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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.

The 11th harmonic, and 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 degree numbers, 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 s14_blue.gif or ^ = quarter-tone sharp or "Jump" or "up"
external image s34_blue.gif or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
external image f14_blue.gif or v = quarter-tone flat or "Drop" or "down"
external image f34_blue.gif 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 quartertonesharp.gif = quarter-tone sharp or Jump
††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp
external image onequarterflat.gif = quarter-tone flat or Drop
external image 9px-Arabic_music_notation_half_flat.svg.png = three quarter-tones flat or drop-flat
For example, the scale 0-5-10-15-20 is written as C-Dexternal image quartertonesharp.gif (or Eexternal image 9px-Arabic_music_notation_half_flat.svg.png) F Gexternal image quartertonesharp.gif (or Aexternal image 9px-Arabic_music_notation_half_flat.svg.png) Bb.

Pros: Very easy to distinguish accidentals from one another
Cons: Not practical, tends to clutter a score

And in Persian music

external image 200px-Koron_sign.svg.pngKoron (en | fa) = quarter-tone flat or Jump
external image 200px-Sori_sign.svg.pngSori (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:
0-5-14 = D Fv A = D.vm = "D downminor"
0-6-14 = D F A = Dm = "D minor"
0-7-14 = D F^ A = D~ = "D mid"
0-8-14 = D F# A = D = "D" or "D major"
0-9-14 = D F#^ A = D.^ = "D dot up" or "D upmajor"


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 Sword_quartertone_stratsm.jpg24-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 Neuter[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 24_tet_Coversm.jpg"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