# 24edo

(Redirected from 24-EDO)
 Prime factorization 23 × 3 Step size 50¢ Fifth 14\24 = 700¢ (→7\12) Major 2nd 4\24 = 200¢ Minor 2nd 2\24 = 100¢ Augmented 1sn 2\24 = 100¢

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 as illustrated in DIY Quartertone Composition with 12 equal tools.

# Theory

Approximation of primary intervals in 24 EDO
Prime number 2 3 5 7 11 13 17 19
Error absolute (¢) +0.0 -2.0 +13.7 -18.8 -1.3 +9.5 -5.0 +2.5
relative (%) +0 -4 +27 -38 -3 +19 -10 +5
Steps (reduced) 24 (0) 38 (14) 56 (8) 67 (19) 83 (11) 89 (17) 98 (2) 102 (6)

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. However, it should be noted that 24edo, like 22edo, does temper out 117440512/117406179, linking the otherwise sub-par 7-limit harmonies with those of the 11-limit, and speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.

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.

# Notation

## Ups and down notation

Degree Cents Approximate Ratios[1] ups and downs notation
0 0 1/1 P1 unison C
1 50 33/32, 34/33 ^P1, vm2 up-unison, downminor 2nd ^C, vDb
2 100 17/16, 18/17 A1, m2 aug unison, minor 2nd C#, Db
3 150 12/11 ~2 mid 2nd vD
4 200 9/8 M2 major 2nd D
5 250 22/19 ^M2, vm3 upmajor 2nd, downminor 3rd ^D, vEb
6 300 19/16 m3 minor 3rd Eb
7 350 11/9, 27/22 ~3 mid 3rd vE
8 400 24/19 M3 major 3rd E
9 450 22/17 ^M3, v4 upmajor 3rd, down-4th ^E, vF
10 500 4/3 P4 fourth F
11 550 11/8 ^4, ~4 up-4th, mid-4th ^F
12 600 17/12 A4, d5 aug 4th, dim 5th F#, Gb
13 650 16/11 v5, ~5 down-5th, mid-5th vG
14 700 3/2 P5 fifth G
15 750 17/11 ^5, vm6 up-fifth, downminor 6th ^G, vAb
16 800 19/12 m6 minor 6th Ab
17 850 18/11, 44/27 ~6 mid 6th vA
18 900 32/19 M6 major 6th A
19 950 19/11 ^M6, vm7 upmajor 6th, downminor 7th ^A, vBb
20 1000 16/9 m7 minor 7th Bb
21 1050 11/6 ~7 mid 7th vB
22 1100 17/9, 32/17 M7 major 7th B
23 1150 33/17, 64/33 ^M7, vP8 upmajor 7th, down-8ve ^B, vC
24 1200 2/1 P8 perfect 8ve C
1. 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 name 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

Ups and downs notation can be used to name chords. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

In many other edos, especially larger ones, 5/4 is downmajor and 11/9 is mid. To agree with this, the term mid is generally preferred over down or downmajor.

edosteps notes written name spoken name JI chord, if any
0-5-14 C vEb G Cvm C downminor 6:7:9
0-6-14 C Eb G Cm C minor 10:12:15 or 6/(6:5:4)
0-7-14 C vE G C~ C mid 18:22:27
0-8-14 C E G C C major or C 4:5:6
0-9-14 C ^E G C^ C upmajor or C up 14:18:21 or 9/(9:7:6)
0-8-13 C E vG C(v5) C down-five
0-7-13 C vE vG C~(v5) C mid down-five 22:27:32
0-4-11-14 C D ^F G C^4,9 C up-four add-nine 8:9:11:12
0-9-14-19 C ^E G ^A C^6 C up-six 14:18:21:24
0-5-14-20 C vEb G Bb Cvm,7 C downminor add-seven
0-5-14-19 C vEb G vBb Cvm7 C downminor-seven 12:14:18:21
0-7-14-21 C vE G vB C~7 C mid-seven 18:22:27:33
0-7-14-21-28 C vE G vB D C~9 C mid-nine 36:44:54:66:81
0-7-14-21-28-35 C vE G vB D ^F C~11 C mid-eleven 36:44:54:66:81:99
0-7-14-21-28-35-42 C vE G vB D ^F A C~13 C mid-thirteen 36:44:54:66:81:99:121
0-8-14-28-35-42 C E G D ^F vA C9,~11~13no7 C nine mid-eleven mid-thirteen no-seven 4:5:6:9:11:13

For a more complete list of chords, see Ups and Downs Notation #Chords and Chord Progressions.

## William Lynch's notation

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. See the full article on 24 Edo intervals.

