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Line 36: Line 36:
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.11.17.19&nbsp;[[subgroup]]}}</ref>
! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19&nbsp;[[subgroup]]}}</ref>
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic intervals in ups and downs notation|EIs]]: vvA1 and d2)
! colspan="3" |[[Ups and downs notation]] ([[Enharmonic intervals in ups and downs notation|EIs]]: vvA1 and d2)
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
! colspan="3" |[[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
! [[24edo solfege|Solfege]]
![[24edo solfege|Solfege]]
|-
|-
| 0
| 0
Line 65: Line 65:
| 2
| 2
| 100
| 100
| 16/15, 17/16, 18/17
| 16/15, 17/16, 18/17, 19/18
| A1, m2
| A1, m2
| aug unison, minor 2nd
| aug unison, minor 2nd
Line 76: Line 76:
| 3
| 3
| 150
| 150
| 12/11
| 13/12, 12/11, 11/10
| ~2
| ~2
| mid 2nd
| mid 2nd
Line 98: Line 98:
| 5
| 5
| 250
| 250
| 22/19
| 15/13, 22/19
| ^M2, vm3
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
Line 109: Line 109:
| 6
| 6
| 300
| 300
| 6/5, 19/16
| 6/5, 13/11, 19/16
| m3
| m3
| minor 3rd
| minor 3rd
Line 120: Line 120:
| 7
| 7
| 350
| 350
| 11/9, 27/22
| 11/9, 16/13, 27/22, 39/32
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 142: Line 142:
| 9
| 9
| 450
| 450
| 22/17
| 13/10, 17/13, 22/17
| ^M3, v4
| ^M3, v4
| upmajor 3rd, down-4th
| upmajor 3rd, down-4th
Line 164: Line 164:
| 11
| 11
| 550
| 550
| 11/8
| 11/8, 15/11
| ^4, ~4
| ^4, ~4
| up-4th, mid-4th
| up-4th, mid-4th
Line 175: Line 175:
| 12
| 12
| 600
| 600
| 17/12
| 17/12, 24/17, 45/32, 64/45
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
Line 186: Line 186:
| 13
| 13
| 650
| 650
| 16/11
| 16/11, 22/15
| v5, ~5
| v5, ~5
| down-5th, mid-5th
| down-5th, mid-5th
Line 208: Line 208:
| 15
| 15
| 750
| 750
| 17/11
| 17/11, 20/13
| ^5, vm6
| ^5, vm6
| up-fifth, downminor 6th
| up-fifth, downminor 6th
Line 230: Line 230:
| 17
| 17
| 850
| 850
| 18/11, 44/27
| 13/8, 18/11, 44/27, 64/39
| ~6
| ~6
| mid 6th
| mid 6th
Line 241: Line 241:
| 18
| 18
| 900
| 900
| 5/3, 32/19
| 5/3, 22/13, 32/19
| M6
| M6
| major 6th
| major 6th
Line 252: Line 252:
| 19
| 19
| 950
| 950
| 19/11
| 19/11, 26/15
| ^M6, vm7
| ^M6, vm7
| upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
Line 274: Line 274:
| 21
| 21
| 1050
| 1050
| 11/6
| 11/6, 20/11
| ~7
| ~7
| mid 7th
| mid 7th
Line 317: Line 317:
|}
|}


<references />
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.


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.


=== Interval qualities in color notation ===
=== Interval qualities in color notation ===
Line 336: Line 335:
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| rowspan="2" | minor
| rowspan="2" |minor
| fourthward wa
| fourthward wa
| style="white-space: nowrap;" | (a, b), b &lt; &minus;1
| style="white-space: nowrap;" | (a, b), b &lt; &minus;1
Line 345: Line 344:
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" | mid
| rowspan="2" |mid
| ilo
| ilo
| style="white-space: nowrap;" | (a, b, 0, 0, 1)
| style="white-space: nowrap;" | (a, b, 0, 0, 1)
Line 354: Line 353:
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| rowspan="2" | major
| rowspan="2" |major
| yo
| yo
| style="white-space: nowrap;" | (a, b, 1)
| style="white-space: nowrap;" | (a, b, 1)
Line 371: Line 370:
Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].  
Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].  


