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}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|24}}  
{{ED intro}}  
{{Wikipedia|Quarter tone}}
{{Wikipedia|Quarter tone}}


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|Arabic]] music.  
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]].  


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]]''.
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]]''.


== Theory ==
== Theory ==
The [[5-limit]] approximations in 24edo are the same as those in 12edo, therefore 24edo offers nothing new as far as approximating the 5-limit is concerned.   
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.   


The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].  
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.  


Speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11/10, 11/9, 11/8, 11/6, 12/11, 15/11, 16/11, 18/11, 20/11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.  
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 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 [[Just intonation subgroup|subgroup]] temperament, on which it is quite accurate.  
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 ===
=== Prime harmonics ===
{{harmonics in equal|24}}
{{Harmonics in equal|24|prec=2}}


=== Subsets and supersets ===
=== Subsets and supersets ===
24edo is the 6th [[highly composite EDO]]. Its divisors are {{EDOs|1, 2, 3, 4, 6, 8, 12}}.
24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}.


== Notation ==
== Intervals ==
There are multiple ways of notating 24edo. While an arguably common form can be seen on [[Wikipedia: Quarter tone #Interval size in equal temperament|Wikipedia's page on quartertones]], there are other forms, and it is these other forms that will be considered here.
{| class="wikitable center-all left-3"
 
=== Ups and down notation ===
 
{| class="wikitable center-all"
|-
|-
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref>based on treating 24-EDO as a 2.3.11.17.19 [[subgroup]]; other approaches are possible.</ref>
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>
! colspan="3" | [[ups and downs notation]]
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and d2)
![[24edo solfege|Solfege]]
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
! [[24edo solfege|Solfege]]
|-
|-
| 0
| 0
Line 46: Line 45:
| P1
| P1
| unison
| unison
| C
| D
|Do
| unison
| P1
| D
| Do
|-
|-
| 1
| 1
Line 54: Line 56:
| ^P1, vm2
| ^P1, vm2
| up-unison, downminor 2nd
| up-unison, downminor 2nd
| ^C, vDb
| ^D, vEb
|Da/Ru
| super unison, uber unison
| S1, U1
| SD, UD
| Da/Ru
|-
|-
| 2
| 2
| 100
| 100
| 17/16, 18/17
| 16/15, 17/16, 18/17, 19/18
| A1, m2
| A1, m2
| aug unison, minor 2nd
| aug unison, minor 2nd
| C#, Db
| D#, Eb
|Ro
| aug unison, minor 2nd
| A1, m2
| D#, Eb
| Ro
|-
|-
| 3
| 3
| 150
| 150
| 12/11
| 13/12, 12/11, 11/10
| ~2
| ~2
| mid 2nd
| mid 2nd
| vD
| vE
|Ra
| neutral 2nd
| N2
| UEb, uE
| Ra
|-
|-
| 4
| 4
| 200
| 200
| 9/8
| 9/8, 10/9
| M2
| M2
| major 2nd
| major 2nd
| D
| E
|Re
| major 2nd
| M2
| E
| Re
|-
|-
| 5
| 5
| 250
| 250
| 22/19
| 15/13, 22/19
| ^M2, vm3
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
| ^D, vEb
| ^E, vF
|Ri/Mu
| supermajor 2nd, subminor 3rd
| SM2, sm3
| SE, sF
| Ri/Mu
|-
|-
| 6
| 6
| 300
| 300
| 19/16
| 6/5, 13/11, 19/16
| m3
| m3
| minor 3rd
| minor 3rd
| Eb
| F
|Mo
| minor 3rd
| m3
| F
| Mo
|-
|-
| 7
| 7
| 350
| 350
| 11/9, 27/22
| 11/9, 16/13, 27/22, 39/32
| ~3
| ~3
| mid 3rd
| mid 3rd
| vE
| vF#
|Ma
| neutral 3rd
| N3
| UF, uF#
| Ma
|-
|-
| 8
| 8
| 400
| 400
| 24/19
| 5/4, 24/19
| M3
| M3
| major 3rd
| major 3rd
| E
| F#
|Me
| major 3rd
| M3
| F#
| Me
|-
|-
| 9
| 9
| 450
| 450
| 22/17
| 13/10, 17/13, 22/17
| ^M3, v4
| ^M3, v4
| upmajor 3rd, down-4th
| upmajor 3rd, down-4th
| ^E, vF
| ^F#, vG
|Mi/Fu
| supermajor 3rd, sub 4th
| SM3, s4
| SF#, sG
| Mi/Fu
|-
|-
| 10
| 10
Line 126: Line 155:
| P4
| P4
| fourth
| fourth
| F
| G
|Fo
| perfect 4th
| P4
| G
| Fo
|-
|-
| 11
| 11
| 550
| 550
| 11/8
| 11/8, 15/11
| ^4, ~4
| ^4, ~4
| up-4th, mid-4th
| up-4th, mid-4th
| ^F
| ^G
|Fa/Su
| uber 4th/neutral 4th
| U4/N4
| UG
| Fa/Su
|-
|-
| 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
| F#, Gb
| G#, Ab
|Fe/So
| aug 4th, dim 5th
| A4, d5
| G#/Ab
| Fe/So
|-
|-
| 13
| 13
| 650
| 650
| 16/11
| 16/11, 22/15
| v5, ~5
| v5, ~5
| down-5th, mid-5th
| down-5th, mid-5th
| vG
| vA
|Fi/Sa
| unter 5th/neutral 5th
| u5/N5
| uA
| Fi/Sa
|-
|-
| 14
| 14
Line 158: Line 199:
| P5
| P5
| fifth
| fifth
| G
| A
|Se
| perfect 5th
| P5
| A
| Se
|-
|-
| 15
| 15
| 750
| 750
| 17/11
| 17/11, 20/13
| ^5, vm6
| ^5, vm6
| up-fifth, downminor 6th
| up-fifth, downminor 6th
| ^G, vAb
| ^A, vBb
|Si/Lu
| super 5th, subminor 6th
| S5, sm6
| SA, sBb
| Si/Lu
|-
|-
| 16
| 16
| 800
| 800
| 19/12
| 8/5, 19/12
| m6
| m6
| minor 6th
| minor 6th
| Ab
| Bb
|Lo
| minor 6th
| m6
| Bb
| Lo
|-
|-
| 17
| 17
| 850
| 850
| 18/11, 44/27
| 13/8, 18/11, 44/27, 64/39
| ~6
| ~6
| mid 6th
| mid 6th
| vA
| vB
|La
| neutral 6th
| N6
| UBb, uB
| La
|-
|-
| 18
| 18
| 900
| 900
| 32/19
| 5/3, 22/13, 32/19
| M6
| M6
| major 6th
| major 6th
| A
| B
|Le
| major 6th
| M6
| B
| Le
|-
|-
| 19
| 19
| 950
| 950
| 19/11
| 19/11, 26/15
| ^M6, vm7
| ^M6, vm7
| upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
| ^A, vBb
| ^B, vC
|Li/Tu
| supermajor 6th, subminor 7th
| SM6, sm7
| SB, sC
| Li/Tu
|-
|-
| 20
| 20
| 1000
| 1000
| 16/9
| 9/5, 16/9
| m7
| m7
| minor 7th
| minor 7th
| Bb
| C
|To
| minor 7th
| m7
| C
| To
|-
|-
| 21
| 21
| 1050
| 1050
| 11/6
| 11/6, 20/11
| ~7
| ~7
| mid 7th
| mid 7th
| vB
| vC#
|Ta
| neutral 7th
| N7
| UC, uC#
| Ta
|-
|-
| 22
| 22
| 1100
| 1100
| 17/9, 32/17
| 15/8, 17/9, 32/17
| M7
| M7
| major 7th
| major 7th
| B
| C#
|Te
| major 7th
| M7
| C#
| Te
|-
|-
| 23
| 23
Line 230: Line 298:
| ^M7, vP8
| ^M7, vP8
| upmajor 7th, down-8ve
| upmajor 7th, down-8ve
| ^B, vC
| ^C#, vD
|Ti/Du
| sub 8ve, unter 8ve
| s8, u8
| C#, uD
| Ti/Du
|-
|-
| 24
| 24
Line 238: Line 309:
| P8
| P8
| perfect 8ve
| perfect 8ve
| C
| D
|Do
| perfect 8ve
| P8
| D
| Do
|}
 
