Xenharmonic Wiki

24edo: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,215 edits
→Brass: Add Schilke B5 quartertone trumpet
 
(149 intermediate revisions by 19 users not shown)
Line 6: Line 6:
}}
}}
{{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 music|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, so 24edo offers nothing new as far as approximating the 5-limit is concerned.
24edo/24-TET, also known as the quarter-tone system, is the double of [[12edo|12edo/12-TET]], so it contains all of the notes of 12edo. It adds to 12edo another circle of it spaced a quarter tone apart, which contains unfamiliar intervals not found in 12edo, such as neutral seconds and thirds. Since it contains 12edo, it is very desirable for microtonalists who want new intervals while still having access to familiar ones.


The 7th harmonic and its intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as inaccurate 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, 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. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].  
The [[5-limit]] approximations in 24edo are the same as those in 12edo, tempering out [[81/80]], [[128/125]], [[648/625]], and [[531441/524288]], so 24edo offers nothing new as far as approximating the 5-limit is concerned. However, it maps the [[7/1|7th harmonic]] differently from 12edo, with [[7/4]] mapped to 950{{c}} rather than 1000{{c}} in 12edo, being 18.8{{c}} flat of just rather than 31.2{{c}} sharp in 12edo. Most intervals of 7 are still approximated quite poorly for its size, though chords like [[6:7:9]] are nonetheless closer to just than in 12edo. Still, if one wishes to approximate intervals of 7 while still having access to the notes of 12edo, it is best to use finer divisions like [[36edo]], [[48edo]], [[72edo]], or [[84edo]].


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 24edo as a 2.3.11.17.19 [[subgroup]] temperament, on which it is quite accurate.  
However, 24edo approximates the [[11/1|11th harmonic]] very accurately at 550{{c}}, only 1.3{{c}} flat of just. Most intervals of 11, such as [[11/8]], [[11/6]], [[11/10]], and [[11/9]], are approximated accurately as well. It is thus usable as an [[2.3.11 subgroup|2.3.11-]] or [[2.3.5.11 subgroup|2.3.5.11-]][[subgroup]] system, notably tempering out [[121/120]], splitting [[6/5]] into two neutral seconds of {{nowrap|[[11/10]][[~]][[12/11]]}}, and [[243/242]], splitting [[3/2]] into two 11/9 neutral thirds. It also has a decent approximation of the [[13/1|13th harmonic]] at 850{{c}}, being 9.5{{c}} sharp of just. Intervals of 13 are thus represented decently, with [[13/10]], [[15/13]], and their [[octave complement]]s being especially close to just due to the cancellation of the sharpness of harmonics 5 and 13. It is thus a good tuning for the 2.3.5.11.13 and 2.3.11.13/5 subgroups, tempering out [[144/143]] in the former, so that [[11/9]] and [[16/13]] are equated, and [[676/675]] in both subgroups, so two 15/13's add up to [[4/3]]. Finally, 24edo shares its tunings of harmonics [[17/1|17]] and [[19/1|19]] with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly. Nonetheless, it is not the best system for approximating JI if one wishes to use prime 7, with other equal temperaments like [[22edo]], [[27edo]], and especially [[31edo]] being more accurate.
 
Aside from harmony, it also preserves the melodic resources of 12edo, containing minor and major seconds and thirds. However, it adds several new intervals, including neutral seconds and thirds, so new melodies can be written in 24edo that aren't possible in 12edo. This also means 24edo contains new scales, most notably the "neutral diatonic" [[3L 4s]] [[MOS]] with step pattern LssLsLs, where ''L'' is a major second and ''s'' is a neutral second. These scales also contain chords unfamiliar to 12edo, such as the [[neutral tetrad]].
 
