Xenharmonic Wiki

24edo: Difference between revisions

← Older edit
VisualWikitext
Jkarimak (talk | contribs)
191 edits
m Made 7:4, 7:5 and 7:6 into links
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Text replacement - "Ups and Downs Notation" to "Ups and downs notation"
 
(285 intermediate revisions by 31 users not shown)
Line 1: Line 1:
__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}}


=Basics=
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 - [[DIY_Quartertone_Composition_with_12_equal_tools|see this page]].


=24edo as a temperament=
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]], [[7:5]] 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 [[36edo|36-tET]], [[72edo|72-tET]], [[84edo|84-tET]] or [[156edo|156-tET]].


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.


{| class="wikitable"
The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth. 24edo is also good at the 13th harmonic, which makes it a good 2.3.5.11.13 system. Specifically, intervals of 13/5 are particularly well approximated. And of course, 24edo shares its 17 and 19 tunings with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.
 
While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically: a supermajor chord is available at [0 9 14] and a subminor chord at [0 5 14], though they're more ultramajor and inframinor. 
 
The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.
 
Its step, at 50 cents, is notable for being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant.
 
=== Prime harmonics ===
{{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 }}.
 
== Intervals ==
{| class="wikitable center-all left-3"
|-
|-
| style="text-align:center;" | Degree
! Degree
| style="text-align:center;" | Cents
! Cents
|pions
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>
|7mus
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and d2)
| style="text-align:center;" | Approximate Ratios*
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
! [[24edo solfege|Solfege]]
|-
|-
| colspan="4" style="text-align:center;" | 0
| 0
| style="text-align:center;" | 1/1
| 0
| style="text-align:center;" | P1
| 1/1
| style="text-align:center;" | unison
| P1
| style="text-align:center;" | C
| unison
| D
| unison
| P1
| D
| Do
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 50
| 50
|53
| 33/32, 34/33
|64 (40<sub>16</sub>)
| ^P1, vm2
| style="text-align:center;" | 33/32, 34/33
| up-unison, downminor 2nd
| style="text-align:center;" | ^P1, vm2
| ^D, vEb
| style="text-align:center;" | up-unison, downminor 2nd
| super unison, uber unison
| style="text-align:center;" | C^, Dbv
| S1, U1
| SD, UD
| Da/Ru
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 100
| 100
|106
| 16/15, 17/16, 18/17, 19/18
|128 (80<sub>16</sub>)
| A1, m2
| style="text-align:center;" | 17/16, 18/17
| aug unison, minor 2nd
| style="text-align:center;" | A1, m2
| D#, Eb
| style="text-align:center;" | aug unison, minor 2nd
| aug unison, minor 2nd
| style="text-align:center;" | C#, Db
| A1, m2
| D#, Eb
| Ro
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 150
| 150
|159
| 13/12, 12/11, 11/10
|192 (C0<sub>16</sub>)
| ~2
| style="text-align:center;" | 12/11
| mid 2nd
| style="text-align:center;" | ~2
| vE
| style="text-align:center;" | mid 2nd
| neutral 2nd
| style="text-align:center;" | Dv
| N2
| UEb, uE
| Ra
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 200
| 200
|212
| 9/8, 10/9
|256 (100<sub>16</sub>)
| M2
| style="text-align:center;" | 9/8
| major 2nd
| style="text-align:center;" | M2
| E
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | D
| M2
| E
| Re
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 250
| 250
|265
| 15/13, 22/19
|320 (140<sub>16</sub>)
| ^M2, vm3
| style="text-align:center;" | 22/19
| upmajor 2nd, downminor 3rd
| style="text-align:center;" | ^M2, vm3
| ^E, vF
| style="text-align:center;" | upmajor 2nd, downminor 3rd
| supermajor 2nd, subminor 3rd
| style="text-align:center;" | D^, Ebv
| SM2, sm3
| SE, sF
| Ri/Mu
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 300
| 300
|318
| 6/5, 13/11, 19/16
|384 (180<sub>16</sub>)
| m3
| style="text-align:center;" | 19/16
| minor 3rd
| style="text-align:center;" | m3
| F
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | Eb
| m3
| F
| Mo
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 350
| 350
|383
| 11/9, 16/13, 27/22, 39/32
|448 (1C0<sub>16</sub>)
| ~3
| style="text-align:center;" | 11/9
| mid 3rd
| style="text-align:center;" | ~3
| vF#
| style="text-align:center;" | mid 3rd
| neutral 3rd
| style="text-align:center;" | Ev
| N3
| UF, uF#
| Ma
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 400
| 400
|424
| 5/4, 24/19
|512 (200<sub>16</sub>)
| M3
| style="text-align:center;" | 24/19
| major 3rd
| style="text-align:center;" | M3
| F#
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | E
| M3
| F#
| Me
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 450
| 450
|477
| 13/10, 17/13, 22/17
|576 (240<sub>16</sub>)
| ^M3, v4
| style="text-align:center;" | 22/17
| upmajor 3rd, down-4th
| style="text-align:center;" | ^M3, vP4
| ^F#, vG
| style="text-align:center;" | upmajor 3rd, down-fourth
| supermajor 3rd, sub 4th
| style="text-align:center;" | E^, Fv
