Solfege]]
|-
|-
|0-5-14
| 0
|C vEb G
| 0
|Cvm
| 1/1
|C downminor
| P1
|6:7:9
| unison
| D
| unison
| P1
| D
| Do
|-
|-
|0-6-14
| 1
|C Eb G
| 50
|Cm
| 33/32, 34/33
|C minor
| ^P1, vm2
|10:12:15 <u>or</u> 6/(6:5:4)
| up-unison, downminor 2nd
| ^D, vEb
| super unison, uber unison
| S1, U1
| SD, UD
| Da/Ru
|-
|-
|0-7-14
| 2
|C vE G
| 100
|C~
| 16/15, 17/16, 18/17, 19/18
|C mid
| A1, m2
|18:22:27
| aug unison, minor 2nd
| D#, Eb
| aug unison, minor 2nd
| A1, m2
| D#, Eb
| Ro
|-
|-
|0-8-14
| 3
|C E G
| 150
|C
| 13/12, 12/11, 11/10
|C major or C
| ~2
|4:5:6
| mid 2nd
| vE
| neutral 2nd
| N2
| UEb, uE
| Ra
|-
|-
|0-9-14
| 4
|C ^E G
| 200
|C^
| 9/8, 10/9
|C upmajor or C up
| M2
|14:18:21 <u>or</u> 9/(9:7:6)
| major 2nd
| E
| major 2nd
| M2
| E
| Re
|-
|-
|0-8-13
| 5
|C E vG
| 250
|C(v5)
| 15/13, 22/19
|C down-five
| ^M2, vm3
|
| upmajor 2nd, downminor 3rd
| ^E, vF
| supermajor 2nd, subminor 3rd
| SM2, sm3
| SE, sF
| Ri/Mu
|-
|-
|0-7-13
| 6
|C vE vG
| 300
|C~(v5)
| 6/5, 13/11, 19/16
|C mid down-five
| m3
|22:27:32
| minor 3rd
| F
| minor 3rd
| m3
| F
| Mo
|-
|-
|0-4-11-14
| 7
|C D ^F G
| 350
|C^4,9
| 11/9, 16/13, 27/22, 39/32
|C up-four add-nine
| ~3
|8:9:11:12
| mid 3rd
| vF#
| neutral 3rd
| N3
| UF, uF#
| Ma
|-
|-
|0-9-14-19
| 8
|C ^E G ^A
| 400
|C^6
| 5/4, 24/19
|C up-six
| M3
|14:18:21:24
| major 3rd
| F#
| major 3rd
| M3
| F#
| Me
|-
|-
|0-5-14-20
| 9
|C vEb G Bb
| 450
|Cvm,7
| 13/10, 17/13, 22/17
|C downminor add-seven
| ^M3, v4
|
| upmajor 3rd, down-4th
| ^F#, vG
| supermajor 3rd, sub 4th
| SM3, s4
| SF#, sG
| Mi/Fu
|-
|-
|0-5-14-19
| 10
|C vEb G vBb
| 500
|Cvm7
| 4/3
|C downminor-seven
| P4
|12:14:18:21
| fourth
| G
| perfect 4th
| P4
| G
| Fo
|-
|-
|0-7-14-21
| 11
|C vE G vB
| 550
|C~7
| 11/8, 15/11
|C mid-seven
| ^4, ~4
|18:22:27:33
| up-4th, mid-4th
| ^G
| uber 4th/neutral 4th
| U4/N4
| UG
| Fa/Su
|-
|-
|0-7-14-21-28
| 12
|C vE G vB D
| 600
|C~9
| 17/12, 24/17, 45/32, 64/45
|C mid-nine
| A4, d5
|36:44:54:66:81
| aug 4th, dim 5th
| G#, Ab
| aug 4th, dim 5th
| A4, d5
| G#/Ab
| Fe/So
|-
|-
|0-7-14-21-28-35
| 13
|C vE G vB D ^F
| 650
|C~11
| 16/11, 22/15
|C mid-eleven
| v5, ~5
|36:44:54:66:81:99
| down-5th, mid-5th
| vA
| unter 5th/neutral 5th
| u5/N5
| uA
| Fi/Sa
|-
|-
|0-7-14-21-28-35-42
| 14
|C vE G vB D ^F A
| 700
|C~13
| 3/2
|C mid-thirteen
| P5
|36:44:54:66:81:99:121
| fifth
| A
| perfect 5th
| P5
| A
| Se
|-
|-
|0-8-14-28-35-42
| 15
|C E G D ^F vA
| 750
|C9,~11~13no7
| 17/11, 20/13
|C nine mid-eleven mid-thirteen no-seven
| ^5, vm6
|4:5:6:9:11:13
| up-fifth, downminor 6th
|}
| ^A, vBb
For a more complete list of chords, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
| super 5th, subminor 6th
 