While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically, though it equates [[7/6]] with [[8/7]] due to vanishing of [[49/48]], leading to [[semaphore]]. Nonetheless, scales of semaphore are quite interesting, especially the 9-note [[5L 4s]] MOS. A supermajor chord is available as [0 9 14], and a subminor chord as [0 5 14]; however, they are better described as ultramajor and inframinor, being interpreted much more accurately as [[10:13:15]] and [[26:30:39|1/(10:13:15)]] respectively, the corresponding temperament being [[barbados]], the 2.3.13/5 temperament tempering out 676/675. These chords are relatively simple and may serve as alternatives to the regular [[4:5:6]] and [[10:12:15|1/(4:5:6)]] triads as bases for harmony; see [[Extraclassical tonality]].
 
A notable superset of 24edo is [[72edo]], which has good approximations up to the [[19-limit]], and especially the [[11-limit]]. The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17.19, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. One will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.


=== Prime harmonics ===
=== 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 nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}.
24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}. Some of its supersets, most notably [[72edo]] and [[96edo]], have been used by a variety of composers.


== Notation ==
=== Miscellaneous properties ===
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.
Its step, at 50{{c}}, is notable for being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). Intervals less than 40{{c}} tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60{{c}} are often perceived as having less "tension", and thus are also considered to be more consonant.


=== Ups and down notation ===
== Intervals ==
 
{| class="wikitable center-all left-3"
{| class="wikitable center-all"
|-
|-
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref>based on treating 24edo 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)
! colspan="3" | [[SKULO interval names|SKULO notation]] (U or S = 1)
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}
! [[24edo solfege|Solfege]]
! [[24edo solfege|Solfege]]
|-
|-
| 0
| 0
| 0
| 0
| 1/1
| [[1/1]]
| P1
| P1
| unison
| unison
Line 53: Line 58:
| 1
| 1
| 50
| 50
| 33/32, 34/33
| [[33/32]], [[34/33]]
| ^P1, vm2
| ^P1, vm2
| up-unison, downminor 2nd
| up-unison, downminor 2nd
Line 64: Line 69:
| 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
Line 75: Line 80:
| 3
| 3
| 150
| 150
| 12/11
| [[13/12]], [[12/11]], [[11/10]]
| ~2
| ~2
| mid 2nd
| mid 2nd
Line 86: Line 91:
| 4
| 4
| 200
| 200
| 9/8
| [[9/8]], [[10/9]]
| M2
| M2
| major 2nd
| major 2nd
Line 97: Line 102:
| 5
| 5
| 250
| 250
| 22/19
| [[15/13]], [[22/19]]
| ^M2, vm3
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
Line 108: Line 113:
| 6
| 6
| 300
| 300
| 19/16
| [[6/5]], [[13/11]], [[19/16]]
| m3
| m3
| minor 3rd
| minor 3rd
Line 119: Line 124:
| 7
| 7
| 350
| 350
| 11/9, 27/22
| [[11/9]], [[16/13]], [[27/22]], [[39/32]]
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 130: Line 135:
| 8
| 8
| 400
| 400
| 24/19
| [[5/4]], [[24/19]]
| M3
| M3
| major 3rd
| major 3rd
Line 141: Line 146:
| 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 152: Line 157:
| 10
| 10
| 500
| 500
| 4/3
| [[4/3]]
| P4
| P4
| fourth
| fourth
Line 163: Line 168:
| 11
| 11
| 550
| 550
| 11/8
| [[11/8]], [[15/11]]
| ^4, ~4
| ^4, ~4
| up-4th, mid-4th
| up-4th, mid-4th
Line 174: Line 179:
| 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 185: Line 190:
| 13
| 13
| 650
| 650
| 16/11
| [[16/11]], [[22/15]]
| v5, ~5
| v5, ~5
| down-5th, mid-5th
| down-5th, mid-5th
Line 196: Line 201:
| 14
| 14
| 700
| 700
| 3/2
| [[3/2]]
| P5
| P5
| fifth
| fifth
Line 207: Line 212:
| 15
| 15
| 750
| 750
| 17/11
| [[17/11]], [[20/13]]
| ^5, vm6
| ^5, vm6
| up-fifth, downminor 6th
| up-fifth, downminor 6th
Line 218: Line 223:
| 16
| 16
| 800
| 800
| 19/12
| [[8/5]], [[19/12]]
| m6
| m6
| minor 6th
| minor 6th
Line 229: Line 234:
| 17
| 17
| 850
| 850
| 18/11, 44/27
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| ~6
| ~6
| mid 6th
| mid 6th
Line 240: Line 245:
| 18
| 18
| 900
| 900
| 32/19
| [[5/3]], [[22/13]], [[32/19]]
| M6
| M6
| major 6th
| major 6th
Line 251: Line 256:
| 19
| 19
| 950
| 950
| 19/11
| [[19/11]], [[26/15]]
| ^M6, vm7
| ^M6, vm7
| upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
Line 262: Line 267:
| 20
| 20
| 1000
| 1000
| 16/9
| [[9/5]], [[16/9]]
| m7
| m7
| minor 7th
| minor 7th
Line 273: Line 278:
| 21
| 21
| 1050
| 1050
| 11/6
| [[11/6]], [[20/11]]
| ~7
| ~7
| mid 7th
| mid 7th
Line 284: Line 289:
| 22
| 22
| 1100
| 1100
| 17/9, 32/17
| [[15/8]], [[17/9]], [[32/17]]
| M7
| M7
| major 7th
| major 7th
Line 295: Line 300:
| 23
| 23
| 1150
| 1150
| 33/17, 64/33
| [[33/17]], [[64/33]]
| ^M7, vP8
| ^M7, vP8
| upmajor 7th, down-8ve
| upmajor 7th, down-8ve
Line 306: Line 311:
| 24
| 24
| 1200
| 1200
| 2/1
| [[2/1]]
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 315: Line 320:
| Do
| Do
|}
|}
<references group="note" />
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.
{{Ups and downs sharpness}}
=== 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|40px|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"
|-
| style="width: 40px;" | [[File:Koron_sign.svg|39px|center]]
| '''Koron''' = quarter-tone flat
|-
| style="width: 40px;" | [[File:Sori_sign.svg|39px|center]]
| '''Sori''' = quarter-tone sharp
|}
'''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.