| SM3, s4
| SF#, sG
| Mi/Fu
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 500
| 500
|530
| 4/3
|640 (280<sub>16</sub>)
| P4
| style="text-align:center;" | 4/3
| fourth
| style="text-align:center;" | P4
| G
| style="text-align:center;" | fourth
| perfect 4th
| style="text-align:center;" | F
| P4
| G
| Fo
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 550
| 550
|583
| 11/8, 15/11
|704 (2C0<sub>16</sub>)
| ^4, ~4
| style="text-align:center;" | 11/8
| up-4th, mid-4th
| style="text-align:center;" | ^P4
| ^G
| style="text-align:center;" | up-fourth
| uber 4th/neutral 4th
| style="text-align:center;" | F^
| U4/N4
| UG
| Fa/Su
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 600
| 600
|636
| 17/12, 24/17, 45/32, 64/45
|768 (300<sub>16</sub>)
| A4, d5
| style="text-align:center;" | 17/12
| aug 4th, dim 5th
| style="text-align:center;" | A4, d5
| G#, Ab
| style="text-align:center;" | aug 4th, dim 5th
| aug 4th, dim 5th
| style="text-align:center;" | F#, Gb
| A4, d5
| G#/Ab
| Fe/So
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 650
| 650
|689
| 16/11, 22/15
|832 (340<sub>16</sub>)
| v5, ~5
| style="text-align:center;" | 16/11
| down-5th, mid-5th
| style="text-align:center;" | vP5
| vA
| style="text-align:center;" | down-fifth
| unter 5th/neutral 5th
| style="text-align:center;" | Gv
| u5/N5
| uA
| Fi/Sa
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 700
| 700
|742
| 3/2
|896 (380<sub>16</sub>)
| P5
| style="text-align:center;" | 3/2
| fifth
| style="text-align:center;" | P5
| A
| style="text-align:center;" | fifth
| perfect 5th
| style="text-align:center;" | G
| P5
| A
| Se
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 750
| 750
|795
| 17/11, 20/13
|960 (3C0<sub>16</sub>)
| ^5, vm6
| style="text-align:center;" | 17/11
| up-fifth, downminor 6th
| style="text-align:center;" | ^P5, vm6
| ^A, vBb
| style="text-align:center;" | up-fifth, downminor 6th
| super 5th, subminor 6th
| style="text-align:center;" | G^, Abv
| S5, sm6
| SA, sBb
| Si/Lu
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 800
| 800
|848
| 8/5, 19/12
|1024 (400<sub>16</sub>)
| m6
| style="text-align:center;" | 19/12
| minor 6th
| style="text-align:center;" | m6
| Bb
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | Ab
| m6
| Bb
| Lo
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 850
| 850
|901
| 13/8, 18/11, 44/27, 64/39
|1088 (340<sub>16</sub>)
| ~6
| style="text-align:center;" | 18/11
| mid 6th
| style="text-align:center;" | ~6
| vB
| style="text-align:center;" | mid 6th
| neutral 6th
| style="text-align:center;" | Av
| N6
| UBb, uB
| La
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 900
| 900
|954
| 5/3, 22/13, 32/19
|1152 (480<sub>16</sub>)
| M6
| style="text-align:center;" | 32/19
| major 6th
| style="text-align:center;" | M6
| B
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | A
| M6
| B
| Le
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 950
| 950
|1007
| 19/11, 26/15
|1216 (4C0<sub>16</sub>)
| ^M6, vm7
| style="text-align:center;" | 19/11
| upmajor 6th, downminor 7th
| style="text-align:center;" | ^M6, vm7
| ^B, vC
| style="text-align:center;" | upmajor 6th, downminor 7th
| supermajor 6th, subminor 7th
| style="text-align:center;" | A^, Bbv
| SM6, sm7
| SB, sC
| Li/Tu
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | 1000
| 1000
|1060
| 9/5, 16/9
|1280 (500<sub>16</sub>)
| m7
| style="text-align:center;" | 16/9
| minor 7th
| style="text-align:center;" | m7
| C
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | Bb
| m7
| C
| To
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 1050
| 1050
|1113
| 11/6, 20/11
|1344 (540<sub>16</sub>)
| ~7
| style="text-align:center;" | 11/6
| mid 7th
| style="text-align:center;" | ~7
| vC#
| style="text-align:center;" | mid 7th
| neutral 7th
| style="text-align:center;" | Bv
| N7
| UC, uC#
| Ta
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:center;" | 1100
| 1100
|1166
| 15/8, 17/9, 32/17
|1408 (580<sub>16</sub>)
| M7
| style="text-align:center;" | 17/9, 32/17
| major 7th
| style="text-align:center;" | M7
| C#
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | B
| M7
| C#
| Te
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | 1150
| 1150
|1219
| 33/17, 64/33
|1472 (%C0<sub>16</sub>)
| ^M7, vP8
| style="text-align:center;" | 33/17, 64/33
| upmajor 7th, down-8ve
| style="text-align:center;" | ^M7, vP8
| ^C#, vD
| style="text-align:center;" | upmajor 7th, down-8ve
| sub 8ve, unter 8ve
| style="text-align:center;" | B^, Cv
| s8, u8
| C#, uD
| Ti/Du
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | 1200
| 1200
|1272
| 2/1
|1536 (600<sub>16</sub>)
| P8
| style="text-align:center;" | 2/1
| perfect 8ve
| style="text-align:center;" | P8
| D
| style="text-align:center;" | perfect 8ve
| perfect 8ve
| style="text-align:center;" | C
| P8
| D
| Do
|}
|}
*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:
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]]
{{sharpness-sharp2a|24}}
! | 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 &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;" | 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 &gt; 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
|}
 