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.
 
== Notation ==
=== Ups and downs notation ===
Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{sharpness-sharp2a|24}}
 
=== Stein–Zimmermann accidentals ===
{{Sharpness-sharp2|24}}
 
{| class="wikitable center-1"
|-
| style="width: 40px;" | [[File:HeQu1.svg|21px|center]]
| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.
|-
| 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.
|-
| 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.
|-
| 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.
|}
 
'''Pros:''' familiar, intuitive, and fairly easy to learn.
 
'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.
 
=== Persian quartertone accidentals ===
{{Wikipedia|Koron (music)|Sori (music)}}
 
{| class="wikitable"
|-
| width="40px" | [[File:Koron_sign.svg|39px|center]]
| '''Koron''' = quarter-tone flat
|-
| width="40px" | [[File:Sori_sign.svg|39px|center]]
| '''Sori''' = quarter-tone sharp
|}
|}


<references/>
'''Pros:''' easy to read.
 
'''Cons:''' hard to write on a computer, does not fit with standard notation well.
 
=== 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]].
 
==== Evo flavor ====
<imagemap>
File:24-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:24-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== 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>
 
==== 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>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to [[#Stein.E2.80.93Zimmermann_accidentals|Stein–Zimmerman notation]].
 
==== 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.
 
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
 
'''Pros:''' easy to read, and less likely to clutter the score.


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.
'''Cons:''' not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality.
 
 
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:
 
[[File:24edo Sagittal.png|800px]]


== 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:
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! quality
! Quality
![[color name]]
! [[Color name]]
! monzo format
! Monzo format
! examples
! Examples
|-
|-
| downminor
| downminor
| zo
| zo
| (a, b, 0, 1)
| {{nowrap|(a, b, 0, 1)}}
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| rowspan="2" | minor
| rowspan="2" | minor
| fourthward wa
| fourthward wa
| (a, b), b &lt; -1
| {{nowrap|(a, b)}}; {{nowrap|b < −1}}
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| gu
| gu
| (a, b, -1)
| {{nowrap|(a, b, −1)}}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" | mid
| rowspan="2" | mid
| ilo
| ilo
| (a, b, 0, 0, 1)
| {{nowrap|(a, b, 0, 0, 1)}}
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| lu
| lu
| (a, b, 0, 0, -1)
| {{nowrap|(a, b, 0, 0, −1)}}
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| rowspan="2" | major
| rowspan="2" | major
| yo
| yo
| (a, b, 1)
| {{nowrap|(a, b, 1)}}
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| fifthward wa
| fifthward wa
| (a, b), b &gt; 1
| {{nowrap|(a, b)}}; {{nowrap|b > 1}}
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| upmajor
| upmajor
| ru
| ru
| (a, b, 0, -1)
| {{nowrap|(a, b, 0, −1)}}
| 9/7, 12/7
| 9/7, 12/7
|}
|}


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 interval and chord names ===
24edo breaks intervals into two sets of five categories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave.  
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.  