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


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.
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 qualities in color notation ===
== 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:


Line 326: Line 422:
|-
|-
! Quality
! Quality
! [[Color name|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 383: Line 479:
|-
|-
| 50
| 50
| quartertone, infra second, wide unison
| Quarter tone, infra second, wide unison
|-
|-
| 150
| 150
| neutral second
| Neutral second
|-
|-
| 250
| 250
| ultra second, infra third
| Ultra second, infra third
|-
|-
| 350
| 350
| neutral third
| Neutral third
|-
|-
| 450
| 450
| minor fourth, ultra third, narrow fourth
| Minor fourth, ultra third, narrow fourth
|-
|-
| 550
| 550
| wide fourth
| Wide fourth
|-
|-
| 650
| 650
| narrow fifth
| Narrow fifth
|-
|-
| 750
| 750
| wide fifth, infra sixth
| Wide fifth, infra sixth
|-
|-
| 850
| 850
| neutral sixth
| Neutral sixth
|-
|-
| 950
| 950
| ultra sixth , infra seventh
| Ultra sixth, infra seventh
|-
|-
| 1050
| 1050
| neutral seventh
| Neutral seventh
|-
|-
| 1150
| 1150
| ultra seventh, narrow octave
| Ultra seventh, narrow octave
|}
|}


=== 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 446: Line 542:
* 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 }}
 
* [[24edo Chord Names]]
* [[Ups and downs notation#Chords and Chord Progressions]].
 