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


Line 447: Line 446:
* 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


=== Quartertone accidentals ===
===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.
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 ====
====Mainstream quartertone notation====
[[Category:IMPORTDEBUG - Change External Images]]
[[Category:IMPORTDEBUG - Change External Images]]
{{Sharpness-sharp2|24}}
{{Sharpness-sharp2|24}}
Line 456: Line 455:
{| class="wikitable center-1"
{| class="wikitable center-1"
|-
|-
| style="width: 40px;" | [[File:HeQu1.svg|21px|center]]
| style="width: 40px;" |[[File:HeQu1.svg|21px|center]]
| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.
| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.
|-
|-
| style="width: 40px;" | [[File:HeQu3.svg|32px|center]]
| style="width: 40px;" |[[File:HeQu3.svg|32px|center]]
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
|-
|-
| style="width: 40px;" | [[File:HeQd1.svg|22px|center]]
| style="width: 40px;" |[[File:HeQd1.svg|22px|center]]
| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
|-
|-
| style="width: 40px;" | [[File:HeQd3.svg|36px|center]]
| style="width: 40px;" |[[File:HeQd3.svg|36px|center]]
| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.
| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.
|}
|}
Line 477: Line 476:
{| class="wikitable"
{| class="wikitable"
|-
|-
| width="40px" | [[File:Koron_sign.svg|39px|center]]
| width="40px" |[[File:Koron_sign.svg|39px|center]]
| '''Koron''' ([[Wikipedia: Koron (music)|Wikipedia]]) = Quarter-tone flat
| '''Koron''' ([[Wikipedia: Koron (music)|Wikipedia]]) = Quarter-tone flat
|-
|-
| width="40px" | [[File:Sori_sign.svg|39px|center]]
| width="40px" |[[File:Sori_sign.svg|39px|center]]
| '''Sori''' ([[Wikipedia: Sori (music)|Wikipedia]]) = Quarter-tone sharp
| '''Sori''' ([[Wikipedia: Sori (music)|Wikipedia]]) = Quarter-tone sharp
|}
|}
Line 506: Line 505:


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


== Approximation to JI ==
== Approximation to JI ==
Line 530: Line 529:
| 2.3.5.11
| 2.3.5.11
| 81/80, 121/120, 128/125
| 81/80, 121/120, 128/125
| {{mapping| 24 38 56 83 }}
|{{mapping| 24 38 56 83 }}
| &minus;1.08
| &minus;1.08
| 2.82
| 2.82
Line 537: Line 536:
| 2.3.5.11.13
| 2.3.5.11.13
| 66/65, 81/80, 128/125, 144/143
| 66/65, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 }}
|{{mapping| 24 38 56 83 89 }}
| &minus;1.37
| &minus;1.37
| 2.59
| 2.59
Line 544: Line 543:
| 2.3.5.11.13.17
| 2.3.5.11.13.17
| 51/50, 66/65, 81/80, 128/125, 144/143
| 51/50, 66/65, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 98 }}
|{{mapping| 24 38 56 83 89 98 }}
| &minus;0.94
| &minus;0.94
| 2.55
| 2.55
Line 551: Line 550:
| 2.3.5.11.13.17.19
| 2.3.5.11.13.17.19
| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143
| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 98 102 }}
|{{mapping| 24 38 56 83 89 98 102 }}
| &minus;0.89
| &minus;0.89
| 2.37
| 2.37
Line 574: Line 573:
| 3
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
|{{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawa
Line 581: Line 580:
| 5
| 5
| [[648/625]]
| [[648/625]]
| {{monzo| 3 4 -4 }}
|{{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadgu
Line 588: Line 587:
| 5
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
|{{monzo| 18 -4 -5 }}
| 60.61
| 60.61
| Saquingu
| Saquingu
Line 595: Line 594:
| 5
| 5
| [[128/125]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
|{{monzo| 7 0 -3 }}
| 41.06
| 41.06
| Trigu
| Trigu
Line 602: Line 601:
| 5
| 5
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
|{{monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
Line 609: Line 608:
| 5
| 5
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
|{{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Sagugu
Line 616: Line 615:
| 5
| 5
| [[67108864/66430125| (16 digits)]]
| [[67108864/66430125| (16 digits)]]
| {{monzo| 26 -12 -3 }}
|{{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-trigu
Line 623: Line 622:
| 5
| 5
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
|{{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layo
Line 630: Line 629:
| 5
| 5
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
|{{monzo| 161 -84 -12 }}
| 0.02
| 0.02
| Sepbisa-quadbigu
| Sepbisa-quadbigu
Line 637: Line 636:
| 7
| 7
| [[1323/1280]]
| [[1323/1280]]
| {{monzo| -8 3 -1 2 }}
|{{monzo| -8 3 -1 2 }}
| 57.20
| 57.20
| Lazozogu
| Lazozogu
Line 644: Line 643:
| 7
| 7
| [[49/48]]
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
|{{monzo| -4 -1 0 2 }}
| 35.70
| 35.70
| Zozo
| Zozo
Line 651: Line 650:
| 7
| 7
| [[245/243]]
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
|{{monzo| 0 -5 1 2 }}
| 14.19
| 14.19
| Zozoyo
| Zozoyo
Line 658: Line 657:
| 7
| 7
| [[19683/19600]]
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
|{{monzo| -4 9 -2 -2 }}
| 7.32
| 7.32
| Labirugu
| Labirugu
Line 665: Line 664:
| 7
| 7
| [[6144/6125]]
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
|{{monzo| 11 1 -3 -2 }}
| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
Line 672: Line 671:
| 11
| 11
| [[56/55]]
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
|{{monzo| 3 0 -1 1 -1 }}
| 31.19
| 31.19
| Luzogu
| Luzogu
Line 679: Line 678:
| 11
| 11
| [[245/242]]
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
|{{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Luluzozoyo
| Luluzozoyo
Line 686: Line 685:
| 11
| 11
| [[121/120]]
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
|{{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
Line 693: Line 692:
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
|{{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 700: Line 699:
| 11
| 11
| [[896/891]]
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
|{{monzo| 7 -4 0 1 -1 }}
| 9.69
| 9.69
| Saluzo
| Saluzo
Line 707: Line 706:
| 11
| 11
| [[243/242]]
| [[243/242]]
| {{monzo| -1 5 0 0 -2 }}
|{{monzo| -1 5 0 0 -2 }}
| 7.14
| 7.14
| Lulu
| Lulu
Line 714: Line 713:
| 11
| 11