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):
 
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.)


===Commas===
=== Stein–Zimmermann accidentals ===
24 EDO tempers out the following commas. (Note: This assumes val &lt; 24 38 56 67 83 89 |.)
{{Sharpness-sharp2|24}}


{| class="wikitable"
{| class="wikitable center-1"
|-
|-
! | Comma
| style="width: 40px;" | [[File:HeQu1.svg|21px|center]]
! | Monzo
| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.
! | Value (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
|-
| style="text-align:center;" | 531441/524288
| style="width: 40px;" | [[File:HeQu3.svg|32px|center]]
| style="text-align:center;" |<nowiki> | -19 12 </nowiki>&gt;
| 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;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 648/625
| style="width: 40px;" | [[File:HeQd1.svg|22px|center]]
| style="text-align:center;" |<nowiki> | 3 4 -4 </nowiki>&gt;
| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
| 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="width: 40px;" | [[File:HeQd3.svg|36px|center]]
| style="text-align:center;" |<nowiki> | 7 0 -3 </nowiki>&gt;
| 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;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 81/80
| style="text-align:center;" |<nowiki> | -4 4 -1 </nowiki>&gt;
| 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:''' familiar, intuitive, and fairly easy to learn.  
 
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.


These are the intervals of 24 EDO that do not exist in 12 EDO: 2
'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.


