# 24edo

(Redirected from 24-edo)
 ← 23edo 24edo 25edo →
Prime factorization 23 × 3
Step size 50¢
Fifth 14\24 (700¢) (→7\12)
Semitones (A1:m2) 2:2 (100¢ : 100¢)
Consistency limit 5
Distinct consistency limit 5
Special properties

24 equal divisions of the octave (abbreviated 24edo or 24ed2), also called 24-tone equal temperament (24tet) or 24 equal temperament (24et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 24 equal parts of exactly 50 ¢ each. Each step represents a frequency ratio of 21/24, or the 24th root of 2.

English Wikipedia has an article on:

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 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

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.

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 36et, 72et, 84et, 156et, or 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. Additionally, like 22edo, 24edo tempers out the quartisma, linking the otherwise sub-par 7-limit harmonies with those of 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 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.

Its step, at 50 cents, is notable for having some of the highest harmonic entropy possible, making it, in theory, one of the most dissonant intervals possible (using the relatively common values of a = 2 and s = 1%, the peak occurs at around 46.4 cents). 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.

### Prime harmonics

Approximation of prime harmonics in 24edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 +13.69 -18.83 -1.32 +9.47 -4.96 +2.49 +21.73 +20.42 +4.96
Relative (%) +0.0 -3.9 +27.4 -37.7 -2.6 +18.9 -9.9 +5.0 +43.5 +40.8 +9.9
Steps
(reduced)
24
(0)
38
(14)
56
(8)
67
(19)
83
(11)
89
(17)
98
(2)
102
(6)
109
(13)
117
(21)
119
(23)

### Subsets and supersets

24edo is the 6th highly composite edo. Its nontrivial divisors are 2, 3, 4, 6, 8, and 12.

## Notation

There are multiple ways of notating 24edo. While an arguably common form can be seen on Wikipedia's page on quartertones, there are other forms, and it is these other forms that will be considered here.

### Ups and down notation

Degree Cents Approximate Ratios[1] Ups and downs notation SKULO notation (U or S = 1) Solfege
0 0 1/1 P1 unison D unison P1 D Do
1 50 33/32, 34/33 ^P1, vm2 up-unison, downminor 2nd ^D, vEb super unison, uber unison S1, U1 SD, UD Da/Ru
2 100 16/15, 17/16, 18/17 A1, m2 aug unison, minor 2nd D#, Eb aug unison, minor 2nd A1, m2 D#, Eb Ro
3 150 12/11 ~2 mid 2nd vE neutral 2nd N2 UEb, uE Ra
4 200 9/8, 10/9 M2 major 2nd E major 2nd M2 E Re
5 250 22/19 ^M2, vm3 upmajor 2nd, downminor 3rd ^E, vF supermajor 2nd, subminor 3rd SM2, sm3 SE, sF Ri/Mu
6 300 6/5, 19/16 m3 minor 3rd F minor 3rd m3 F Mo
7 350 11/9, 27/22 ~3 mid 3rd vF# neutral 3rd N3 UF, uF# Ma
8 400 5/4, 24/19 M3 major 3rd F# major 3rd M3 F# Me
9 450 22/17 ^M3, v4 upmajor 3rd, down-4th ^F#, vG supermajor 3rd, sub 4th SM3, s4 SF#, sG Mi/Fu
10 500 4/3 P4 fourth G perfect 4th P4 G Fo
11 550 11/8 ^4, ~4 up-4th, mid-4th ^G uber 4th/neutral 4th U4/N4 UG Fa/Su
12 600 17/12 A4, d5 aug 4th, dim 5th G#, Ab aug 4th, dim 5th A4, d5 G#/Ab Fe/So
13 650 16/11 v5, ~5 down-5th, mid-5th vA unter 5th/neutral 5th u5/N5 uA Fi/Sa
14 700 3/2 P5 fifth A perfect 5th P5 A Se
15 750 17/11 ^5, vm6 up-fifth, downminor 6th ^A, vBb super 5th, subminor 6th S5, sm6 SA, sBb Si/Lu
16 800 8/5, 19/12 m6 minor 6th Bb minor 6th m6 Bb Lo
17 850 18/11, 44/27 ~6 mid 6th vB neutral 6th N6 UBb, uB La
18 900 5/3, 32/19 M6 major 6th B major 6th M6 B Le
19 950 19/11 ^M6, vm7 upmajor 6th, downminor 7th ^B, vC supermajor 6th, subminor 7th SM6, sm7 SB, sC Li/Tu
20 1000 9/5, 16/9 m7 minor 7th C minor 7th m7 C To
21 1050 11/6 ~7 mid 7th vC# neutral 7th N7 UC, uC# Ta
22 1100 15/8, 17/9, 32/17 M7 major 7th C# major 7th M7 C# Te
23 1150 33/17, 64/33 ^M7, vP8 upmajor 7th, down-8ve ^C#, vD sub 8ve, unter 8ve s8, u8 C#, uD Ti/Du
24 1200 2/1 P8 perfect 8ve D perfect 8ve P8 D Do
1. based on treating 24edo as a 2.3.5.11.17.19 subgroup; other approaches are possible.