| S5, sm6
==William Lynch's notation==
| SA, sBb
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. See the [[24_EDO_Interval_names_and_Harmonies|full article on 24 Edo intervals.]]
| Si/Lu
 
These are the intervals of 24 EDO that do not exist in 12 EDO:
 
{| class="wikitable"
|-
|-
! colspan="4" | The twelve new intervals in 24edo
| 16
! colspan="3" | some nearby JI intervals
| 800
| 8/5, 19/12
| m6
| minor 6th
| Bb
| minor 6th
| m6
| Bb
| Lo
|-
|-
! | cents
| 17
!pions
| 850
!7mus
| 13/8, 18/11, 44/27, 64/39
! | common names
| ~6
! | frequency ratio
| mid 6th
! | cents
| vB
! | common name
| neutral 6th
| N6
| UBb, uB
| La
|-
|-
| style="text-align:center;" | 50
| 18
|53
| 900
|64 (40<sub>16</sub>)
| 5/3, 22/13, 32/19
| style="text-align:center;" | quartertone
| M6
 
| major 6th
infra second, wide unison
| B
| style="text-align:center;" | [[36/35|36/35]]
| major 6th
 
| M6
[[35/34|35/34]]
| B
 
| Le
[[34/33|34/33]]
 
[[33/32|33/32]]
| style="text-align:center;" | 48.77
 
50.184
 
51.682
 
53.273
| style="text-align:center;" | large septimal quarter-tone (Archytas)
 
large 17-limit quartertone
 
small 17-limit quartertone
 
33rd harmonic
|-
|-
| style="text-align:center;" | 150
| 19
|159
| 950
|192 (C0<sub>16</sub>)
| 19/11, 26/15
| style="text-align:center;" | neutral second
| ^M6, vm7
| style="text-align:center;" | [[12/11|12/11]]
| upmajor 6th, downminor 7th
| style="text-align:center;" | 150.637
| ^B, vC
| style="text-align:center;" | large undecimal neutral second
| supermajor 6th, subminor 7th
| SM6, sm7
| SB, sC
| Li/Tu
|-
|-
| style="text-align:center;" | 250
| 20
|265
| 1000
|320 (140<sub>16</sub>)
| 9/5, 16/9
| style="text-align:center;" | ultra second
| m7
 
| minor 7th
<span style="line-height: 1.5;">infra third</span>
| C
| style="text-align:center;" | [[144/125|144/125]]
| minor 7th
 
| m7
[[15/13|15/13]]
| C
 
| To
[[52/45|52/45]]
| style="text-align:center;" | 244.969
 
247.741
 
250.304
| style="text-align:center;" | diminished third (6/5 x 24/25)
 
..
 
..
|-
|-
| style="text-align:center;" | 350
| 21
|383
| 1050
|448 (1C0<sub>16</sub>)
| 11/6, 20/11
| style="text-align:center;" | neutral third
| ~7
| style="text-align:center;" | [[11/9|11/9]]
| mid 7th
 
| vC#
[[27/22|27/22]]
| neutral 7th
 
| N7
[[16/13|16/13]]
| UC, uC#
| style="text-align:center;" | 347.408
| Ta
 
354.547
 
359.472
| style="text-align:center;" | undecimal neutral third
 
..
 