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.
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.
Line 304: Line 470:
{| class="wikitable center-all right-6"
{| class="wikitable center-all right-6"
|-
|-
! colspan="4" | The twelve new intervals in 24edo
! [[Cent]]s
! colspan="3" | some nearby JI intervals
! Names
|-
! [[cent]]s
! [[pion]]s
! [[7mu]]s ([[Wikipedia: Hexadecimal |hex]])
! common names
! frequency ratio
! cents
! common name
|-
|-
| 50
| 50
| 53
| Quarter tone, infra second, wide unison
| 64 (40<sub>16</sub>)
| quartertone <br> infra second, wide unison
| [[36/35]] <br> [[35/34]] <br> [[34/33]] <br> [[33/32]]
| 48.770 <br> 50.184 <br> 51.682 <br> 53.273
| large septimal quarter-tone (Archytas) <br> large 17-limit quartertone <br> small 17-limit quartertone <br> 33rd harmonic
|-
|-
| 150
| 150
| 159
| Neutral second
| 192 (C0<sub>16</sub>)
| neutral second
| [[12/11]]
| 150.637
| large undecimal neutral second
|-
|-
| 250
| 250
| 265
| Ultra second, infra third
| 320 (140<sub>16</sub>)
| ultra second <br> infra third
| [[144/125]] <br> [[15/13]] <br> [[52/45]]
| 244.969 <br> 247.741 <br> 250.304
| diminished third (6/5 x 24/25) <br> .. <br> ..
|-
|-
| 350
| 350
| 383
| Neutral third
| 448 (1C0<sub>16</sub>)
| neutral third
| [[11/9]] <br> [[27/22]] <br> [[16/13]]
| 347.408 <br> 354.547 <br> 359.472
| undecimal neutral third <br> .. <br> tridecimal neutral third
|-
|-
| 450
| 450
| 477
| Minor fourth, ultra third, narrow fourth
| 576 (240<sub>16</sub>)
| minor fourth, ultra third, narrow fourth
| [[22/17]] <br> [[35/27]] <br> [[13/10]]
| 446.363 <br> 449.275 <br> 454.214
| 17-limit supermajor third <br> .. <br> tridecimal subfourth
|-
|-
| 550
| 550
| 583
| Wide fourth
| 704 (2C0<sub>16</sub>)
| wide fourth
| [[11/8]]
| 551.318
| undecimal superfourth, harmonic 11th
|-
|-
| 650
| 650
| 689
| Narrow fifth
| 832 (340<sub>16</sub>)
| narrow fifth
| [[16/11]]
| 648.682
| undecimal subfifth, 11th subharmonic
|-
|-
| 750
| 750
| 795
| Wide fifth, infra sixth
| 960 (3C0<sub>16</sub>)
| wide fifth, infra sixth
| [[20/13]] <br> [[54/35]] <br> [[17/11]]
| 745.786 <br> 750.725 <br> 753.637
| tridecimal superfifth <br> .. <br> 17-limit subminor sixth
|-
|-
| 850
| 850
| 901
| Neutral sixth
| 1088 (340<sub>16</sub>)
| neutral sixth
| [[13/8]] <br> [[44/27]] <br> [[18/11]]
| 840.528 <br> 845.453 <br> 852.592
| overtone sixth, 13th harmonic <br> .. <br> undecimal neutral sixth
|-
|-
| 950
| 950
| 1007
| Ultra sixth, infra seventh
| 1216 (4C0<sub>16</sub>)
| ultra sixth , infra seventh
| [[45/26]] <br> [[26/15]] <br> [[125/72]]
| 949.696 <br> 952.259 <br> 955.031
| .. <br> .. <br> ..
|-
|-
| 1050
| 1050
| 1113
| Neutral seventh
| 1344 (540<sub>16</sub>)
| neutral seventh
| [[11/6]]
| 1049.363
| undecimal neutral seventh
|-
|-
| 1150
| 1150
| 1219
| Ultra seventh, narrow octave
| 1472 (5C0<sub>16</sub>)
| ultra seventh, narrow octave
| [[31/16]] <br> [[33/17]] <br> [[35/18]]
| 1145.036 <br> 1148.318 <br> 1151.230
| 31st harmonic <br> .. <br> ..
|}
|}


=== Interval alterations ===
==== Interval alterations ====
The special alterations of the intervals and chords of 12edo can be notated like this:
The special alterations of the intervals and chords of 12edo can be notated like this:


Line 439: Line 537:
* 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 ===
=== Further discussion of interval and chord naming ===
Besides ups and downs, there are various systems for notating quarter tones. Here are some of them, along with their pros and cons.
{{main|{{PAGENAME}}/Interval names and harmonies }}


==== Mainstream quartertone notation ====
* [[24edo Chord Names]]
[[Category:IMPORTDEBUG - Change External Images]]
* [[Ups and downs notation#Chords and Chord Progressions]].