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.

### Interval qualities in color notation

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. See 24edo Chord Names and Ups and Downs Notation #Chords and Chord Progressions.

### William Lynch's notation

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.

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 24edo that do not exist in 12edo:

Cents Names
50 Quarter tone, infra second, wide unison
150 Neutral second
250 Ultra second, infra third
350 Neutral third
450 Minor fourth, ultra third, narrow fourth
550 Wide fourth
650 Narrow fifth
750 Wide fifth, infra sixth
850 Neutral sixth
950 Ultra sixth , infra seventh
1050 Neutral seventh
1150 Ultra seventh, narrow octave

### Interval alterations

The special alterations of the intervals and chords of 12edo 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 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

### 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

 Semitones Symbol −2 −1+1⁄2 −1 −1⁄2 0 +1⁄2 +1 +1+1⁄2 +2
 A "semisharp" accidental comprising one half of a regular musical sharp symbol.
 A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.
 A "semiflat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
 A "flat and a half" or "sesquiflat" accidental, comprising the above half-flat symbol and a regular flat symbol placed back to back.

Pros: Familiar, fairly easy to learn

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

#### Persian accidentals

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

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 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.

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 A. Barton, a diagram of how to notate 24edo in the Revo flavor of Sagittal:

## JI approximation

Selected 19-limit intervals approximated in 24edo

### Interval mappings

The following tables show how 15-odd-limit intervals are represented in 24edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 24edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.637 1.3
11/8, 16/11 1.318 2.6
3/2, 4/3 1.955 3.9
15/13, 26/15 2.259 4.5
11/9, 18/11 2.592 5.2
9/8, 16/9 3.910 7.8
13/10, 20/13 4.214 8.4
13/8, 16/13 9.472 18.9
13/11, 22/13 10.790 21.6
13/12, 24/13 11.427 22.9
15/8, 16/15 11.731 23.5
15/11, 22/15 13.049 26.1
13/9, 18/13 13.382 26.8
5/4, 8/5 13.686 27.4
9/7, 14/9 14.916 29.8
11/10, 20/11 15.004 30.0
5/3, 6/5 15.641 31.3
7/6, 12/7 16.871 33.7
7/5, 10/7 17.488 35.0
11/7, 14/11 17.508 35.0
9/5, 10/9 17.596 35.2
7/4, 8/7 18.826 37.7
15/14, 28/15 19.443 38.9
13/7, 14/13 21.702 43.4
15-odd-limit intervals in 24edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.637 1.3
11/8, 16/11 1.318 2.6
3/2, 4/3 1.955 3.9
15/13, 26/15 2.259 4.5
11/9, 18/11 2.592 5.2
9/8, 16/9 3.910 7.8
13/10, 20/13 4.214 8.4
13/8, 16/13 9.472 18.9
13/11, 22/13 10.790 21.6
13/12, 24/13 11.427 22.9
15/8, 16/15 11.731 23.5
15/11, 22/15 13.049 26.1
13/9, 18/13 13.382 26.8
5/4, 8/5 13.686 27.4
9/7, 14/9 14.916 29.8
11/10, 20/11 15.004 30.0
5/3, 6/5 15.641 31.3
7/6, 12/7 16.871 33.7
11/7, 14/11 17.508 35.0
9/5, 10/9 17.596 35.2
7/4, 8/7 18.826 37.7
13/7, 14/13 28.298 56.6
15/14, 28/15 30.557 61.1
7/5, 10/7 32.512 65.0