== Approximation to JI ==
[[File:24ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]


==== Mainstream quartertone notation ====
=== Interval mappings ===
[[Category:IMPORTDEBUG - Change External Images]]
{{Q-odd-limit intervals|24}}


{| class="wikitable center-all"
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| width="40px" |[[File:HeQu1.svg|21px|center]]
! rowspan="2" | [[Subgroup]]
| A "semisharp" accidental comprising one half of a regular musical sharp symbol.
! rowspan="2" | [[Comma list]]
|}
! rowspan="2" | [[Mapping]]
 
! rowspan="2" | Optimal<br>8ve stretch (¢)
{| class="wikitable center-all"
! colspan="2" | Tuning error
|-
|-
| width="40px" |[[File:HeQu3.svg|30px|center]]
! [[TE error|Absolute]] (¢)
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
! [[TE simple badness|Relative]] (%)
|}
 
{| class="wikitable center-all"
|-
|-
| width="40px" |[[File:HeQd1.svg|22px|center]]
| 2.3.5.11
| A "semiflat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
| 81/80, 121/120, 128/125
|}
| {{mapping| 24 38 56 83 }}
 
| −1.08
{| class="wikitable center-all"
| 2.82
| 5.63
|-
|-
| width="40px" |[[File:HeQd3.svg|36px|center]]
| 2.3.5.11.13
| A "flat and a half" or "sesquiflat" accidental, comprising the above half-flat symbol and a regular flat symbol placed back to back.
| 66/65, 81/80, 128/125, 144/143
|}
| {{mapping| 24 38 56 83 89 }}
 
| −1.37
'''Pros:''' Familiar, fairly easy to learn
| 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]]
[[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"
|-
|-
| width="40px" | [[File:200px-Koron_sign.svg.png|39px|center]]
| 2.3.5.11.13.17
| '''Koron''' ([[wikipedia:Koron_(music)|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
|-
|-
| width="40px" | [[File:200px-Sori_sign.svg.png|39px|center]]
| 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
=== Uniform maps ===
 
{{Uniform map|edo=24}}
'''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:
=== 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 }}.  
[[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==
[[File:24ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]
===Uniform maps===
{{Uniform map|13|23.5|24.5}}
 
=== Commas===
This is a partial list of the [[commas]] that 24edo [[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
Line 557: Line 622:
| 62.57
| 62.57
| Quadgu
| Quadgu
| Greater diesis, diminished comma
| Diminished comma, greater diesis
|-
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
|-
| 5
| 5
Line 564: Line 636:
| 41.06
| 41.06
| Trigu
| Trigu
| Lesser diesis, augmented comma
| Augmented comma, lesser diesis
|-
|-
| 5
| 5
Line 571: Line 643:
| 21.51
| 21.51
| Gu
| Gu
| Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| 5
| 5
Line 613: Line 685:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 642: Line 714:
| Luzogu
| Luzogu
| Undecimal tritonic comma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
|-
| 11
| 11
Line 670: Line 749:
| Lulu
| Lulu
| Rastma
| Rastma
|-
| 11
| <abbr title="214990848/214358881">(18 digits)</abbr>
| {{monzo| 15 8 0 0 -8 }}
| 5.10
| Quadbilu
| [[Octatonic comma]]
|-
|-
| 11
| 11
Line 691: Line 777:
| Bilorugu
| Bilorugu
| Kalisma, Gauss' comma
| Kalisma, Gauss' comma
|-
| 11
| <abbr title="1771561/1771470">(14 digits)</abbr>
| {{monzo| -1 -11 -1 0 6 }}
| 0.089
| Satribilo-agu
| [[Parimo]]
|-
|-
| 13
| 13
Line 704: Line 797:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 726: Line 819:
| Thulu
| Thulu
| Grossma
| 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
| 13
Line 733: Line 840:
| Bithogu
| Bithogu
| Island comma, parizeksma
| Island comma, parizeksma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
|-
| 17
| 17
Line 775: Line 896:
| Subizoyo
| Subizoyo
| Noellisma
| 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
| 19
Line 781: Line 916:
| 18.13
| 18.13
| Nugu
| Nugu
| 19th Partial chroma
| 19th-partial chroma
|-
|-
| 19
| 19
Line 789: Line 924:
| Noluzo
| Noluzo
| Minithirdma
| 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
|-
|-
| 19
| 19
Line 810: Line 959:
| Nonogu
| Nonogu
| Go comma
| Go comma
|-
| 19
| [[5776/5775]]
| {{monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.30
| Nonolurugugu
| Neovish comma
|}
|}
<references />
<references group="note" />
 
