24edo

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Basics

The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as quarter-tone tuning, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in Arabic music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - see this page.

24edo as a temperament

The 5-limit approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.

The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit 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 subgroup temperament, on which it is quite accurate.

Degree Cents Approximate Ratios* ups and downs notation
0 0 1/1 P1 unison C
1 50 33/32, 34/33 ^P1, vm2 up-unison, downminor 2nd C^, Dbv
2 100 17/16, 18/17 A1, m2 aug unison, minor 2nd C#, Db
3 150 12/11 ~2 mid 2nd Dv
4 200 9/8 M2 major 2nd D
5 250 22/19 ^M2, vm3 upmajor 2nd, downminor 3rd D^, Ebv
6 300 19/16 m3 minor 3rd Eb
7 350 11/9 ~3 mid 3rd Ev
8 400 24/19 M3 major 3rd E
9 450 22/17 ^M3, vP4 upmajor 3rd, down-fourth E^, Fv
10 500 4/3 P4 fourth F
11 550 11/8 ^P4 up-fourth F^
12 600 17/12 A4, d5 aug 4th, dim 5th F#, Gb
13 650 16/11 vP5 down-fifth Gv
14 700 3/2 P5 fifth G
15 750 17/11 ^P5, vm6 up-fifth, downminor 6th G^, Abv
16 800 19/12 m6 minor 6th Ab
17 850 18/11 ~6 mid 6th Av
18 900 32/19 M6 major 6th A
19 950 19/11 ^M6, vm7 upmajor 6th, downminor 7th A^, Bbv
20 1000 16/9 m7 minor 7th Bb
21 1050 11/6 ~7 mid 7th Bv
22 1100 17/9, 32/17 M7 major 7th B
23 1150 33/17, 64/33 ^M7, vP8 upmajor 7th, down-8ve B^, Cv
24 1200 2/1 P8 perfect 8ve C
  • based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.

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

quality color monzo format examples
downminor zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, b < -1 32/27, 16/9
" gu {a, b, -1} 6/5, 9/5
mid ilo {a, b, 0, 0, 1} 11/9, 11/6
" lu {a, b, 0, 0, -1} 12/11, 18/11
major yo {a, b, 1} 5/4, 5/3
" fifthward wa {a, b}, b > 1 9/8, 27/16
upmajor ru {a, b, 0, -1} 9/7, 12/7

The 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth. Some good chords in 24-tET are (the numbers are edosteps, e.g. 4 is a major second, 8 is a major third):

0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)

Its inversion, 0-3-6-10-14 ("minor")

0-7-14 ("neutral")

0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 (godzilla), 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.)

Commas

24 EDO tempers out the following commas. (Note: This assumes val < 24 38 56 67 83 89 |.)

Comma Monzo Value (Cents) Name 1 Name 2 Name 3
531441/524288 | -19 12 > 23.46 Pythagorean Comma
648/625 | 3 4 -4 > 62.57 Major Diesis Diminished Comma
128/125 | 7 0 -3 > 41.06 Diesis Augmented Comma
81/80 | -4 4 -1 > 21.51 Syntonic Comma Didymos Comma Meantone Comma
2048/2025 | 11 -4 -2 > 19.55 Diaschisma
5201701/5149091 | 26 -12 -3 > 17.60 Misty Comma
32805/32768 | -15 8 1 > 1.95 Schisma
1465155/1465142 | 161 -84 -12 > 0.02 Atom
49/48 | -4 -1 0 2 > 35.70 Slendro Diesis
245/243 | 0 -5 1 2 > 14.19 Sensamagic
19683/19600 | -4 9 -2 -2 > 7.32 Cataharry
6144/6125 | 11 1 -3 -2 > 5.36 Porwell
121/120 | -3 -1 -1 0 2 > 14.37 Biyatisma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
896/891 | 7 -4 0 1 -1 > 9.69 Pentacircle
243/242 | -1 5 0 0 -2 > 7.14 Rastma
385/384 | -7 -1 1 1 1 > 4.50 Keenanisma
9801/9800 | -3 4 -2 -2 2 > 0.18 Kalisma Gauss' Comma
91/90 | -1 -2 -1 1 0 1 > 19.13 Superleap
676/675 | 2 -3 -2 0 0 2 > 2.56 Parizeksma

Intervals

24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.