tridecimal neutral third
|-
|-
| style="text-align:center;" | 450
| 22
|477
| 1100
|576 (240<sub>16</sub>)
| 15/8, 17/9, 32/17
| style="text-align:center;" | minor fourth, ultra third, narrow fourth
| M7
| style="text-align:center;" | [[22/17|22/17]]
| major 7th
 
| C#
[[35/27|35/27]]
| major 7th
 
| M7
[[13/10|13/10]]
| C#
| style="text-align:center;" | 446.363
| Te
 
449.275
 
454.214
| style="text-align:center;" | 17-limit supermajor third
 
..
 
tridecimal subfourth
|-
|-
| style="text-align:center;" | 550
| 23
|583
| 1150
|704 (2C0<sub>16</sub>)
| 33/17, 64/33
| style="text-align:center;" | wide fourth
| ^M7, vP8
| style="text-align:center;" | [[11/8|11/8]]
| upmajor 7th, down-8ve
| style="text-align:center;" | 551.318
| ^C#, vD
| style="text-align:center;" | undecimal superfourth, harmonic 11th
| sub 8ve, unter 8ve
| s8, u8
| C#, uD
| Ti/Du
|-
|-
| style="text-align:center;" | 650
| 24
|689
| 1200
|832 (340<sub>16</sub>)
| 2/1
| style="text-align:center;" | narrow fifth
| P8
| style="text-align:center;" | [[16/11|16/11]]
| perfect 8ve
| style="text-align:center;" | 648.682
| D
| style="text-align:center;" | undecimal subfifth, 11th subharmonic
| perfect 8ve
|-
| P8
| style="text-align:center;" | 750
| D
|795
| Do
|960 (3C0<sub>16</sub>)
|}
| style="text-align:center;" | wide fifth, infra sixth
| style="text-align:center;" | [[20/13|20/13]]
 
[[54/35|54/35]]
 
[[17/11|17/11]]
| style="text-align:center;" | 745.786


750.725
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.


753.637
== Notation ==
| style="text-align:center;" | tridecimal superfifth
=== Ups and downs notation ===
Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{sharpness-sharp2a|24}}


..
=== Stein–Zimmermann accidentals ===
{{Sharpness-sharp2|24}}


17-limit subminor sixth
{| class="wikitable center-1"
|-
|-
| style="text-align:center;" | 850
| style="width: 40px;" | [[File:HeQu1.svg|21px|center]]
|901
| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.
|1088 (340<sub>16</sub>)
| style="text-align:center;" | neutral sixth
| 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
| style="width: 40px;" | [[File:HeQu3.svg|32px|center]]
|1007
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
|1216 (4C0<sub>16</sub>)
| style="text-align:center;" | ultra sixth , infra seventh
| 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
| style="width: 40px;" | [[File:HeQd1.svg|22px|center]]
|1113
| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
|1344 (540<sub>16</sub>)
| style="text-align:center;" | neutral seventh
| 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
| style="width: 40px;" | [[File:HeQd3.svg|36px|center]]
|1219
| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.
|1472 (5C0<sub>16</sub>)
| style="text-align:center;" | ultra seventh, narrow octave
| 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
 
..
 
..
|}
|}


===Interval Alterations===
'''Pros:''' familiar, intuitive, and fairly easy to learn.  
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 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.
 
They are:
 
Super + perfect interval such as "perfect fifth" means to raise it by 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
 
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:
 
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
 
=Quartertone Accidentals=
Besides ups and downs, there are various systems for notating quarter tones. Here are some of them, along with their 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">
'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.  
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
=== Persian quartertone accidentals ===
 
{{Wikipedia|Koron (music)|Sori (music)}}
<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
 
==Persian Accidentals==


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


Pros: Easy to read
'''Pros:''' easy to read.
 
Cons: Hard to write on a computer, doesn't fit with standard notation well


==Sagittal Notation==
'''Cons:''' hard to write on a computer, does not fit with standard notation well.
[[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.
=== 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]].


A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them
==== 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>


with Sagittal signs for sharp and flat.
==== Revo flavor ====
<imagemap>
File:24-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:24-EDO_Revo_Sagittal.svg]]
</imagemap>


[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
==== Evo-SZ flavor ====
<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>


Pros: Easy to read, and less likely to clutter the score
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein.E2.80.93Zimmermann_accidentals|Stein–Zimmerman notation]].