[[File:Accidental semisharp.png|thumb|alt=A "semisharp" accidental comprising one half of a regular musical sharp symbol.|left]]
== 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]]


or ^ = quarter-tone sharp or "Jump" or "up"
=== Interval mappings ===
{{Q-odd-limit intervals|24}}


 
== Regular temperament properties ==
[[File:Accidental_sesquisharp.png|thumb|alt=A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.|left]]
{| class="wikitable center-4 center-5 center-6"
 
|-
 
! rowspan="2" | [[Subgroup]]
or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
! rowspan="2" | [[Comma list]]
 
! rowspan="2" | [[Mapping]]
 
! rowspan="2" | Optimal<br>8ve stretch (¢)
[[File:Accidental semiflat.png|thumb|alt=A "semiflat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.|left]]
! colspan="2" | Tuning error
 
|-
 
! [[TE error|Absolute]] (¢)
or v = quarter-tone flat or "Drop" or "down"
! [[TE simple badness|Relative]] (%)
 
|-
 
| 2.3.5.11
[[File:Accidental_sesquiflat.png|thumb|alt=A "flat and a half" or "sesquiflat" accidental, comprising the above half-flat symbol and a regular flat symbol placed back to back.|left]]
| 81/80, 121/120, 128/125
 
| {{mapping| 24 38 56 83 }}
 
| −1.08
or bv or vb = three-quarter-tone sharp or "Drop-Flat" or "downflat"
| 2.82
 
| 5.63
 
|-
 
| 2.3.5.11.13
 
| 66/65, 81/80, 128/125, 144/143
'''Pros:''' Familiar, fairly easy to learn
| {{mapping| 24 38 56 83 89 }}
[[Category:IMPORTDEBUG - Change External Images]]
| −1.37
[[Category:IMPORTDEBUG - Change External Images]]
| 2.59
[[Category:IMPORTDEBUG - Change External Images]]
| 5.19
[[Category:IMPORTDEBUG - Change External Images]]
[[Category:IMPORTDEBUG - Change External Images]]
[[Category:IMPORTDEBUG - Change External Images]]
[[Category:IMPORTDEBUG - Change External Images]]
 
'''Cons:''' Clutters a score easily, can get confusing when sight read at faster paces
 
==== Persian accidentals ====
 
{| class="wikitable"
|-
|-
|[[File:200px-Koron_sign.svg.png|40px|left]]
| 2.3.5.11.13.17
|'''Koron''' ([https://en.wikipedia.org/wiki/Koron_%28music%29 en] <nowiki>|</nowiki> [https://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86 fa]) = quarter-tone flat
| 51/50, 66/65, 81/80, 128/125, 144/143
|}
| {{mapping| 24 38 56 83 89 98 }}
 
| −0.94
{| class="wikitable"
| 2.55
| 5.11
|-
|-
|[[File:200px-Sori_sign.svg.png|40px|left]]
| 2.3.5.11.13.17.19
|'''Sori''' ([https://fa.wikipedia.org/wiki/%D8%B3%D8%B1%DB%8C_%28%D9%85%D9%88%D8%B3%DB%8C%D9%82%DB%8C%29 fa]) = quarter-tone sharp
| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143
| {{mapping| 24 38 56 83 89 98 102 }}
| −0.89
| 2.37
| 4.74
|}
|}


'''Pros:''' Easy to read
'''Cons:''' Hard to write on a computer, doesn't fit with standard notation well
==== Sagittal notation ====
[[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.
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
'''Pros:''' Easy to read, and less likely to clutter the score
'''Cons:''' Not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality
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:
[[File:24edo Sagittal.png|800px]]
=== Further discussion of 24edo notation ===
* [[24edo interval names and harmonies]]
* [[24edo Chord Names]]
* [[Ups and Downs Notation#Chords and Chord Progressions]].
== Regular temperament properties ==
=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|23.5|24.5}}
{{Uniform map|edo=24}}


=== Commas ===
=== Commas ===
24edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 24 38 56 67 83 89 }}.  
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 }}.  