[[24_EDO_Interval_names_and_Harmonies|See full article on 24 Edo intervals.]]
=== Persian quartertone accidentals ===
{{Wikipedia|Koron (music)|Sori (music)}}


{| class="wikitable"
{| class="wikitable"
|-
|-
! colspan="4" | The twelve new intervals in 24edo
| width="40px" | [[File:Koron_sign.svg|39px|center]]
! colspan="3" | some nearby JI intervals
| '''Koron''' = quarter-tone flat
|-
|-
! | cents
| width="40px" | [[File:Sori_sign.svg|39px|center]]
!pions
| '''Sori''' = quarter-tone sharp
!7mus
|}
! | common names
! | frequency ratio
! | cents
! | common name
|-
| style="text-align:center;" | 50
|53
|64 (40<sub>16</sub>)
| style="text-align:center;" | quartertone


infra second, wide unison
'''Pros:''' easy to read.
| style="text-align:center;" | [[36/35|36/35]]


[[35/34|35/34]]
'''Cons:''' hard to write on a computer, does not fit with standard notation well.


[[34/33|34/33]]
=== 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]].


[[33/32|33/32]]
==== Evo flavor ====
| style="text-align:center;" | 48.77
<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>


50.184
==== 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>


51.682
==== 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>


53.273
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein.E2.80.93Zimmermann_accidentals|Stein–Zimmerman notation]].
| style="text-align:center;" | large septimal quarter-tone (Archytas)


large 17-limit quartertone
==== 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.


small 17-limit quartertone
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
 
33rd harmonic
|-
| style="text-align:center;" | 150
|159
|192 (C0<sub>16</sub>)
| style="text-align:center;" | neutral second
| style="text-align:center;" | [[12/11|12/11]]
| style="text-align:center;" | 150.637
| style="text-align:center;" | large undecimal neutral second
|-
| style="text-align:center;" | 250
|265
|320 (140<sub>16</sub>)
| style="text-align:center;" | ultra second


<span style="line-height: 1.5;">infra third</span>
'''Pros:''' easy to read, and less likely to clutter the score.  
| style="text-align:center;" | [[144/125|144/125]]


[[15/13|15/13]]
'''Cons:''' not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality.


[[52/45|52/45]]
| style="text-align:center;" | 244.969


247.741
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:


250.304
[[File:24edo Sagittal.png|800px]]
| style="text-align:center;" | diminished third (6/5 x 24/25)


..
== 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:


..
{| class="wikitable center-all"
|-
|-
| style="text-align:center;" | 350
! Quality
|383
! [[Color name]]
|448 (1C0<sub>16</sub>)
! Monzo format
| style="text-align:center;" | neutral third
! Examples
| 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
| downminor
|477
| zo
|576 (240<sub>16</sub>)
| {{nowrap|(a, b, 0, 1)}}
| style="text-align:center;" | minor fourth, ultra third, narrow fourth
| 7/6, 7/4
| 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
| rowspan="2" | minor
|583
| fourthward wa
|704 (2C0<sub>16</sub>)
| {{nowrap|(a, b)}}; {{nowrap|b < −1}}
| style="text-align:center;" | wide fourth
| 32/27, 16/9
| 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
| gu
|689
| {{nowrap|(a, b, −1)}}
|832 (340<sub>16</sub>)
| 6/5, 9/5
| style="text-align:center;" | narrow fifth
| 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
| rowspan="2" | mid
|795
| ilo
|960 (3C0<sub>16</sub>)
| {{nowrap|(a, b, 0, 0, 1)}}
| style="text-align:center;" | wide fifth, infra sixth
| 11/9, 11/6
| 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
| lu
|901
| {{nowrap|(a, b, 0, 0, −1)}}
|1088 (340<sub>16</sub>)
| 12/11, 18/11
| 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
| rowspan="2" | major
|1007
| yo
|1216 (4C0<sub>16</sub>)
| {{nowrap|(a, b, 1)}}
| style="text-align:center;" | ultra sixth , infra seventh
| 5/4, 5/3
| 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
| fifthward wa
|1113
| {{nowrap|(a, b)}}; {{nowrap|b > 1}}
|1344 (540<sub>16</sub>)
| 9/8, 27/16
| 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
| upmajor
|1219
| ru
|1472 (%C0<sub>16</sub>)
| {{nowrap|(a, b, 0, −1)}}
| style="text-align:center;" | ultra seventh, narrow octave
| 9/7, 12/7
| 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=
Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].
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==
=== 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.