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


=Chord Types=
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
24edo features a rich variety of 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, 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.
 
More good chords in 24-tET:
 
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")


0-5-10 (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.)
'''Pros:''' easy to read, and less likely to clutter the score.  


William Lynch considers these as some possible good tetrads:
'''Cons:''' not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality.


[[File:Three_chords.PNG|alt=Three chords.PNG|Three chords.PNG]]


{| class="wikitable"
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:
|-
! | Chord name
! | Degrees of 24edo
! | Chord spelling
! | Audio example
|-
| | neutral
| | 0 7 14 21
| | 1 v3 5 v7
| | [[File:Neutral_Tetrad_on_C.mp3]]
|-
| | arto
| | 0 5 14 20
| | 1 vb3 5 b7
| | [[File:arto_tetrad_on_C.mp3]]
|-
| | tendo
| | 0 9 14 19
| | 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.
[[File:24edo Sagittal.png|800px]]


=Commas=
== Interval and chord naming ==
24 EDO tempers out the following [[commas]]. (Note: This assumes val &lt; 24 38 56 67 83 89 |.)
==== Combining ups and downs with color notation ====
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable" style="text-align:center;"
{| class="wikitable center-all"
|-
|-
! | [[Ratio]]
! Quality
! | [[Monzo]]
! [[Color name]]
! | [[Cents]]
! Monzo format
![[Color notation/Temperament Names|Color Name]]
! Examples
! | Name 1
! | Name 2
! | Name 3
|-
|-
| style="text-align:center;" | 531441/524288
| downminor
| style="text-align:center;" |<nowiki> | -19 12 </nowiki>&gt;
| zo
| style="text-align:center;" | 23.46
| {{nowrap|(a, b, 0, 1)}}
|Lalawa
| 7/6, 7/4
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 648/625
| rowspan="2" | minor
| style="text-align:center;" |<nowiki> | 3 4 -4 </nowiki>&gt;
| fourthward wa
| style="text-align:center;" | 62.57
| {{nowrap|(a, b)}}; {{nowrap|b < −1}}
|Quadgu
| 32/27, 16/9
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 128/125
| gu
| style="text-align:center;" |<nowiki> | 7 0 -3 </nowiki>&gt;
| {{nowrap|(a, b, −1)}}
| style="text-align:center;" | 41.06
| 6/5, 9/5
|Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 81/80
| rowspan="2" | mid
| style="text-align:center;" |<nowiki> | -4 4 -1 </nowiki>&gt;
| ilo
| style="text-align:center;" | 21.51
| {{nowrap|(a, b, 0, 0, 1)}}
|Gu
| 11/9, 11/6
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
|-
| style="text-align:center;" | 2048/2025
| lu
| style="text-align:center;" |<nowiki> | 11 -4 -2 </nowiki>&gt;
| {{nowrap|(a, b, 0, 0, −1)}}
| style="text-align:center;" | 19.55
| 12/11, 18/11
|Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 5201701/5149091
| rowspan="2" | major
| style="text-align:center;" |<nowiki> | 26 -12 -3 </nowiki>&gt;
| yo
| style="text-align:center;" | 17.60
| {{nowrap|(a, b, 1)}}
|Sasa-trigu
| 5/4, 5/3
| style="text-align:center;" | Misty Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 32805/32768
| fifthward wa
| style="text-align:center;" |<nowiki> | -15 8 1 </nowiki>&gt;
| {{nowrap|(a, b)}}; {{nowrap|b > 1}}
| style="text-align:center;" | 1.95
| 9/8, 27/16
|Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 1465155/1465142
| upmajor
| style="text-align:center;" |<nowiki> | 161 -84 -12 </nowiki>&gt;
| ru
| style="text-align:center;" | 0.02
| {{nowrap|(a, b, 0, −1)}}
|Sepbisa-quadbigu
| 9/7, 12/7
| 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
|Zozo
| 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
|Zozoyo
| 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
|Labirugu
| 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
|Sarurutrigu
| 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
|Lologu
| 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
|Lorugugu
| 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
|Saluzo
| 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
|Lulu
| 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
|Lozoyo
| 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
|Bilorugu
| 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
|Thozogu
| 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
|Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| style="text-align:center;" |
|}
|}


=Rank two temperaments=
Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].
[[List_of_24et_rank_two_temperaments_by_badness|List of 24et rank two temperaments by badness]]
 
=== 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.