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
![[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
![[Monzo]]
! [[Monzo]]
![[Cent]]s
! [[Cent]]s
![[Color name]]
! [[Color name]]
!Name(s)
! Name(s)
|-
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
|-
|3
| 5
|[[531441/524288|(12 digits)]]
| [[648/625]]
|{{monzo| -19 12 }}
| {{monzo| 3 4 -4 }}
|23.46
| 62.57
|Lalawa
| Quadgu
|[[Pythagorean comma]]
| Diminished comma, greater diesis
|-
|-
|5
| 5
|[[648/625]]
| <abbr title="262144/253125">(12 digits)</abbr>
|{{monzo| 3 4 -4 }}
| {{monzo| 18 -4 -5 }}
|62.57
| 60.61
|Quadgu
| Saquingu
|Major diesis, diminished domma
| [[Passion comma]]
|-
|-
| 5
| 5
|[[128/125]]
| [[128/125]]
|{{monzo| 7 0 -3 }}
| {{monzo| 7 0 -3 }}
|41.06
| 41.06
|Trigu
| Trigu
|Diesis, augmented comma
| Augmented comma, lesser diesis
|-
|-
|5
| 5
|[[81/80]]
| [[81/80]]
|{{monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
| 21.51
| 21.51
|Gu
| Gu
|Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
|5
| 5
|[[2048/2025]]
| [[2048/2025]]
|{{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
|19.55
| 19.55
| Sagugu
| Sagugu
|Diaschisma
| Diaschisma
|-
|-
|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
|[[Misty comma]]
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
|-
| 5
| 5
|[[32805/32768]]
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
|{{monzo| -15 8 1 }}
| {{monzo| 161 -84 -12 }}
|1.95
| 0.02
|Layo
| Sepbisa-quadbigu
|Schisma
| [[Kirnberger's atom]]
|-
|-
|5
| 7
|<abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| [[1323/1280]]
|{{monzo| 161 -84 -12 }}
| {{monzo| -8 3 -1 2 }}
|0.02
| 57.20
|Sepbisa-quadbigu
| Lazozogu
|[[Atom]]
| Septimal two-seventh tone
|-
|-
|7
| 7
|[[49/48]]
| [[49/48]]
|{{monzo| -4 -1 0 2 }}
| {{monzo| -4 -1 0 2 }}
|35.70
| 35.70
|Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
|7
| 7
|[[245/243]]
| [[245/243]]
|{{monzo| 0 -5 1 2 }}
| {{monzo| 0 -5 1 2 }}
|14.19
| 14.19
|Zozoyo
| Zozoyo
|Sensamagic
| Sensamagic comma
|-
|-
|7
| 7
|[[19683/19600]]
| [[19683/19600]]
|{{monzo| -4 9 -2 -2 }}
| {{monzo| -4 9 -2 -2 }}
|7.32
| 7.32
| Labirugu
| Labirugu
| Cataharry
| Cataharry comma
|-
|-
|7
| 7
|[[6144/6125]]
| [[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
| {{monzo| 11 1 -3 -2 }}
|5.36
| 5.36
|Sarurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|-
|-
|11
| 11
|[[121/120]]
| [[56/55]]
|{{monzo| -3 -1 -1 0 2 }}
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
|Lologu
| Lologu
| Biyatisma
| Biyatisma
|-
|-
|11
| 11
|[[176/175]]
| [[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
|9.86
| 9.86
| Lorugugu
| Lorugugu
|Valinorsma
| Valinorsma
|-
|-
|11
| 11
|[[896/891]]
| [[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
| {{monzo| 7 -4 0 1 -1 }}
|9.69
| 9.69
|Saluzo
| Saluzo
|Pentacircle
| Pentacircle comma
|-
|-
|11
| 11
|[[243/242]]
| [[243/242]]
|{{monzo| -1 5 0 0 -2 }}
| {{monzo| -1 5 0 0 -2 }}
| 7.14
| 7.14
|Lulu
| Lulu
|Rastma
| Rastma
|-
|-
|11
| 11
|[[385/384]]
| <abbr title="214990848/214358881">(18 digits)</abbr>
|{{monzo| -7 -1 1 1 1 }}
| {{monzo| 15 8 0 0 -8 }}
|4.50
| 5.10
|Lozoyo
| Quadbilu
|Keenanisma
| [[Octatonic comma]]
|-
|-
|11
| 11
|[[9801/9800]]
| [[385/384]]
|{{monzo| -3 4 -2 -2 2 }}
| {{monzo| -7 -1 1 1 1 }}
|0.18
| 4.50
|Bilorugu
| Lozoyo
|Kalisma, Gauss' comma
| Keenanisma
|-
|-
|13
| 11
|[[91/90]]
| <abbr title="117440512/117406179">(18 digits)</abbr>
|{{monzo| -1 -2 -1 1 0 1 }}
| {{monzo| 24 -6 0 1 -5 }}
|19.13
| 0.51
| Saquinlu-azo
| [[Quartisma]]
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 11
| <abbr title="1771561/1771470">(14 digits)</abbr>
| {{monzo| -1 -11 -1 0 6 }}
| 0.089
| Satribilo-agu
| [[Parimo]]
|-
| 13
| [[66/65]]
| {{monzo| 1 1 -1 0 1 -1 }}
| 26.43
| Thulogu
| Winmeanma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Thozogu
|Superleap
| Superleap comma, biome comma
|-
| 13
| [[512/507]]
| {{monzo| 9 -1 0 0 0 -2 }}
| 16.99
| Thuthu
| Tridecimal neutral thirds comma
|-
| 13
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| Thuzoyo
| Animist comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Thorugugu
| Ratwolfsma
|-
| 13
| [[352/351]]
| {{monzo| 5 -3 0 0 1 -1 }}
| 4.93
| Thulo
| Minor minthma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[170/169]]
| {{monzo| 1 0 1 0 0 -2 1 }}
| 10.21
| Sothuthuyo
| Major naiadma
|-
| 17
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| Sotholugu
| Minor naiadma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[1225/1224]]
| {{monzo| -3 -2 2 2 0 0 -1 }}
| 1.41
| Subizoyo
| Noellisma
|-
| 19
| [[76/75]]
| {{monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.93
| Nogugu
| Large undevicesimal ninth tone
|-
| 19
| [[77/76]]
| {{monzo| -2 0 0 1 1 0 0 -1 }}
| 22.63
| Nulozo
| Small undevicesimal ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[133/132]]
| {{monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.07
| Noluzo
| Minithirdma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
|-
|13
| 19
|[[676/675]]
| [[209/208]]
|{{monzo| 2 -3 -2 0 0 2 }}
| {{monzo| -4 0 0 0 1 -1 0 1 }}
|2.56
| 8.30
|Bithogu
| Nothulo
|Island comma, parizeksma
| Yama comma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|-
| 19
| [[5776/5775]]
| {{monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.30
| Nonolurugugu
| Neovish comma
|}
|}
<references />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
*[[List of 24et rank two temperaments by badness]]
* [[List of 24et rank two temperaments by badness]]
*[[List of edo-distinct 24et rank two temperaments]]
* [[List of edo-distinct 24et rank two temperaments]]
 