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
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.
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"
These are the intervals of 24edo that do not exist in 12edo:


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
{| class="wikitable center-all right-6"
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>
! [[Cent]]s
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
! Names
</div>
|-
[[Category:IMPORTDEBUG - Change External Images]]
| 50
 
| Quarter tone, infra second, wide unison
or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
|-
 
| 150
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
| Neutral second
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>
| 250
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
| Ultra second, infra third
</div>
|-
[[Category:IMPORTDEBUG - Change External Images]]
| 350
 
| Neutral third
or v = quarter-tone flat or "Drop" or "down"
|-
 
| 450
<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
| Minor fourth, ultra third, narrow fourth
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>
| 550
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
| Wide fourth
</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
 
==In Persian music==
 
{| class="wikitable"
|-
|-
| [[File:200px-Koron_sign.svg.png|40px|left]]
| 650
| '''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
| Narrow fifth
|}
 
{| class="wikitable"
|-
|-
| [[File:200px-Sori_sign.svg.png|40px|left]]
| 750
| '''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
| Wide fifth, infra sixth
|}
 
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:


=Naming Chords in 24edo=
* 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.
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, ilo, 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;" | ilo
| 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]] (or [[maqamic]] with 24d val)
|-
| | 1
| | 11\24
| |
|-
| | 2
| | 1\24
| | [[Shrutar]]
|-
| | 2
| | 5\24
| | [[Sruti]], [[Anguirus]], [[Decimal]]
|-
|-
| | 3
! rowspan="2" | [[Subgroup]]
| | 1\24
! rowspan="2" | [[Comma list]]
| | [[Semiaug]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 3
! [[TE error|Absolute]] (¢)
| | 3\24
! [[TE simple badness|Relative]] (%)
| | [[Triforce]]
|-
|-
| | 4
| 2.3.5.11
| | 1\24
| 81/80, 121/120, 128/125
| |  
| {{mapping| 24 38 56 83 }}
| −1.08
| 2.82
| 5.63
|-
|-
| | 6
| 2.3.5.11.13
| | 1\24
| 66/65, 81/80, 128/125, 144/143
| |  
| {{mapping| 24 38 56 83 89 }}
| −1.37
| 2.59
| 5.19
|-
|-
| | 8
| 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
|-
|-
| | 12
| 2.3.5.11.13.17.19
| | 1\24
| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143
| | [[Catler]]
| {{mapping| 24 38 56 83 89 98 102 }}
| −0.89
| 2.37
| 4.74
|}
|}


=Scales / Modes=
=== Uniform maps ===
{{Uniform map|edo=24}}


{| 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" |
====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)]]
|-
| | <tt>'''3 6 3 3 6'''</tt>
| | [[MOS]] of type [[3L 2s]] (father)
|}


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


{| class="wikitable"
=== Rank-2 temperaments ===
|-
* [[List of 24et rank two temperaments by badness]]
| colspan="2" |
* [[List of edo-distinct 24et rank two temperaments]]
====Octatonic:====
 