[[List_of_edo-distinct_24et_rank_two_temperaments|List of edo-distinct 24et rank two temperaments]]
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.


{| class="wikitable"
These are the intervals of 24edo that do not exist in 12edo:
 
{| class="wikitable center-all right-6"
|-
|-
! | Periods per octave
! [[Cent]]s
! | Generator
! Names
! | Name
|-
|-
| | 1
| 50
| | 1\24
| Quarter tone, infra second, wide unison
| |
|-
|-
| | 1
| 150
| | 5\24
| Neutral second
| | [[Semaphore_and_Godzilla|Semaphore/godzilla]] / [[Chromatic_pairs#Bridgetown|Bridgetown]]
|-
|-
| | 1
| 250
| | 7\24
| Ultra second, infra third
| | [[Mohajira]] (or [[maqamic]] with 24d val)
|-
|-
| | 1
| 350
| | 11\24
| Neutral third
| | [[Chromatic_pairs#Barton|Barton]]
|-
|-
| | 2
| 450
| | 1\24
| Minor fourth, ultra third, narrow fourth
| | [[Shrutar]]
|-
|-
| | 2
| 550
| | 5\24
| Wide fourth
| | [[Sruti]], [[Anguirus]], [[Decimal]]
|-
|-
| | 3
| 650
| | 1\24
| Narrow fifth
| | [[Semiaug]]
|-
|-
| | 3
| 750
| | 3\24
| Wide fifth, infra sixth
| | [[Triforce]]
|-
|-
| | 4
| 850
| | 1\24
| Neutral sixth
| |
|-
|-
| | 6
| 950
| | 1\24
| Ultra sixth, infra seventh
| |
|-
|-
| | 8
| 1050
| | 1\24
| Neutral seventh
| |
|-
|-
| | 12
| 1150
| | 1\24
| Ultra seventh, narrow octave
| | [[Catler]]
|}
|}


=Scales / Modes=
==== 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.
 
==== 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.
 
They are:
 
* Super + perfect interval such as "perfect fifth" means to raise it by 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
* 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:
 
* 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
 
=== Further discussion of interval and chord naming ===
{{main|{{PAGENAME}}/Interval names and harmonies }}
 
* [[24edo Chord Names]]
* [[Ups and downs notation#Chords and Chord Progressions]].
 
== 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]]
 
=== Interval mappings ===
{{Q-odd-limit intervals|24}}


{| class="wikitable"
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| colspan="2" |  
! [[TE error|Absolute]] (¢)
====Pentatonic:====
! [[TE simple badness|Relative]] (%)
|-
|-
| | <tt>'''2 8 3 6 5'''</tt>
| 2.3.5.11
| | Anchihoye: Ethiopia
| 81/80, 121/120, 128/125
| {{mapping| 24 38 56 83 }}
| −1.08
| 2.82
| 5.63
|-
|-
| | <tt>'''5 5 4 5 5'''</tt>
| 2.3.5.11.13
| | Quasi-equal Pentatonic - [[MOSScales|MOS]] of type [[4L_1s|4L 1s (bug)]]
| 66/65, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 }}
| −1.37
| 2.59
| 5.19
|-
|-
| | <tt>'''5 5 5 5 4'''</tt>
| 2.3.5.11.13.17
| | Hába's Pentatonic - [[MOSScales|MOS]] of type [[4L_1s|4L 1s (bug)]]
| 51/50, 66/65, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 98 }}
| −0.94
| 2.55
| 5.11
|-
|-
| | <tt>'''6 3 6 6 3'''</tt>
| 2.3.5.11.13.17.19
| | [[MOS]] of type [[3L 2s]] (father)
| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 98 102 }}
| −0.89
| 2.37
| 4.74
|}
|}


{| class="wikitable"
=== Uniform maps ===
|-
{{Uniform map|edo=24}}
| colspan="2" |
 