Important MOSes include:


*semaphore 4L1s 55455 (generator: 5\24)
{| class="wikitable center-1 center-2"
*semaphore 5L4s 414144141 (generator: 5\24)
*mohajira 3L4s 3434343 (generator: 7\24)
*mohajira 7L3s 3313313313 (generator: 7\24)
 
{| class="wikitable right-2"
|-
|-
!Periods per octave
! Periods<br>per 8ve
!Generator
! Generator
!Name
! Name
|-
|-
|1
| 1
|1\24
| 1\24
|
| [[Hemiripple]] (24)
|-
|-
|1
| 1
|5\24
| 5\24
|[[Semaphore_and_Godzilla|Semaphore/godzilla]] / [[Chromatic_pairs#Bridgetown|Bridgetown]]
| [[Godzilla]] (24) / [[baragon]] (24) / [[semaphoresmic clan #Varan|varan]] (24)
|-
|-
|1
| 1
|7\24
| 7\24
|[[Mohajira]] (or [[maqamic]] with 24d val)
| [[Mohajira]] (24) / [[neutrominant]] (24d) / [[migration]] (24d)
|-
|-
| 1
| 1
| 11\24
| 11\24
|[[Chromatic pairs#Barton|Barton]]
| [[Cohemiripple]] (24), [[freivald]] (24)
|-
|-
|2
| 2
| 1\24
| 1\24
|[[Shrutar]]
| [[Shrutar]] (24)
|-
|-
|2
| 2
| 5\24
| 5\24
|[[Sruti]], [[Anguirus]], [[Decimal]]
| [[Sruti]] (24), [[anguirus]] (24), [[decimal]] (24c)
|-
|-
|3
| 3
|1\24
| 1\24
|[[Semiaug]]
| [[Hemiaug]] (24)
|-
|-
|3
| 3
|3\24
| 3\24
|[[Triforce]]
| [[Triforce]] (24)
|-
|-
|4
| 4
|1\24
| 1\24
|[[Dimipent_family#Hemidim|Hemidim]]
| [[Hemidim]] (24)
|-
|-
|6
| 6
|1\24
| 1\24
|[[Augmented_family#Hemisemiaug|Hemisemiaug]]
| [[Hemisemiaug]] (24)
|-
|-
|8
| 8
| 1\24
| 1\24
|[[Dimipent_family#Semidim|Semidim]]
| [[Semidim]] (24)
|-
|-
|12
| 12
|1\24
| 1\24
|[[Catler]]
| [[Catler]]
|}
|}


== Scales / modes ==
Important MOSes include:
''See [[24edo modes]].''
* 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 ==
== Tetrachords ==
Line 752: Line 1,035:


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


As for entirely new chords, the most obvious is the neutral or mid triad 0-7-14. However there are other options such as:
As for entirely new chords, there are three new fundamental options, giving five basic triads over 12edo's two:


* 0-9-14 (Ultra Triad or upmajor triad)
{| class="wikitable center-all"
* 0-5-14 (Infra Triad or downminor triad)
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental triads of 24edo
|-
! JI Chord
! Edosteps
! Notes of C Chord
! Written name
! Spoken name
|-
| 6:7:9, 26:30:39
| {{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
|-
| 10:12:15
| {{dash|0, 6, 14|hair}}
| {{dash|C, E♭, G|hair}}
| Cm, Cmin
| C minor
|-
| 18:22:27, 22:27:33
| 0-7-14
| {{dash|C, E{{demiflat2}}, G|hair}}
| C~, Cneu
| C neutral
|-
| 4:5:6
| {{dash|0, 8, 14|hair}}
| {{dash|C, E, G|hair}}
| C, Cmaj
| C, C major
|-
| 14:18:21, 10:13:15
| {{dash|0, 9, 14|hair}}
| {{dash|C, E{{demisharp2}}, G|hair}}
| C^<br>C({{demisharp2}}3), Cmaj({{demisharp2}}3)
| C ultramajor<br>C major semisharp-three
|}


These chords tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads. For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.  
These chords tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads. For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: {{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 barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.
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:
More good chords in 24edo:


* 0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
* {{dash|0, 4, 8, 11, 14|hair}} ("major" chord with a 9:8 and a 11:8 above the root)
* Its inversion, 0-3-6-10-14 ("minor")
* Its inversion, {{dash|0, 3, 6, 10, 14|hair}} ("minor")
* 0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 ([[godzilla]]), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[mohajira]]), a heptatonic scale close to several Arabic scales.)
* 0-5-10 (another kind of "neutral", splitting the fourth in two. The {{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:
William Lynch considers these as some possible good tetrads:
Line 773: Line 1,093:
[[File:Three_chords.PNG|alt=Three chords.PNG|Three chords.PNG]]
[[File:Three_chords.PNG|alt=Three chords.PNG|Three chords.PNG]]