These are the intervals of 24 EDO that do not exist in 12 EDO: 2

See full article on 24 Edo intervals.

The twelve new intervals in 24edo some nearby JI intervals
cents common names frequency ratio cents common name
50 quartertone

infra second, wide unison

36/35

35/34

34/33

33/32

48.770

50.184

51.682

53.273

large septimal quarter-tone (Archytas)

large 17-limit quartertone

small 17-limit quartertone

33rd harmonic

150 neutral second 12/11 150.637 large undecimal neutral second
250 ultra second

infra third

144/125

15/13

52/45

244.969

247.741

250.304

diminished third (6/5 x 24/25)

..

..

350 neutral third 11/9

27/22

16/13

347.408

354.547

359.472

undecimal neutral third

..

tridecimal neutral third

450 minor fourth, ultra third, narrow fourth 22/17

35/27

13/10

446.363

449.275

454.214

17-limit supermajor third

..

tridecimal subfourth

550 wide fourth 11/8 551.318 undecimal superfourth, harmonic 11th
650 narrow fifth 16/11 648.682 undecimal subfifth, 11th subharmonic
750 wide fifth, infra sixth 20/13

54/35

17/11

745.786

750.725

753.637

tridecimal superfifth

..

17-limit subminor sixth

850 neutral sixth 13/8

44/27

18/11

840.528

845.453

852.592

overtone sixth, 13th harmonic

..

undecimal neutral sixth

950 ultra sixth , infra seventh 45/26

26/15

125/72

949.696

952.259

955.031

..

..

..

1050 neutral seventh 11/6 1049.363 undecimal neutral seventh
1150 ultra seventh, narrow octave 31/16

33/17

35/18

1145.036

1148.318

1151.230

31st harmonic

..

..

Notation

There have been disputes about disadvantages of various systems for notating quarter tones. Here are some of the few systems along with pros and cons.

Mainstream Quartertone Notation

External image: http://www.wfg.woodwind.org/n/s14_blue.gif

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or ^ = quarter-tone sharp or "Jump" or "up"

External image: http://www.wfg.woodwind.org/n/s34_blue.gif

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or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"

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or v = quarter-tone flat or "Drop" or "down"

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or bv or vb = three-quarter-tone flat or "Drop-Flat" or "downflat"

Pros: Familiar, fairly easy to learn

Cons: Clutters a score easily, can get confusing when sight read at faster paces

Alternate Quartertone Accidentals

External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif

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= quarter-tone sharp or Jump

††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp

External image: http://www.dolmetsch.com/onequarterflat.gif

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= quarter-tone flat or Drop

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= three quarter-tones flat or drop-flat

For example, the scale 0-5-10-15-20 is written as C-D

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(or E

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) F G

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(or A

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) Bb.

Pros: Very easy to distinguish accidentals from one another

Cons: Not practical, tends to clutter a score

And in Persian music

External image: http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png

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Koron (en | fa) = quarter-tone flat or Jump

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Sori (fa) = quarter-tone sharp or Drop

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 24 Edo notation as well as other systems.

It's easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.

A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them

with Sagittal signs for sharp and flat.

sagittal 24.PNG

Interval Alterations

The special alterations of the intervals and chords of 12 equal can be notated like this:

Supermajor or "Tendo" is a major interval raised a quarter tone

Subminor or "Arto" is a minor interval lowered a quarter tone

Neutral are intervals that exist between the major and minor version of an interval

The prefix under indicates a perfect interval lowered by one quarter tone

The prefix over indicates a perfect interval raised by a quarter tone

The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.

Chord Structures

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

As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as

0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.

These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.

For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 William Lynch considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.

