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== 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]], [[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.  
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]].


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.
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.


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.
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]].


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.
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 ===
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=== 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.
 
=== Miscellaneous properties ===
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.


== Intervals ==
== Intervals ==
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| 0
| 0
| 0
| 0
| 1/1
| [[1/1]]
| P1
| P1
| unison
| unison
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| 1
| 1
| 50
| 50
| 33/32, 34/33
| [[33/32]], [[34/33]]
| ^P1, vm2
| ^P1, vm2
| up-unison, downminor 2nd
| up-unison, downminor 2nd
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| 2
| 2
| 100
| 100
| 16/15, 17/16, 18/17, 19/18
| [[16/15]], [[17/16]], [[18/17]], [[19/18]]
| A1, m2
| A1, m2
| aug unison, minor 2nd
| aug unison, minor 2nd
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| 3
| 3
| 150
| 150
| 13/12, 12/11, 11/10
| [[13/12]], [[12/11]], [[11/10]]
| ~2
| ~2
| mid 2nd
| mid 2nd
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| 4
| 4
| 200
| 200
| 9/8, 10/9
| [[9/8]], [[10/9]]
| M2
| M2
| major 2nd
| major 2nd
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| 5
| 5
| 250
| 250
| 15/13, 22/19
| [[15/13]], [[22/19]]
| ^M2, vm3
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
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| 6
| 6
| 300
| 300
| 6/5, 13/11, 19/16
| [[6/5]], [[13/11]], [[19/16]]
| m3
| m3
| minor 3rd
| minor 3rd
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| 7
| 7
| 350
| 350
| 11/9, 16/13, 27/22, 39/32
| [[11/9]], [[16/13]], [[27/22]], [[39/32]]
| ~3
| ~3
| mid 3rd
| mid 3rd
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| 8
| 8
| 400
| 400
| 5/4, 24/19
| [[5/4]], [[24/19]]
| M3
| M3
| major 3rd
| major 3rd
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| 9
| 9
| 450
| 450
| 13/10, 17/13, 22/17
| [[13/10]], [[17/13]], [[22/17]]
| ^M3, v4
| ^M3, v4
| upmajor 3rd, down-4th
| upmajor 3rd, down-4th
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| 10
| 10
| 500
| 500
| 4/3
| [[4/3]]
| P4
| P4
| fourth
| fourth
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| 11
| 11
| 550
| 550
| 11/8, 15/11
| [[11/8]], [[15/11]]
| ^4, ~4
| ^4, ~4
| up-4th, mid-4th
| up-4th, mid-4th
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| 12
| 12
| 600
| 600
| 17/12, 24/17, 45/32, 64/45
| [[17/12]], [[24/17]], [[45/32]], [[64/45]]
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
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| 13
| 13
| 650
| 650
| 16/11, 22/15
| [[16/11]], [[22/15]]
| v5, ~5
| v5, ~5
| down-5th, mid-5th
| down-5th, mid-5th
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| 14
| 14
| 700
| 700
| 3/2
| [[3/2]]
| P5
| P5
| fifth
| fifth
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| 15
| 15
| 750
| 750
| 17/11, 20/13
| [[17/11]], [[20/13]]
| ^5, vm6
| ^5, vm6
| up-fifth, downminor 6th
| up-fifth, downminor 6th
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| 16
| 16
| 800
| 800
| 8/5, 19/12
| [[8/5]], [[19/12]]
| m6
| m6
| minor 6th
| minor 6th
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| 17
| 17
| 850
| 850
| 13/8, 18/11, 44/27, 64/39
| [[13/8]], [[18/11]], [[44/27]], [[64/39]]
| ~6
| ~6
| mid 6th
| mid 6th
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| 18
| 18
| 900
| 900
| 5/3, 22/13, 32/19
| [[5/3]], [[22/13]], [[32/19]]
| M6
| M6
| major 6th
| major 6th
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| 19
| 19
| 950
| 950
| 19/11, 26/15
| [[19/11]], [[26/15]]
| ^M6, vm7
| ^M6, vm7
| upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
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| 20
| 20
| 1000
| 1000
| 9/5, 16/9
| [[9/5]], [[16/9]]
| m7
| m7
| minor 7th
| minor 7th
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| 21
| 21
| 1050
| 1050
| 11/6, 20/11
| [[11/6]], [[20/11]]
| ~7
| ~7
| mid 7th
| mid 7th
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| 22
| 22
| 1100
| 1100
| 15/8, 17/9, 32/17
| [[15/8]], [[17/9]], [[32/17]]
| M7
| M7
| major 7th
| major 7th
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| 23
| 23
| 1150
| 1150
| 33/17, 64/33
| [[33/17]], [[64/33]]
| ^M7, vP8
| ^M7, vP8
| upmajor 7th, down-8ve
| upmajor 7th, down-8ve
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| 24
| 24
| 1200
| 1200
| 2/1
| [[2/1]]
| P8
| P8
| perfect 8ve
| perfect 8ve
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| 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.
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.
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=== Ups and downs 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 are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{sharpness-sharp2a|24}}
{{Ups and downs sharpness}}