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

The twelve new intervals in 24edo some nearby JI intervals
cents pions 7mus common names frequency ratio cents common name
50 53 64 (4016) 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
150 159 192 (C016) neutral second 12/11 150.637 large undecimal neutral second
250 265 320 (14016) ultra second
infra third
144/125
15/13
52/45
244.969
247.741
250.304
diminished third (6/5 x 24/25)
..
..
350 383 448 (1C016) neutral third 11/9
27/22
16/13
347.408
354.547
359.472
undecimal neutral third
..
tridecimal neutral third
450 477 576 (24016) minor fourth, ultra third, narrow fourth 22/17
35/27
13/10
446.363
449.275
454.214
17-limit supermajor third
..
tridecimal subfourth
550 583 704 (2C016) wide fourth 11/8 551.318 undecimal superfourth, harmonic 11th
650 689 832 (34016) narrow fifth 16/11 648.682 undecimal subfifth, 11th subharmonic
750 795 960 (3C016) wide fifth, infra sixth 20/13
54/35
17/11
745.786
750.725
753.637
tridecimal superfifth
..
17-limit subminor sixth
850 901 1088 (34016) neutral sixth 13/8
44/27
18/11
840.528
845.453
852.592
overtone sixth, 13th harmonic
..
undecimal neutral sixth
950 1007 1216 (4C016) ultra sixth , infra seventh 45/26
26/15
125/72
949.696
952.259
955.031
..
..
..
1050 1113 1344 (54016) neutral seventh 11/6 1049.363 undecimal neutral seventh
1150 1219 1472 (5C016) ultra seventh, narrow octave 31/16
33/17
35/18
1145.036
1148.318
1151.230
31st harmonic
..
..

### 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 Names

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, 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

# Quartertone Accidentals

Besides ups and downs, there are various systems for notating quarter tones. Here are some of them, along with their pros and cons.

## Mainstream Quartertone Notation

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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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

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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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

External image: http://www.wfg.woodwind.org/n/f14_blue.gif

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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

External image: http://www.wfg.woodwind.org/n/f34_blue.gif

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

= 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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

= quarter-tone flat or Drop

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

= three quarter-tones flat or drop-flat

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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

(or E

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

) F G

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

(or A

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

) Bb.

Pros: Very easy to distinguish accidentals from one another

Cons: Not practical, tends to clutter a score

## Persian Accidentals

 Koron (en | fa) = quarter-tone flat
 Sori (fa) = quarter-tone sharp

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.

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

# Chord Types

24edo features a rich variety of 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, 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.

More good chords in 24-tET:

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")

0-5-10 (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.)

William Lynch considers these as some possible good tetrads:

Chord name Degrees of 24edo Chord spelling Audio example
neutral 0 7 14 21 1 v3 5 v7
arto 0 5 14 20 1 vb3 5 b7
tendo 0 9 14 19 1 ^3 5 vb7 ...

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.

# Commas

24 EDO tempers out the following commas. This assumes val 24 38 56 67 83 89].

Prime
Limit
Ratio[1] Monzo Cents Color name Name(s)
3 (12 digits) [-19 12 23.46 Lalawa Pythagorean comma
5 648/625 [3 4 -4 62.57 Quadgu Major diesis, diminished domma
5 128/125 [7 0 -3 41.06 Trigu Diesis, augmented comma
5 81/80 [-4 4 -1 21.51 Gu Syntonic comma, Didymus comma, meantone comma
5 2048/2025 [11 -4 -2 19.55 Sagugu Diaschisma
5 (16 digits) [26 -12 -3 17.60 Sasa-trigu Misty comma
5 32805/32768 [-15 8 1 1.95 Layo Schisma
5 (98 digits) [161 -84 -12 0.02 Sepbisa-quadbigu Atom
7 49/48 [-4 -1 0 2 35.70 Zozo Slendro diesis
7 245/243 [0 -5 1 2 14.19 Zozoyo Sensamagic
7 19683/19600 [-4 9 -2 -2 7.32 Labirugu Cataharry
7 6144/6125 [11 1 -3 -2 5.36 Sarurutrigu Porwell
11 121/120 [-3 -1 -1 0 2 14.37 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.86 Lorugugu Valinorsma
11 896/891 [7 -4 0 1 -1 9.69 Saluzo Pentacircle
11 243/242 [-1 5 0 0 -2 7.14 Lulu Rastma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 9801/9800 [-3 4 -2 -2 2 0.18 Bilorugu Kalisma, Gauss' comma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2 2.56 Bithogu Island comma, parizeksma
1. Ratios longer than 10 digits are presented by placeholders with informative hints

# Rank two temperaments

Important MOSes include:

• semaphore 4L1s 55455 (generator: 5\24)
• semaphore 5L4s 414144141 (generator: 5\24)
• mohajira 3L4s 3434343 (generator: 7\24)
• mohajira 7L3s 3313313313 (generator: 7\24)
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 Barton
2 1\24 Shrutar
2 5\24 Sruti, Anguirus, 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)
6 3 6 6 3 MOS of type 3L 2s (father)

#### Hexatonic:

1 1 8 4 2 8 Spondeiakos
5 5 2 5 5 2 MOS of type 4L 2s
3 5 3 5 3 5 MOS of type 3L 3s (triforce)

#### 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 L'ottava nota)

#### 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) (godzilla/semaphore)
3 4 2 2 2 2 2 4 3 Maqam Huzam
4 4 2 1 3 2 2 4 2 Maqam Shawq Afza
3 3 2 3 3 2 3 3 2 MOS of type 6L 3s (triforce)

#### 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
3 2 2 2 2 3 2 2 2 2 2 MOS of type 2L 9s

#### 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

2 2 1 2 2 2 1 2 2 1 2 2 2 1 Young Half-Octave Diatonic - MOS of type 10L 4s
1 1 3 1 1 3 1 1 3 1 1 3 1 3 MOS of type 5L 9s (godzilla/semaphore)

1 2 1 2 2 1 2 1 2 2 1 2 1 2 2 MOS of type 9L 6s (triforce)

1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2 MOS of type 7L 10s (mohajira)

# 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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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.

# Music

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

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

"3 Hommages" by Georg Friedrich Haas

# Practical Theory / Books

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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

"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.