| <abbr title="214990848/214358881">(18 digits)</abbr>
| <abbr title="214990848/214358881">(18 digits)</abbr>
| {{monzo| 15 8 0 0 -8 }}
|{{monzo| 15 8 0 0 -8 }}
| 5.10
| 5.10
| Quadbilu
| Quadbilu
Line 721: Line 720:
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
|{{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
Line 728: Line 727:
| 11
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{monzo| 24 -6 0 1 -5 }}
|{{monzo| 24 -6 0 1 -5 }}
| 0.51
| 0.51
| Saquinlu-azo
| Saquinlu-azo
Line 735: Line 734:
| 11
| 11
| [[9801/9800]]
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
|{{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Bilorugu
Line 742: Line 741:
| 13
| 13
| [[66/65]]
| [[66/65]]
| {{monzo| 1 1 -1 0 1 -1 }}
|{{monzo| 1 1 -1 0 1 -1 }}
| 26.43
| 26.43
| Thulogu
| Thulogu
Line 749: Line 748:
| 13
| 13
| [[91/90]]
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
Line 756: Line 755:
| 13
| 13
| [[512/507]]
| [[512/507]]
| {{monzo| 9 -1 0 0 0 -2 }}
|{{monzo| 9 -1 0 0 0 -2 }}
| 16.99
| 16.99
| Thuthu
| Thuthu
Line 763: Line 762:
| 13
| 13
| [[105/104]]
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
|{{monzo| -3 1 1 1 0 -1 }}
| 16.57
| 16.57
| Thuzoyo
| Thuzoyo
Line 770: Line 769:
| 13
| 13
| [[144/143]]
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
|{{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| 12.06
| Thulu
| Thulu
Line 777: Line 776:
| 13
| 13
| [[676/675]]
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
|{{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Bithogu
| Bithogu
Line 784: Line 783:
| 13
| 13
| [[4096/4095]]
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
|{{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| 0.42
| Sathurugu
| Sathurugu
Line 791: Line 790:
| 17
| 17
| [[51/50]]
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
|{{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| 34.28
| Sogugu
| Sogugu
Line 798: Line 797:
| 17
| 17
| [[136/135]]
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
|{{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| 12.78
| Sogu
| Sogu
Line 805: Line 804:
| 17
| 17
| [[170/169]]
| [[170/169]]
| {{monzo| 1 0 1 0 0 -2 1 }}
|{{monzo| 1 0 1 0 0 -2 1 }}
| 10.21
| 10.21
| Sothuthuyo
| Sothuthuyo
Line 812: Line 811:
| 17
| 17
| [[221/220]]
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
|{{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| 7.85
| Sotholugu
| Sotholugu
Line 819: Line 818:
| 17
| 17
| [[256/255]]
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
|{{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| 6.78
| Sugu
| Sugu
Line 826: Line 825:
| 17
| 17
| [[289/288]]
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
|{{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| 6.00
| Soso
| Soso
Line 833: Line 832:
| 17
| 17
| [[1225/1224]]
| [[1225/1224]]
| {{monzo| -3 -2 2 2 0 0 -1 }}
|{{monzo| -3 -2 2 2 0 0 -1 }}
| 1.41
| 1.41
| Subizoyo
| Subizoyo
Line 840: Line 839:
| 19
| 19
| [[76/75]]
| [[76/75]]
| {{monzo| 2 -1 -2 0 0 0 0 1 }}
|{{monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.93
| 22.93
| Nogugu
| Nogugu
Line 847: Line 846:
| 19
| 19
| [[77/76]]
| [[77/76]]
| {{monzo| -2 0 0 1 1 0 0 -1 }}
|{{monzo| -2 0 0 1 1 0 0 -1 }}
| 22.63
| 22.63
| Nulozo
| Nulozo
Line 854: Line 853:
| 19
| 19
| [[96/95]]
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
|{{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| 18.13
| Nugu
| Nugu
Line 861: Line 860:
| 19
| 19
| [[133/132]]
| [[133/132]]
| {{monzo| -2 -1 0 1 -1 0 0 1 }}
|{{monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.07
| 13.07
| Noluzo
| Noluzo
Line 868: Line 867:
| 19
| 19
| [[153/152]]
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
|{{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| 11.35
| Nuso
| Nuso
Line 875: Line 874:
| 19
| 19
| [[171/170]]
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
|{{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| 10.15
| Nosugu
| Nosugu
Line 882: Line 881:
| 19
| 19
| [[209/208]]
| [[209/208]]
| {{monzo| -4 0 0 0 1 -1 0 1 }}