{| class="wikitable"
{| 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
|-
| {{dash|0, 5, 14, 19|hair}}
| {{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
|-
| {{dash|0, 6, 14, 20|hair}}
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E♭, G, B♭|hair}}
| m7, min7
| Minor seven
|-
| {{dash|0, 7, 14, 21|hair}}
| {{dash|1, v3, 5, v7|hair}}
| {{dash|C, E{{demiflat2}}, G, B{{demiflat2}}|hair}}
| n7, neu7
| Neutral seven
| [[File:Neutral Tetrad on C.mp3]]
|-
| {{dash|0, 8, 14, 22|hair}}
| {{dash|1, b3, 5, b7|hair}}
| {{dash|C, E, G, B|hair}}
| maj7
| Major seven
|-
| {{dash|0, 8, 14, 22|hair}}
| {{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
|-
|-
!Chord name
| {{dash|0, 8, 14, 20|hair}}
!Degrees of 24edo
| {{dash|1, 3, 5, b7|hair}}
!Chord spelling
| {{dash|C, E, G, B♭|hair}}
!Audio example
| 7, dom7
| Dominant seven
|-
|-
|neutral
| {{dash|0, 8, 14, 19|hair}}
|0 7 14 21
| {{dash|1, 3, 5, vb7|hair}}
| 1 v3 5 v7
| {{dash|C, E, G, B{{sesquiflat2}}|hair}}
|[[File:Neutral Tetrad on C.mp3]]
| h7<br>7({{demiflat2}}7)
| Harmonic seven<br>Dominant 7 semiflat-seven
|-
|-
|arto
| {{dash|0, 5, 14, 20|hair}}
|0 5 14 20
| {{dash|1, vb3, 5, b7|hair}}
|1 vb3 5 b7
| {{dash|C, E{{sesquiflat2}}, G, B♭|hair}}
|[[File:arto tetrad on C.mp3]]
| min7({{demiflat2}}3)
| Arto<br>Minor seven semiflat-three
| [[File:arto tetrad on C.mp3]]
|-
|-
|tendo
| {{dash|0, 9, 14, 19|hair}}
| 0 9 14 19
| {{dash|1, ^3, 5, vb7|hair}}
|1 ^3 5 vb7
| {{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
|  
|}
|}


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


== Counterpoint ==
== Counterpoint ==
Line 802: Line 1,167:


Furthermore, in the same fashion, every sequence of intervals available in 12edo are reachable by equal contrary motion in 24edo.
Furthermore, in the same fashion, every sequence of intervals available in 12edo are reachable by equal contrary motion in 24edo.
{| class="wikitable"
[[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}}
|[[File:Strict-contrary-motion-24edo.png|left|frame|Every sequence of 12edo intervals are reachable by strict contrary motion in 24edo.]]
|}


== 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.


<div class="external-image-warning" style="background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block">
=== Guitar ===
External image: http://www.swordguitars.com/Sword_quartertone_stratsm.jpg<br>
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.
:<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
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:


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


== Music ==
While these are playable, the extra frets can make playing chords and navigating the fretboard significantly more challenging for [[12edo]] chords and scales.
{{Wikipedia|List of quarter tone pieces}}


; [[5 hideya]]
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.
* [https://www.youtube.com/watch?v=mYgHB0L96Gc Like Stairs]


; [[Bazil Müzik]]
[[File:Eastwood-guitars-phase-4-mt-2307179.jpg|500px]]
* [https://www.youtube.com/watch?v=_r30TcsRLRg Bye Dydimus]


; [[Nathan BeDell]]
=== Harp, Harpsichord, and Piano ===
* [http://micro.soonlabel.com/gene_ward_smith/Others/BeDell/Quarter%20Tone%20Prelude%20for%20two%20Harps.mp3 Quarter Tone Prelude For Two Harps]


; [[Dawson Berry]]
==== Scordatura tuning of 12edo instruments ====
* "Folly of a Drunk" ([[:File:Folly of a Drunk.mp3|mp3]]) ([[:File:Folly of a Drunk (Musescore 3).pdf|score]])


; [[User:Francium|Francium]]
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.
* [https://www.youtube.com/watch?v=pTflHXcCBKo A Microtonal Prelude]
* [https://www.youtube.com/watch?v=VFvENIKxZH4 Melancholie]
* [https://www.youtube.com/watch?v=VFvENIKxZH4 Reclusão]
* [https://www.youtube.com/watch?v=mkmb5gZtC2k Partyopfer]
* [https://www.youtube.com/watch?v=AhtiKa612tw Languid]
* [https://www.youtube.com/watch?v=cN8oOVW4HI8 An 80's Theme]
* [https://www.youtube.com/watch?v=rfXZ5Utn6ig Pendulum]
* [https://www.youtube.com/watch?v=vlxj3nd5NCI Relaxing Holidays]
* [https://www.youtube.com/watch?v=nt-Ls1v7ZAU Party At The Mysterious End]
* [https://www.youtube.com/watch?v=YvKMbOHaWmI Piano Duet No. 3 for two by a quarter tone detuned pianos]
* [https://www.youtube.com/watch?v=n-i8pN1hj7Y Schlepping]
* [https://www.youtube.com/watch?v=Vbmsnd63hFc Neutral Thirds Circle]
* [https://www.youtube.com/watch?v=AMhkr1UWXXk A Microtonal Invention]
* [https://www.youtube.com/watch?v=5HDJ1qKpbV8 When Mom Isn't Home]
* [https://www.youtube.com/watch?v=7U00_fnf5Rc wonky #3]
* [https://www.youtube.com/watch?v=AfImRjuPY_M Bizaram Az Dime Shoma]