=== Stein–Zimmermann accidentals ===
=== Stein–Zimmermann accidentals ===
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| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
| A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
|-
|-
| style="width: 40px;" | [[File:HeQd3.svg|36px|center]]
| style="width: 40px;" | [[File:HeQd3.svg|40px|center]]
| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.
| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.
|}
|}
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'''Pros:''' familiar, intuitive, and fairly easy to learn.  
'''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.  
'''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 ===
=== Persian quartertone accidentals ===
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{| class="wikitable"
{| class="wikitable"
|-
|-
| width="40px" | [[File:Koron_sign.svg|39px|center]]
| style="width: 40px;" | [[File:Koron_sign.svg|39px|center]]
| '''Koron''' = quarter-tone flat
| '''Koron''' = quarter-tone flat
|-
|-
| width="40px" | [[File:Sori_sign.svg|39px|center]]
| style="width: 40px;" | [[File:Sori_sign.svg|39px|center]]
| '''Sori''' = quarter-tone sharp
| '''Sori''' = quarter-tone sharp
|}
|}
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'''Cons:''' not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality.  
'''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:
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:
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* [[24edo Chord Names]]
* [[24edo Chord Names]]
* [[Ups and Downs Notation#Chords and Chord Progressions]].
* [[Ups and downs notation#Chords and Chord Progressions]].


== Approximation to JI ==
== Approximation to JI ==
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| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|-
|-
| 17
| 17
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| Neovish comma
| Neovish comma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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Important MOSes include:  
Important MOSes include:  
* Semaphore 4L1s 55455 (generator: 5\24)
* Semaphore 4L&nbsp;1s 55455 (generator: 5\24)
* Semaphore 5L4s 414141414 (generator: 5\24)
* Semaphore 5L&nbsp;4s 414141414 (generator: 5\24)
* Mohajira 3L4s 3434343 (generator: 7\24)
* Mohajira 3L&nbsp;4s 3434343 (generator: 7\24)
* Mohajira 7L3s 3313313313 (generator: 7\24)
* Mohajira 7L&nbsp;3s 3313313313 (generator: 7\24)
 
== 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 ==
== Scales and modes ==
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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.
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 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:
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]]
[[File:24edo_guitar.jpg|500px]]
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While these are playable, the extra frets can make playing chords and navigating the fretboard significantly more challenging for [[12edo]] chords and scales.
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 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.
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]]
[[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 ===
=== Harp, Harpsichord, and Piano ===
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==== Scordatura tuning of 12edo instruments ====
==== 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 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.
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 differrent scordatura tuning), followed by the scordatura tuning of the harp.
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 ====
==== Quarter-tone instruments ====
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; Quarter-tone flute, made by Eva Kingma
; Quarter-tone flute, made by Eva Kingma
* [https://www.youtube.com/watch?v=F3GD0Omr4Z0 Visit to the workshop of Eva Kingma, followed by test by Manuel Luis Cochofel] (2010) (demonstration of fingering starts at 06:56)
* [https://www.youtube.com/watch?v=F3GD0Omr4Z0 Visit to the workshop of Eva Kingma, followed by test by Manuel Luis Cochofel] (2010) (demonstration of fingering starts at 06:56)
=== 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 ==
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{{Todo| cleanup }}
{{Todo| cleanup }}
== Notes ==
<references group="note" />


[[Category:Semaphore]]
[[Category:Semaphore]]
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[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Twentuning]]
[[Category:Twentuning]]
[[Category:Lists of intervals]]
Retrieved from "https://en.xen.wiki/w/24edo"