===Rank-2 temperaments ===
*[[List of 24et rank two temperaments by badness]]
*[[List of edo-distinct 24et rank two temperaments]]
 
Important MOSes include:


*semaphore 4L1s 55455 (generator: 5\24)
=== Rank-2 temperaments ===
*semaphore 5L4s 414144141 (generator: 5\24)
* [[List of 24et rank two temperaments by badness]]
*mohajira 3L4s 3434343 (generator: 7\24)
* [[List of edo-distinct 24et rank two temperaments]]
*mohajira 7L3s 3313313313 (generator: 7\24)


{| class="wikitable right-2"
{| class="wikitable center-1 center-2"
|-
|-
! Periods per octave
! Periods<br>per 8ve
! Generator
! Generator
! Name
! Name
Line 832: Line 981:
| 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]] (patent val), [[Neutrominant]] (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
Line 879: Line 1,028:
|}
|}


==Scales / modes==
Important MOSes include:
''See [[24edo scales]].''
* Semaphore 4L&nbsp;1s 55455 (generator: 5\24)
* Semaphore 5L&nbsp;4s 414141414 (generator: 5\24)
* Mohajira 3L&nbsp;4s 3434343 (generator: 7\24)
* Mohajira 7L&nbsp;3s 3313313313 (generator: 7\24)


==Tetrachords==
== Octave stretch or compression ==
If one wishes to use 24edo as a full 19-or-lower-limit tuning, then it benefits from slight [[octave stretching]], mostly to improve its [[prime]] 7. If one wishes to use 24edo as a no-7s 19-or-lower-limit tuning, then it benefits from slight [[octave shrinking]], mostly to improve its primes 5 and 13.
* Stretched-octave tunings (least to most stretch): [[ed12|86ed12]], [[ed6|62ed6]], [[38edt]]
* Compressed-octave tunings (least to most compression): [[zpi|90zpi]], [[equal tuning|80ed10]], [[ed5|56ed5]]
 
== Scales and modes ==
''See: [[24edo scales]] and [[List of MOS scales in 24edo]].''
 
== Tetrachords ==
''See [[24edo tetrachords]]''.
''See [[24edo tetrachords]]''.


==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, there are three new fundamental options, giving five basic triads over 12edo's three:
As for entirely new chords, there are three new fundamental options, giving five basic triads over 12edo's two:


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="white-space: nowrap; font-size: 105%;" | Fundamental triads of 24edo
|-
|-
! JI Chord
! JI Chord
Line 899: Line 1,060:
|-
|-
| 6:7:9, 26:30:39
| 6:7:9, 26:30:39
| 0–5–14
| {{dash|0, 5, 14|hair}}
| C–E{{sesquiflat2}}–G
| {{dash|C, E{{sesquiflat2}}, G|hair}}
| Cvm
| Cvm<br>Cm({{demiflat2}}3), Cmin({{demiflat2}}3)
| C subminor
| C inframinor<br>C minor semiflat-three
|-
|-
| 10:12:15
| 10:12:15
| 0–6–14
| {{dash|0, 6, 14|hair}}
| C–E♭–G
| {{dash|C, E♭, G|hair}}
| Cm
| Cm, Cmin
| C minor
| C minor
|-
|-
| 18:22:27, 22:27:33
| 18:22:27, 22:27:33
| 0-7-14
| 0-7-14
| C–E{{demiflat2}}–G
| {{dash|C, E{{demiflat2}}, G|hair}}
| C~
| C~, Cneu
| C neutral
| C neutral
|-
|-
| 4:5:6
| 4:5:6
| 0–8–14
| {{dash|0, 8, 14|hair}}
| C–E–G
| {{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
| 0–9–14
| {{dash|0, 9, 14|hair}}
| C–E{{demisharp2}}–G
| {{dash|C, E{{demisharp2}}, G|hair}}
| C^
| C^<br>C({{demisharp2}}3), Cmaj({{demisharp2}}3)
| C supermajor
| 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 943: Line 1,104:
[[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
|-
|-
! Chord name
! Degrees of 24edo
! Degrees of 24edo
! Chord spelling
! Chord spelling
! Notes of C chord
! Written name
! Spoken name
! Audio example
! Audio example
|-
|-
| neutral
| {{dash|0, 5, 14, 19|hair}}
| 0 7 14 21
| {{dash|1, vb3, 5, vb7|hair}}
| 1 v3 5 v7
| {{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]]
| [[File:Neutral Tetrad on C.mp3]]
|-
|-
| arto
| {{dash|0, 8, 14, 22|hair}}
| 0 5 14 20
| {{dash|1, b3, 5, b7|hair}}
| 1 vb3 5 b7
| {{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
|-
| {{dash|0, 8, 14, 20|hair}}
| {{dash|1, 3, 5, b7|hair}}
| {{dash|C, E, G, B♭|hair}}
| 7, dom7
| Dominant seven
|-
| {{dash|0, 8, 14, 19|hair}}
| {{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
|-
| {{dash|0, 5, 14, 20|hair}}
| {{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]]
| [[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 ==
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.
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.