William Lynch considers these as some possible good tetrads:

Three chords.PNG

Chord name Degrees of 24edo Chord spelling Audio example
neutral 0 7 14 21 1 v3 5 v7
arto 0 5 14 20 1 bv3 5 b7
tendo 0 9 14 19 1 ^3 5 bv7 ...

Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.

Naming Chords in 24edo

Naming chords in 24edo can be achieved by adding a few things to the already existing set of terms that are used to name 12edo chords.

They are:

Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone

Sub + perfect interval means to lower a quarter tone

Sharp is to raise by one half tone

Flat is to raise by a half tone

Neutral, arto and tendo refer to triads or tetrads

Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively

Examples:

Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top

Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11

Alternatively, ups and downs notation can be used. Here are the zo, gu, ilo, yo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-5-14 C Ebv G C.vm C downminor
gu 10:12:15 0-6-14 C Eb G Cm C minor
ilo 18:22:27 0-7-14 C Ev G C~ C mid
yo 4:5:6 0-8-14 C E G C C major or C
ru 14:18:27 0-9-14 C E^ G C.^ C upmajor or C dot up

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Rank two temperaments

List of 24et rank two temperaments by badness

List of edo-distinct 24et rank two temperaments

Periods per octave Generator Name
1 1\24
1 5\24 Semaphore/godzilla / Bridgetown
1 7\24 Mohajira (or maqamic with 24d val)
1 11\24
2 1\24 Shrutar
2 5\24 Similar to decimal
3 1\24 Semiaug
3 3\24 Triforce
4 1\24
6 1\24
8 1\24
12 1\24 Catler

Scales / Modes

Pentatonic:

2 8 3 6 5 Anchihoye: Ethiopia
5 5 4 5 5 Quasi-equal Pentatonic - MOS of type 4L 1s (bug)
5 5 5 5 4 Hába's Pentatonic - MOS of type 4L 1s (bug)

Hexatonic:

1 1 8 4 2 8 Spondeiakos

Heptatonic:

1 1 8 1 1 8 4 Enharmonic Mixolydian
1 8 1 1 8 4 1 Enharmonic Lydian
8 1 1 8 4 1 1 Enharmonic Phrygian
1 1 8 4 1 1 8 Enharmonic Dorian
1 8 4 1 1 8 1 Enharmonic Hypolydian
8 4 1 1 8 1 1 Enharmonic Hypophrygian
4 1 1 8 1 1 8 Enharmonic Hypodorian
2 3 5 2 3 5 4 Soft Diatonic Mixolydian
3 5 2 3 5 4 2 Soft Diatonic Lydian
5 2 3 5 4 2 3 Soft Diatonic Phrygian
2 3 5 4 2 3 5 Soft Diatonic Dorian
3 5 4 2 3 5 2 Soft Diatonic Hypolydian
5 4 2 3 5 2 3 Soft Diatonic Hypophrygian
4 2 3 5 2 3 5 Soft Diatonic Hypodorian
3 3 4 3 3 4 4 Maqam Ouchairan-Hussaini, Bayatan, Neutral Diatonic Mixolydian - MODMOS of type 3L 4s (mosh)
3 4 3 3 4 4 3 Dastgah-e Sehgah, Neutral Diatonic Lydian - MODMOS of type 3L 4s (mosh)
4 3 3 4 4 3 3 Arabic Diatonic, Maqam Rast, Quasi-equal Heptatonic, Neutral Diatonic Phrygian - MODMOS of type 3L 4s (mosh)
3 3 4 4 3 3 4 Maqam Hussaini, Ushaq, Neutral Diatonic Dorian - MODMOS of type 3L 4s (mosh)
3 4 4 3 3 4 3 Maqam Sikah (Segah), Neutral Diatonic Hypolydian - MODMOS of type 3L 4s (mosh)
4 4 3 3 4 3 3 Neutral Diatonic Hypophrygian - MODMOS of type 3L 4s (mosh)
4 3 3 4 3 3 4 Miha'il Musaqa's mode: Egypt, Neutral Diatonic Hypodorian, Dastgah-e Sehgah, Maqam Nairuz - MODMOS of type 3L 4s (mosh)
1 5 4 1 5 4 4 Diatonic + Enharmonic Diesis Mixolydian
5 4 1 5 4 4 1 Diatonic + Enharmonic Diesis Lydian
4 1 5 4 4 1 5 Diatonic + Enharmonic Diesis Phrygian
1 5 4 4 1 5 4 Diatonic + Enharmonic Diesis Dorian
5 4 4 1 5 4 1 Diatonic + Enharmonic Diesis Hypolydian
4 4 1 5 4 1 5 Diatonic + Enharmonic Diesis Hypophrygian
4 1 5 4 1 5 4 Diatonic + Enharmonic Diesis Hypodorian
1 3 6 1 3 6 4 Chromatic/Enharmonic Mixolydian
3 6 1 3 6 4 1 Chromatic/Enharmonic Lydian
6 1 3 6 4 1 3 Chromatic/Enharmonic Phrygian
1 3 6 4 1 3 6 Chromatic/Enharmonic Dorian
3 6 4 1 3 6 1 Chromatic/Enharmonic Hypolydian
6 4 1 3 6 1 3 Chromatic/Enharmonic Hypophrygian
4 1 3 6 1 3 6 Chromatic/Enharmonic Hypodorian
3 4 3 3 4 3 4 Neutral Mixolydian - MOS of type 3L 4s (mosh)
4 3 3 4 3 4 3 Neutral Lydian - MOS of type 3L 4s (mosh)
3 3 4 3 4 3 4 Neutral Phrygian - MOS of type 3L 4s (mosh)
3 4 3 4 3 4 3 Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi - MOS of type 3L 4s (mosh)
4 3 4 3 4 3 3 Neutral Hypolydian - MOS of type 3L 4s (mosh)
3 4 3 4 3 3 4 Neutral Hypophrygian - MOS of type 3L 4s (mosh)
4 3 4 3 3 4 3 Neutral Hypodorian - MOS of type 3L 4s (mosh)
3 5 2 4 3 5 2 Athanasopoulos' Byzantine Liturgical Chromatic, Dastgah-e Chahargah
4 2 4 4 3 3 4 Dastgah-e Nava, Maqam Ushaq Masri
2 7 1 4 2 7 1 Second plagal Byzantine Liturgical mode
3 3 4 4 2 4 4 Maqam 'Ushshaq Turki, Urfa, Isfahan, Dastgah-e Shur
3 3 4 4 4 2 4 Maqam Nahfat
3 3 2 6 2 4 4 Maqam Saba
3 3 2 6 2 6 2 Maqam Sabr Jadid
4 3 3 4 2 6 2 Maqam Suznak (Soznak)
4 3 3 4 4 4 2 Maqam Mahur
3 3 4 2 6 2 4 Maqam Qarjighar, Bayati Shuri
3 4 2 6 2 4 3 Maqam Hizam (Huzam, El Houzam), Rahat al Arouah
2 4 4 4 3 3 4 Maqam Nawa
2 5 3 4 2 5 3 Maqam Higaz-kar
3 4 4 2 4 4 3 Maqam Su'ar, Naghmeh Abuata, Naghmeh Afshari
4 4 2 4 4 3 3 Maqam Jahargah (Jiharkah), Naghmeh Bayat-e Tork, Naghmeh Dashti
3 5 2 4 2 4 4 Dastgah-e Homayun
4 2 4 4 3 5 2 Naghmeh Esfahan
3 6 1 5 2 6 1 Maqam 'Awg 'ara (Aug-ara)
4 1 5 4 2 6 2 Maqam Buselik
4 2 6 2 2 5 3 Maqam Neuter
4 3 3 4 4 2 4 Dance scale of Yi people: China
4 4 2 3 1 4 6 Daniel-mode of Spanish-Arab Jews

Octatonic:

3 3 3 3 3 3 3 3 8-equal, Wyschnegradsky's octatonic
3 3 4 4 2 1 3 4 Maqam Bayati
3 3 2 6 2 4 2 2 Maqam Saba
4 3 1 2 4 4 2 4 Maqam Suzidil 'ara
3 3 2 2 4 3 3 4 Maqam Mansuri
4 3 3 4 4 2 1 3 Maqam Rast, Dilkashidah, Dilnishin
3 4 2 6 2 4 2 1 Maqam Rahat al-Arwah
3 4 3 3 4 4 2 1 Maqam Iraq
2 6 2 4 2 1 3 4 Maqam Hijaz
3 4 4 2 1 3 4 3 Maqam Musta'ar
3 4 4 4 2 4 2 1 Maqam Farahnak
3 4 3 3 2 6 2 1 Maqam Bastanikar, Tarz Nuin
4 2 6 2 4 2 1 3 Maqam Farah Faza, Maqam Nakriz
3 1 2 4 4 2 4 4 Maqam Jabburi
1 4 4 2 4 4 4 1 Giancarlo Dalmonte's new quarter-tone scale (see http://www.ottavanota.info)

Enneatonic:

1 2 3 4 4 1 2 3 4 Progressive Enneatonic
4 1 4 1 4 1 4 1 4 de Vries 9-tone - MOS of type 5L 4s (unfair bug)
3 4 2 2 2 2 2 4 3 Maqam Huzam
4 4 2 1 3 2 2 4 2 Maqam Shawq Afza

Decatonic:

3 1 3 3 3 1 3 3 1 3 Breed Decatonic - MOS of type 7L 3s (unfair mosh)
2 3 2 2 3 2 3 2 2 3 Oljare Decatonic - MOS of type 4L 6s (fair bicycle)
2 1 3 2 2 4 2 4 2 2 Maqam Shawq Tarab
4 2 1 3 2 2 4 2 1 3 Maqam Basandida
4 3 1 2 1 3 4 2 1 3 Maqam Yakah

Hendecatonic:

4 2 1 3 2 2 2 2 3 1 2 Maqam Hayyan

Tridecatonic:

1 2 2 2 2 2 2 1 2 2 2 2 2 de Vries 13-tone - MOS of type 11L 2s
2 2 2 2 2 1 2 2 2 2 2 2 1 Agmon Diatonic DS5, Ivan Wyschnegradsky's diatonicized chromatic scale - MOS of type 11L 2s

Tetradecatonic:

2 2 1 2 2 2 1 2 2 1 2 2 2 1 Young Half-Octave Diatonic - MOS of type 10L 4s

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 "12 note octave scales" approach is used in a wide part of the existing literature - see below.

External image: http://www.swordguitars.com/Sword_quartertone_stratsm.jpg

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24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars

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.

Compositions

Music

Microhex3 Microhex4 and Microhex5 by Shaahin Mohajeri

Quarterpicnic by Chris Vaisvil

Quarter Tone Prelude For Two Harps by Nathan BeDell

Fretless Chrome 1 and Fretless Chrome 2 by Chris Vaisvil

Lament by Jake Freivald. In the freivaldneutral24 scale.

Mo - Happy - Happy play by Jake Freivald in Neutral[7] (2.3.11 mohajira), 24et tuning

Autumn Winds, Easter Time at Nine, Waters of Persia by William Lynch in mohajira, 24et tuning.

Serena, by Mason Green (intro and coda in 24edo, the rest is in 12edo)

Autumn Girl, by Mason Green

About

"Prométhée enchaîné" by Fromental Halévy (considered the first mainstream western orchestral composition to use quarter tones.)

"3 Hommages" by Georg Friedrich Haas

List of quartertone pieces on Wikipedia

Practical Theory / Books

External image: http://ronsword.com/images/24_tet_Coversm.jpg

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"Icosakaiteraphonic Scales for Guitar" - A Book for Twenty-Four Equal Divisions of the Octave on guitar, or 'Quarter-tones'. Features a practical approach to understanding the tuning, and over 550 Scale Examples on Nine-String finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the Scales / Modes from the list above.

External links

quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory Permalink

About 24-EDO by Shaahin Mohajeri Permalink

Notation and Chord Names for 24-EDO by William Lynch

The place of QUARTERTONES in Today's Xenharmonics by Ivor Darreg Permalink