====Hexatonic:====
=== 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 }}.
| | <tt>'''1 1 8 4 2 8'''</tt>
| | Spondeiakos
|-
| | <tt>'''5 5 2 5 5 2'''</tt>
| | [[MOS]] of type [[4L 2s]]
|-
| | <tt>'''3 5 3 5 3 5'''</tt>
| | [[MOS]] of type [[3L 3s]] (triforce)
|}


{| 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
| Schismina
|-
|-
| | <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
|}
 
=== 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 4L1s 55455 (generator: 5\24)
* Semaphore 5L4s 414141414 (generator: 5\24)
* Mohajira 3L4s 3434343 (generator: 7\24)
* Mohajira 7L3s 3313313313 (generator: 7\24)
 
== 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}}
| | <tt>'''4 3 3 4 4 2 1 3'''</tt>
| C^<br>C({{demisharp2}}3), Cmaj({{demisharp2}}3)
| | Maqam Rast, Dilkashidah, Dilnishin
| C ultramajor<br>C major semisharp-three
|-
| | <tt>'''3 4 2 6 2 4 2 1'''</tt>
| | Maqam Rahat al-Arwah
|-
| | <tt>'''3 4 3 3 4 4 2 1'''</tt>
| | Maqam Iraq
|-
| | <tt>'''2 6 2 4 2 1 3 4'''</tt>
| | Maqam Hijaz
|-
| | <tt>'''3 4 4 2 1 3 4 3'''</tt>
| | Maqam Musta'ar
|-
| | <tt>'''3 4 4 4 2 4 2 1'''</tt>
| | Maqam Farahnak
|-
| | <tt>'''3 4 3 3 2 6 2 1'''</tt>
| | Maqam Bastanikar, Tarz Nuin
|-
| | <tt>'''4 2 6 2 4 2 1 3'''</tt>
| | Maqam Farah Faza, Maqam Nakriz
|-
| | <tt>'''3 1 2 4 4 2 4 4'''</tt>
| | Maqam Jabburi
|-
| | <tt>'''1 4 4 2 4 4 4 1'''</tt>
| | Giancarlo Dalmonte's new quarter-tone scale (see [https://www.amazon.co.uk/Lottava-nota-Giancarlo-Dalmonte/dp/8865981385 L'ottava nota])
|}
|}


{| class="wikitable"
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.
|-
 
| colspan="2" |
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]]


====Enneatonic:====
{| class="wikitable center-all"
|-
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental tetrads of 24edo
| | <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)]] (godzilla/semaphore)
|-
| | <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
|-
|-
| | <tt>'''3 3 2 3 3 2 3 3 2'''</tt>
! Degrees of 24edo
| | [[MOS]] of type [[6L 3s]] (triforce)
! Chord spelling
|}
! Notes of C chord
 
! Written name
{| class="wikitable"
! Spoken name
! Audio example
|-
|-
| colspan="2" |  
| {{dash|0, 5, 14, 19|hair}}
====Decatonic:====
| {{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 1 3 3 3 1 3 3 1 3'''</tt>
| {{dash|0, 6, 14, 20|hair}}
| | Breed Decatonic - [[MOSScales|MOS]] of type [[7L_3s|7L 3s (unfair mosh)]]
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E♭, G, B♭|hair}}
| m7, min7
| Minor seven
|-
|-
| | <tt>'''2 3 2 2 3 2 3 2 2 3'''</tt>
| {{dash|0, 7, 14, 21|hair}}
| | Oljare Decatonic - [[MOSScales|MOS]] of type [[4L_6s|4L 6s (fair bicycle)]]
| {{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>'''2 1 3 2 2 4 2 4 2 2'''</tt>
| {{dash|0, 8, 14, 22|hair}}
| | Maqam Shawq Tarab
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E, G, B|hair}}
| maj7
| Major seven
|-
|-
| | <tt>'''4 2 1 3 2 2 4 2 1 3'''</tt>
| {{dash|0, 8, 14, 22|hair}}
| | Maqam Basandida
| {{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>'''4 3 1 2 1 3 4 2 1 3'''</tt>
| {{dash|0, 8, 14, 20|hair}}
| | Maqam Yakah
| {{dash|1, 3, 5, b7|hair}}
|}
| {{dash|C, E, G, B♭|hair}}
 