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. [[File:24-EDO_Contrary_Motion.flac]]]]
 
|}
== 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.
 
=== 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.
 
Guitars with 24 frets per octave are also an option, although only [https://eastwoodguitars.com/products/hi-flier-edo-24-electric-microtonal-guitar Eastwood] offer this as a standard production model at the time of writing. Other luthiers you can commission custom microtonal instruments from, including 24edo ones, include [https://www.etsy.com/uk/listing/1154683769 JLJ instruments] and [https://meantoneguitar.com Meantone Guitar].
 
[[File:24edo_guitar.jpg|500px]]
 
While these are playable, the extra frets can make playing chords and navigating the fretboard significantly more challenging for [[12edo]] chords and scales.
 
More common is the "Sazocaster" tuning popularised by Australian band [[King Gizzard & the Lizard Wizard]], which adds quarter tones between approximately half the regular frets. Multiple guitar makers, including [https://eastwoodguitars.com/products/sg2c-flying-banana-mt Eastwood] and [https://salamuzik.com/collections/guitar/products/professional-microtonal-electric-classical-guitar-with-equalizer-kg-5 Sala Muzik] have produced Sazocaster variations.
 
[[File:Eastwood-guitars-phase-4-mt-2307179.jpg|500px]]
 
It is also possible to create a guitar that has quartertones on all the lower frets, then switches to regular 12edo at some point on the neck to keep the upper notes easily acessible, as demonstrated by the band [[Angine de Poitrine]]. Guitars using this layout are available at [https://www.microtonalguitar.org/product-page/angine-de-poitrine-style-fixed-microtonal-electric-guitar-ap24 microtonalguitar.org] and doubleneck guitar/basses are available from [https://eastwoodguitars.com/products/eastwood-microtonal-doubleneck-electric-guitar-bass Eastwood].
 
=== Harp, Harpsichord, and Piano ===
 
==== Scordatura tuning of 12edo instruments ====
 
Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface tuning]]. Iceface tuning is one type of scordatura piano (or other keyboard instrument) tuning. A simpler version of this (fewer notes retuned) is demonstrated in [https://www.youtube.com/watch?v=KS-mmj5kuxw ''when it blooms (24edo)''] (2021). A more complex type of [[Wikipedia:scordatura|scordatura]] tuning was required for a performance of Charles Ives' 4th Symphony which calls for a quarter-tone piano, but for which no quarter-tone piano was available, as described by Thomas Broadhead in [https://www.youtube.com/watch?v=T1G2XFVtnXU this video]. For this composition the gamut of notes needed would not be met using a simple transformation such as Iceface.
 