|{{monzo| -4 0 0 0 1 -1 0 1 }}
| 8.30
| 8.30
| Nothulo
| Nothulo
Line 889: Line 888:
| 19
| 19
| [[324/323]]
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
|{{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| 5.35
| Nusu
| Nusu
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| 19
| 19
| [[361/360]]
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
|{{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| 4.80
| Nonogu
| Nonogu
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| 19
| 19
| [[5776/5775]]
| [[5776/5775]]
| {{monzo| 4 -1 -2 -1 -1 0 0 2 }}
|{{monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.30
| 0.30
| Nonolurugugu
| Nonolurugugu
| Neovish comma
| Neovish comma
|}
|}
<references />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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Important MOSes include:  
Important MOSes include:  
* Semaphore 4L1s 55455 (generator: 5\24)
* Semaphore 4L1s 55455 (generator: 5\24)
* Semaphore 5L4s 414144141 (generator: 5\24)
* Semaphore 5L4s 414141414 (generator: 5\24)
* Mohajira 3L4s 3434343 (generator: 7\24)
* Mohajira 3L4s 3434343 (generator: 7\24)
* Mohajira 7L3s 3313313313 (generator: 7\24)
* Mohajira 7L3s 3313313313 (generator: 7\24)
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{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental triads of 24edo
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental triads of 24edo
|-
! JI Chord
! JI Chord
! Edosteps
! Edosteps
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|-
|-
| 6:7:9, 26:30:39
| 6:7:9, 26:30:39
| {{dash|0, 5, 14|hair}}
|{{dash|0, 5, 14|hair}}
| {{dash|C, E{{sesquiflat2}}, G|hair}}
|{{dash|C, E{{sesquiflat2}}, G|hair}}
| Cvm<br />Cm({{demiflat2}}3), Cmin({{demiflat2}}3)
| Cvm<br />Cm({{demiflat2}}3), Cmin({{demiflat2}}3)
| C subminor<br />C minor semiflat-three
| C subminor<br />C minor semiflat-three
|-
|-
| 10:12:15
| 10:12:15
| {{dash|0, 6, 14|hair}}
|{{dash|0, 6, 14|hair}}
| {{dash|C, E♭, G|hair}}
|{{dash|C, E♭, G|hair}}
| Cm, Cmin
| Cm, Cmin
| C minor
| C minor
Line 1,009: Line 1,007:
| 18:22:27, 22:27:33
| 18:22:27, 22:27:33
| 0-7-14
| 0-7-14
| {{dash|C, E{{demiflat2}}, G|hair}}
|{{dash|C, E{{demiflat2}}, G|hair}}
| C~, Cneu
| C~, Cneu
| C neutral
| C neutral
|-
|-
| 4:5:6
| 4:5:6
| {{dash|0, 8, 14|hair}}
|{{dash|0, 8, 14|hair}}
| {{dash|C, E, G|hair}}
|{{dash|C, E, G|hair}}
| C, Cmaj
| C, Cmaj
| C, C major
| C, C major
|-
|-
| 14:18:21, 10:13:15
| 14:18:21, 10:13:15
| {{dash|0, 9, 14|hair}}
|{{dash|0, 9, 14|hair}}
| {{dash|C, E{{demisharp2}}, G|hair}}
|{{dash|C, E{{demisharp2}}, G|hair}}
| C^<br />C({{demisharp2}}3), Cmaj({{demisharp2}}3)
| C^<br />C({{demisharp2}}3), Cmaj({{demisharp2}}3)
| C supermajor<br />C major semisharp-three
| C supermajor<br />C major semisharp-three
Line 1,042: Line 1,040:
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental tetrads of 24edo
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental tetrads of 24edo
|-
! Degrees of 24edo
! Degrees of 24edo
! Chord spelling
! Chord spelling
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! Audio example
! Audio example
|-
|-
| {{dash|0, 5, 14, 19|hair}}
|{{dash|0, 5, 14, 19|hair}}
| {{dash|1, vb3, 5, vb7|hair}}
|{{dash|1, vb3, 5, vb7|hair}}
| {{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|hair}}
|{{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|hair}}
| smin7<br />min7({{demiflat2}}3, {{demiflat2}}7)
| smin7<br />min7({{demiflat2}}3, {{demiflat2}}7)
| Subminor seven<br />Minor seven semiflat-three semiflat-seven
| Subminor seven<br />Minor seven semiflat-three semiflat-seven
|  
| |-
|-
|{{dash|0, 6, 14, 20|hair}}
| {{dash|0, 6, 14, 20|hair}}
|{{dash|1, b3, 5, b7|hair}}
| {{dash|1, b3, 5, b7|hair}}
|{{dash|C, E♭, G, B♭|hair}}
| {{dash|C, E♭, G, B♭|hair}}
| m7, min7
| m7, min7
| Minor seven
| Minor seven
|  
| |-
|-
|{{dash|0, 7, 14, 21|hair}}
| {{dash|0, 7, 14, 21|hair}}
|{{dash|1, v3, 5, v7|hair}}
| {{dash|1, v3, 5, v7|hair}}
|{{dash|C, E{{demiflat2}}, G, B{{demiflat2}}|hair}}
| {{dash|C, E{{demiflat2}}, G, B{{demiflat2}}|hair}}