; [[Jake Freivald]]
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] in the [[freivaldneutral24]] scale.
* [https://soundcloud.com/jdfreivald/mo-happy-happy Mo - Happy - Happy] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/mo-happy-happy.mp3 play] in Neutral[7] (2.3.11 mohajira), 24et tuning


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


; [[wikipedia:Georg_Friedrich_Haas|Georg Friedrich Haas]]
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):
* "3 Hommages"


; [[Fromental Halévy]]
; 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)
* "Prométhée enchaîné" (considered the first mainstream western orchestral composition to use quarter tones.)
* [https://www.youtube.com/shorts/Ieqi54XE2lI Demonstration short video by Nahre Sol] (2024)


; [[William Lynch]]
; 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=q1Lp8AtKK9o Autumn Winds], [https://www.youtube.com/watch?v=Igxe3DwbFJ4&feature=c4-overview&list=UUvq5bg-LvOS6adpB5efPTyQ Easter Time at Nine], [https://www.youtube.com/watch?v=z_3uhA9Cq08&list=UUvq5bg-LvOS6adpB5efPTyQ Waters of Persia] in mohajira, 24et tuning.
* [https://www.youtube.com/watch?v=PdP4epQIUrU Demonstration video by Steve Cohn] (2011)


; [[Mandrake]]
=== Electronic Keyboards ===
* [https://www.youtube.com/watch?v=M5jkXpa5q2s Pink Noise]


; [[Shaahin Mohajeri]]
24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see [[Lumatone mapping for 24edo]]
* <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>


; [[Joseph Monzo]]
=== Flute ===
* [https://m.youtube.com/watch?v=VwV9aLXElLA 24-equal Tune] (1994)
Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:


; [[norokusi]]
; Quarter-tone flute, made by Eva Kingma
* [https://www.youtube.com/watch?v=cJFwZ24SIZw Adagio for strings]
* [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=87MmRej2dJQ Route 24－1. Jupiter's rebuttal]
* [https://www.youtube.com/watch?v=35Vy722jvw8 Route 24－2. The Steel Holiday]
* [https://www.youtube.com/watch?v=IJIKCXEah4Y Serenade for Harp and Strings]
* [https://www.youtube.com/watch?v=ns3Vv9fFow0 Lyric Pieces for strings I. Romanza]
* [https://www.youtube.com/watch?v=keVfOQVPpdM Lyric Pieces for strings II. Cantabile]
* [https://www.youtube.com/watch?v=MtRrMYxEd6M Lyric Pieces for strings III. Elegie]
* [https://www.youtube.com/watch?v=KpFv0LnFsBQ Lyric Pieces for strings IV. Melancolia]
* [https://www.youtube.com/watch?v=y6LvsVEhFhM Lyric Pieces for strings V. Canon]
* [https://www.youtube.com/watch?v=Qkw-4-289bA Lyric Pieces for strings VI. Song of Summer]
* [https://www.youtube.com/watch?v=l9dWATsJDWc Menuett for String Orchestra n. 1007]
* [https://www.youtube.com/watch?v=wH26-Z3nR0I Lyric Pieces for strings Book 2 I. Autumn]
* [https://www.youtube.com/watch?v=VAScKV4iXEQ Lyric Pieces for strings Book 2 II. Dry Leaves]
* [https://www.youtube.com/watch?v=pSLmBDrcOGU Lyric Pieces for strings Book 2 III. The Farewell]
* [https://www.youtube.com/watch?v=HxdnurdWNt0 Lyric Pieces for strings Book 2 IV. Eclogue]
* [https://www.youtube.com/watch?v=jtKt4CgWnm8 Lyric Pieces for strings Book 2 V. Silent Woods]
* [https://www.youtube.com/watch?v=JPvL0nLC6sk Lyric Pieces for strings Book 2 VI. Nocturne]


; [[Chris Vaisvil]]
== Music ==
* <span style="">[http://micro.soonlabel.com/24et/quarterpicnic.mp3 Quarterpicnic]</span>
{{Wikipedia|List of quarter tone pieces}}
* <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>
{{Main|{{ROOTPAGENAME}}/Music}}{{clear}}
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=qUmcQbGPZGg Stygian Gates]
 
; Yeah Gore
* [https://www.youtube.com/watch?v=goIEjHjQ8UE Headaches plus nausea] (2021)
 
; [[Zewen Senpai]]
* [https://www.youtube.com/watch?v=icLPCbYbJtQ A Short Walk At Twilight]
* [https://www.youtube.com/watch?v=0VQahowP0Wk Simple Folk Etude]


== 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 ==
* [[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] [https://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] by Shaahin Mohajeri [https://www.webcitation.org/5xeFBNdQW Permalink]
* [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
* [http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]  
* [http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]
* [http://tonalsoft.com/enc/q/quarter-tone.aspx Tonalsoft Encyclopedia | ''quarter-tone / 24-edo / 24-ed2'']
 
{{Todo| cleanup }}
 
== Notes ==
<references group="note" />


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[[Category:Godzilla]]
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Quartertone]]
[[Category:Quartertone]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Subgroup]]
[[Category:Subgroup temperaments]]
 
[[Category:Twentuning]]
[[Category:Twentuning]]
{{Todo| cleanup }}
[[Category:Lists of intervals]]
[[Category:Lists of intervals]]
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