|-
{| class="wikitable center-1 center-2"
| | <tt>'''3 3 3 3 3 3 3 3'''</tt>
| | 8-equal, Wyschnegradsky's octatonic
|-
|-
| | <tt>'''3 3 4 4 2 1 3 4'''</tt>
! Periods<br>per 8ve
| | Maqam Bayati
! Generator
! Name
|-
|-
| | <tt>'''3 3 2 6 2 4 2 2'''</tt>
| 1
| | Maqam Saba
| 1\24
| [[Hemiripple]] (24)
|-
|-
| | <tt>'''4 3 1 2 4 4 2 4'''</tt>
| 1
| | Maqam Suzidil 'ara
| 5\24
| [[Godzilla]] (24) / [[baragon]] (24) / [[semaphoresmic clan #Varan|varan]] (24)
|-
|-
| | <tt>'''3 3 2 2 4 3 3 4'''</tt>
| 1
| | Maqam Mansuri
| 7\24
| [[Mohajira]] (24) / [[neutrominant]] (24d) / [[migration]] (24d)
|-
|-
| | <tt>'''4 3 3 4 4 2 1 3'''</tt>
| 1
| | Maqam Rast, Dilkashidah, Dilnishin
| 11\24
| [[Cohemiripple]] (24), [[freivald]] (24)
|-
|-
| | <tt>'''3 4 2 6 2 4 2 1'''</tt>
| 2
| | Maqam Rahat al-Arwah
| 1\24
| [[Shrutar]] (24)
|-
|-
| | <tt>'''3 4 3 3 4 4 2 1'''</tt>
| 2
| | Maqam Iraq
| 5\24
| [[Sruti]] (24), [[anguirus]] (24), [[decimal]] (24c)
|-
|-
| | <tt>'''2 6 2 4 2 1 3 4'''</tt>
| 3
| | Maqam Hijaz
| 1\24
| [[Hemiaug]] (24)
|-
|-
| | <tt>'''3 4 4 2 1 3 4 3'''</tt>
| 3
| | Maqam Musta'ar
| 3\24
| [[Triforce]] (24)
|-
|-
| | <tt>'''3 4 4 4 2 4 2 1'''</tt>
| 4
| | Maqam Farahnak
| 1\24
| [[Hemidim]] (24)
|-
|-
| | <tt>'''3 4 3 3 2 6 2 1'''</tt>
| 6
| | Maqam Bastanikar, Tarz Nuin
| 1\24
| [[Hemisemiaug]] (24)
|-
|-
| | <tt>'''4 2 6 2 4 2 1 3'''</tt>
| 8
| | Maqam Farah Faza, Maqam Nakriz
| 1\24
| [[Semidim]] (24)
|-
|-
| | <tt>'''3 1 2 4 4 2 4 4'''</tt>
| 12
| | Maqam Jabburi
| 1\24
|-
| [[Catler]]
| | <tt>'''1 4 4 2 4 4 4 1'''</tt>
| | Giancarlo Dalmonte's new quarter-tone scale (see [http://www.ottavanota.info/ http://www.ottavanota.info])
|}
|}


{| 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
====Enneatonic:====
! Edosteps
! Notes of C Chord
! Written name
! Spoken name
|-
|-
| | <tt>'''1 2 3 4 4 1 2 3 4'''</tt>
| 6:7:9, 26:30:39
| | Progressive Enneatonic
| {{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>'''4 1 4 1 4 1 4 1 4'''</tt>
| 10:12:15
| | de Vries 9-tone - [[MOSScales|MOS]] of type [[5L_4s|5L 4s (unfair bug)]] (godzilla/semaphore)
| {{dash|0, 6, 14|hair}}
| {{dash|C, E♭, G|hair}}
| Cm, Cmin
| C minor
|-
|-
| | <tt>'''3 4 2 2 2 2 2 4 3'''</tt>
| 18:22:27, 22:27:33
| | Maqam Huzam
| 0-7-14
| {{dash|C, E{{demiflat2}}, G|hair}}
| C~, Cneu
| C neutral
|-
|-
| | <tt>'''4 4 2 1 3 2 2 4 2'''</tt>
| 4:5:6
| | Maqam Shawq Afza
| {{dash|0, 8, 14|hair}}
| {{dash|C, E, G|hair}}
| C, Cmaj
| C, C major
|-
|-
| | <tt>'''3 3 2 3 3 2 3 3 2'''</tt>
| 14:18:21, 10:13:15
| | [[MOS]] of type [[6L 3s]] (triforce)
| {{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
|}
|}


{| 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.
 
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
|-
! Degrees of 24edo
! Chord spelling
! Notes of C chord
! Written name
! 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>
: <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
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.


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


==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 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 differrent 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 Neutral[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=
== Music ==
{{Wikipedia|List of quarter tone pieces}}
{{Main|{{ROOTPAGENAME}}/Music}}{{clear}}


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
== Further reading ==
External image: http://ronsword.com/images/24_tet_Coversm.jpg<br>
* Ellis, Don. ''[https://archive.org/details/don-ellis-quarter-tones/ Quarter Tones: A Text with Musical Examples, Exercises and Etudes]''. 1975.
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
* [[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.)
: <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]]


'''"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.
== See also ==
* [[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)


=External links=
== External links ==
[http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory] [http://www.webcitation.org/5xeFMH6cd Permalink]
* [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'']  


[http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [http://www.webcitation.org/5xeFBNdQW Permalink]
{{Todo| cleanup }}


[https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
== Notes ==
<references group="note" />


[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:Lists of intervals]]
[[Category:todo:unify_precision]]
[[Category:twentuning]]
Retrieved from "https://en.xen.wiki/w/24edo"