| n7, neu7
| n7, neu7
| Neutral seven
| Neutral seven
| [[File:Neutral Tetrad on C.mp3]]
| [[File:Neutral Tetrad on C.mp3]]
|-
|-
| {{dash|0, 8, 14, 22|hair}}
|{{dash|0, 8, 14, 22|hair}}
| {{dash|1, b3, 5, b7|hair}}
|{{dash|1, b3, 5, b7|hair}}
| {{dash|C, E, G, B|hair}}
|{{dash|C, E, G, B|hair}}
| maj7
| maj7
| Major seven
| Major seven
|  
| |-
|-
|{{dash|0, 8, 14, 22|hair}}
| {{dash|0, 8, 14, 22|hair}}
|{{dash|1, b3, 5, b7|hair}}
| {{dash|1, b3, 5, b7|hair}}
|{{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|hair}}
| {{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|hair}}
| smaj7<br />maj7({{demisharp2}}3, {{demisharp2}}7)
| smaj7<br />maj7({{demisharp2}}3, {{demisharp2}}7)
| Supermajor seven<br />Major seven semisharp-three semisharp-seven
| Supermajor seven<br />Major seven semisharp-three semisharp-seven
|  
| |-
|-
|{{dash|0, 8, 14, 20|hair}}
| {{dash|0, 8, 14, 20|hair}}
|{{dash|1, 3, 5, b7|hair}}
| {{dash|1, 3, 5, b7|hair}}
|{{dash|C, E, G, B♭|hair}}
| {{dash|C, E, G, B♭|hair}}
| 7, dom7
| 7, dom7
| Dominant seven
| Dominant seven
|  
| |-
|-
|{{dash|0, 8, 14, 19|hair}}
| {{dash|0, 8, 14, 19|hair}}
|{{dash|1, 3, 5, vb7|hair}}
| {{dash|1, 3, 5, vb7|hair}}
|{{dash|C, E, G, B{{sesquiflat2}}|hair}}
| {{dash|C, E, G, B{{sesquiflat2}}|hair}}
| h7<br />7({{demiflat2}}7)
| h7<br />7({{demiflat2}}7)
| Harmonic seven<br />Dominant 7 semiflat-seven
| Harmonic seven<br />Dominant 7 semiflat-seven
|  
| |-
|-
|{{dash|0, 5, 14, 20|hair}}
| {{dash|0, 5, 14, 20|hair}}
|{{dash|1, vb3, 5, b7|hair}}
| {{dash|1, vb3, 5, b7|hair}}
|{{dash|C, E{{sesquiflat2}}, G, B♭|hair}}
| {{dash|C, E{{sesquiflat2}}, G, B♭|hair}}
| min7({{demiflat2}}3)
| min7({{demiflat2}}3)
| Arto<br />Minor seven semiflat-three
| Arto<br />Minor seven semiflat-three
| [[File:arto tetrad on C.mp3]]
| [[File:arto tetrad on C.mp3]]
|-
|-
| {{dash|0, 9, 14, 19|hair}}
|{{dash|0, 9, 14, 19|hair}}
| {{dash|1, ^3, 5, vb7|hair}}
|{{dash|1, ^3, 5, vb7|hair}}
| {{dash|C, E{{demisharp2}}, G, B{{sesquiflat2}}|hair}}
|{{dash|C, E{{demisharp2}}, G, B{{sesquiflat2}}|hair}}
| h7({{demisharp2}}3)<br />7({{demisharp2}}3, {{demiflat2}}7)
| h7({{demisharp2}}3)<br />7({{demisharp2}}3, {{demiflat2}}7)
| Tendo<br />Harmonic seven semisharp-three<br />Dominant seven semisharp-three semiflat-seven
| Tendo<br />Harmonic seven semisharp-three<br />Dominant seven semisharp-three semiflat-seven
Line 1,123: Line 1,116:
|}
|}


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


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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):
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):


; 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)
;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/shorts/Ieqi54XE2lI Demonstration short video by Nahre Sol] (2024)


; 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)
;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=PdP4epQIUrU Demonstration video by Steve Cohn] (2011)


Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:
Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:


; Quarter-tone flute, made by Eva Kingma
;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)
*[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)


24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see [[Lumatone mapping for 24edo]]
24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see [[Lumatone mapping for 24edo]]
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{{Main|Music in 24edo}}
{{Main|Music in 24edo}}


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


==See also==
== See also ==
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[48edo]] (2mu tuning)
** [[48edo]] (2mu tuning)
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== External links ==
== 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://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
* [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
* [https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
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