| 7, dom7
{| class="wikitable"
| Dominant seven
|-
|-
| colspan="2" |  
| {{dash|0, 8, 14, 19|hair}}
====Hendecatonic:====
| {{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>'''4 2 1 3 2 2 2 2 3 1 2'''</tt>
| {{dash|0, 5, 14, 20|hair}}
| | Maqam Hayyan
| {{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>'''3 2 2 2 2 3 2 2 2 2 2'''</tt>
| {{dash|0, 9, 14, 19|hair}}
| | [[MOS]] of type [[2L 9s]]
| {{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" |


====Tridecatonic:====
== 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.
| | <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"
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" |
====Tetradecatonic:====
|-
| | '''<tt>2 2 1 2 2 2 1 2 2 1 2 2 2 1<tt>'''
| | Young Half-Octave Diatonic - [[MOSScales|MOS]] of type [[10L_4s|10L 4s]]
|-
| | <tt>'''1 1 3 1 1 3 1 1 3 1 1 3 1 3'''</tt>
| | [[MOS]] of type [[5L 9s]] (godzilla/semaphore)
|}


{| 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" |
====Pentadecatonic:====
|-
| | <tt>'''1 2 1 2 2 1 2 1 2 2 1 2 1 2 2'''</tt>
| | [[MOS]] of type [[9L 6s]] (triforce)
|}


{| 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" |
====Heptadecatonic:====
|-
| | <tt>'''1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2'''</tt>
| | [[MOS]] of type [[7L 10s]] (mohajira)
|}


=Instruments=
Guitars with 24 frets per octave are also an option and some guitar makers, such as Ron Sword's [http://metatonalmusic.com Metatonal Music], can make custom instruments and perform re-fretting, with an example 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.


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
[[File:24edo_guitar.jpg|500px]]
External image: http://www.swordguitars.com/Sword_quartertone_stratsm.jpg<br>
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24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars
While these are playable, the extra frets can make playing chords and navigating the fretboard significantly more challenging for [[12edo]] chords and scales.


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]].
More common is the "Sazocaster" tuning popularised by Australian band King Gizzard and the Lizard Wizard, which adds quarter tones between approximately half the regular frets. Multiple guitar makers, including Eastwood and Revelation, have produced Sazocaster variations.


=Music=
[[File:Eastwood-guitars-phase-4-mt-2307179.jpg|500px]]
<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]]
=== Harp, Harpsichord, and Piano ===


[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]
==== Scordatura tuning of 12edo instruments ====


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


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


[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 Neutral[7] (2.3.11 mohajira), 24et tuning
==== Quarter-tone instruments ====


[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.
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):


[https://soundcloud.com/mason-l-green/serena Serena], by Mason Green (intro and coda in 24edo, the rest is in 12edo)
; 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://www.youtube.com/watch?v=yzvXEMYgHCY Autumn Girl], by Mason Green
; 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)


"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.)
=== Electronic Keyboards ===


"3 Hommages" by [http://en.wikipedia.org/wiki/Georg_Friedrich_Haas Georg Friedrich Haas]
24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see [[Lumatone mapping for 24edo]]


[http://en.wikipedia.org/wiki/List_of_quarter_tone_pieces List of quartertone pieces on Wikipedia]
=== Flute ===
Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:


=Practical Theory / Books=
; 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)


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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
{{Todo| cleanup }}


[http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor_Darreg|Ivor Darreg]]   
== Notes ==
<references group="note" />


[[Category:24edo| ]] <!-- main article -->
[[Category:Semaphore]]
[[Category:Edo]]
[[Category:Godzilla]]
[[Category:Intervals]]
[[Category:Meantone]]
[[Category:Listen]]
[[Category:Quartertone]]
[[Category:Quartertone]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Subgroup]]
[[Category:Subgroup temperaments]]
[[Category:Table]]
[[Category:Theory]]
[[Category:Twentuning]]
[[Category:Twentuning]]
 
[[Category:Lists of intervals]]
[[Category:Todo:unify precision]]
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