Although no recording using the above tuning is currently legally freely available, [[Paweł Mykietyn]] has used a similar idea with harp and harpsichord. A score video of this is available as [https://www.youtube.com/watch?v=_7o0uwPrYas ''Klave for Microtonal Harpsichord and Chamber Orchestra (Score-Video)''] (2004, performed by Elżbieta Chojnacka with Marek Moś conducting the AUKSO chamber orchestra of the city of Tychy, uploaded by Quinone Bob with permission from Paweł Mykietyn); the video starts with slides explaining the scordatura tuning of each manual of the Revival harpsichord (with each manual having a different scordatura tuning), followed by the scordatura tuning of the harp.
 
==== Quarter-tone instruments ====
 
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)
* [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)
* [https://www.youtube.com/watch?v=PdP4epQIUrU Demonstration video by Steve Cohn] (2011)


== Instruments==
=== Electronic Keyboards ===
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">
24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see [[Lumatone mapping for 24edo]]
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
=== Flute ===
Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:


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]].
; 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)


24edo can also be played on the Lumatone: see [[Lumatone mapping for 24edo]]
=== Brass ===
Since the trombone is a free-pitched instrument, playing quartertones, or any other edo simply requires increased precision in moving the slide. If you want a brass instrument with fixed steps, [https://www.a-courtois.com/en/instruments/trompettes-2/t-o-m-a Courtois] and [https://www.vanlaartrumpets.nl/en/trumpets/quartertone Van Laar] both produce trumpets with an additional valve that enable you to easily play quartertones. In addition, {{W|Renold Schilke|Schilke Music Products Incorporated}} built quartertone trumpets (model B5), as shown in All Things Brass And Technology's [https://www.youtube.com/watch?v=1ip0lOlQ2Xo&list=WL&index=257 ''Schilke B5 Quartertone Trumpet from 1971''] (2023).


==Music==
== Music ==
{{Wikipedia|List of quarter tone pieces}}
{{Wikipedia|List of quarter tone pieces}}
{{Main|Music in 24edo}}
{{Main|{{ROOTPAGENAME}}/Music}}{{clear}}


== 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)
**[[96edo]] (3mu tuning)
** [[96edo]] (3mu tuning)
**[[192edo]] (4mu tuning)
** [[192edo]] (4mu tuning)
**[[384edo]] (5mu tuning)
** [[384edo]] (5mu tuning)
**[[768edo]] (6mu tuning)
** [[768edo]] (6mu tuning)
**[[1536edo]] (7mu tuning)
** [[1536edo]] (7mu tuning)
**[[3072edo]] (8mu tuning)
** [[3072edo]] (8mu tuning)
**[[6144edo]] (9mu tuning)
** [[6144edo]] (9mu tuning)
**[[12288edo]] (10mu tuning)
** [[12288edo]] (10mu tuning)
**[[24576edo]] (11mu tuning)
** [[24576edo]] (11mu tuning)
**[[49152edo]] (12mu tuning)
** [[49152edo]] (12mu tuning)
**[[98304edo]] (13mu tuning)
** [[98304edo]] (13mu tuning)
**[[196608edo]] (14mu tuning)
** [[196608edo]] (14mu tuning)


== External links ==
* [http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo / 24-ed2] [https://www.webcitation.org/5xeFMH6cd Permalink] on [[Tonalsoft Encyclopedia]]
* [http://www.96edo.com/24_EDO.html About 24-EDO] [https://www.webcitation.org/5xeFBNdQW Permalink] by Shaahin Mohajeri
* [https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
* [http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]
* [http://tonalsoft.com/enc/q/quarter-tone.aspx Tonalsoft Encyclopedia | ''quarter-tone / 24-edo / 24-ed2'']


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


[[Category:Semaphore]]
[[Category:Godzilla]]
[[Category:Meantone]]
[[Category:Quartertone]]
[[Category:Quartertone]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Twentuning]]
[[Category:Twentuning]]
{{Todo| cleanup }}
[[Category:Lists of intervals]]
Retrieved from "https://en.xen.wiki/w/24edo"