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5.11 81/80, 121/120, 128/125 [24 38 56 83]] −1.08 2.82 5.63
2.3.5.11.13 66/65, 81/80, 128/125, 144/143 [24 38 56 83 89]] −1.37 2.59 5.19
2.3.5.11.13.17 51/50, 66/65, 81/80, 128/125, 144/143 [24 38 56 83 89 98]] −0.94 2.55 5.11
2.3.5.11.13.17.19 51/50, 66/65, 76/75, 81/80, 128/125, 144/143 [24 38 56 83 89 98 102]] −0.89 2.37 4.74

### Uniform maps

13-limit uniform maps between 23.5 and 24.5
Min. size Max. size Wart notation Map
23.5000 23.5588 24bbcddeeeefff 24 37 55 66 81 87]
23.5588 23.6458 24bbcddeefff 24 37 55 66 82 87]
23.6458 23.6599 24bbcddeef 24 37 55 66 82 88]
23.6599 23.6878 24cddeef 24 38 55 66 82 88]
23.6878 23.8478 24ceef 24 38 55 67 82 88]
23.8478 23.9025 24cf 24 38 55 67 83 88]
23.9025 23.9161 24f 24 38 56 67 83 88]
23.9161 24.0440 24 24 38 56 67 83 89]
24.0440 24.1369 24d 24 38 56 68 83 89]
24.1369 24.1863 24de 24 38 56 68 84 89]
24.1863 24.2908 24deff 24 38 56 68 84 90]
24.2908 24.3332 24bdeff 24 39 56 68 84 90]
24.3332 24.4002 24bccdeff 24 39 57 68 84 90]
24.4002 24.4260 24bccdddeff 24 39 57 69 84 90]
24.4260 24.4566 24bccdddeeeff 24 39 57 69 85 90]
24.4566 24.5000 24bccdddeeeffff 24 39 57 69 85 91]

### Commas

This is a partial list of the commas that 24edo tempers out with its patent val, 24 38 56 67 83 89].

Prime
Limit
Ratio[note 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 Greater diesis, diminished comma
5 (12 digits) [18 -4 -5 60.61 Saquingu Passion comma
5 128/125 [7 0 -3 41.06 Trigu Lesser 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 Kirnberger's atom
7 1323/1280 [-8 3 -1 2 57.20 Lazozogu Septimal two-seventh tone
7 49/48 [-4 -1 0 2 35.70 Zozo Slendro diesis
7 245/243 [0 -5 1 2 14.19 Zozoyo Sensamagic comma
7 19683/19600 [-4 9 -2 -2 7.32 Labirugu Cataharry comma
7 6144/6125 [11 1 -3 -2 5.36 Sarurutrigu Porwell comma
11 56/55 [3 0 -1 1 -1 31.19 Luzogu Undecimal tritonic comma
11 245/242 [-1 0 1 2 -2 21.33 Luluzozoyo Frostma
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 comma
11 243/242 [-1 5 0 0 -2 7.14 Lulu Rastma
11 (18 digits) [15 8 0 0 -8 5.10 Quadbilu Octatonic comma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 (18 digits) [24 -6 0 1 -5 0.51 Saquinlu-azo Quartisma
11 9801/9800 [-3 4 -2 -2 2 0.18 Bilorugu Kalisma, Gauss' comma
13 66/65 [1 1 -1 0 1 -1 26.43 Thulogu Winmeanma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap comma, biome comma
13 512/507 [9 -1 0 0 0 -2 16.99 Thuthu Tridecimal neutral thirds comma
13 105/104 [-3 1 1 1 0 -1 16.57 Thuzoyo Animist comma
13 144/143 [4 2 0 0 -1 -1 12.06 Thulu Grossma
13 676/675 [2 -3 -2 0 0 2 2.56 Bithogu Island comma, parizeksma
13 4096/4095 [12 -2 -1 -1 0 -1 0.42 Sathurugu Schismina
17 51/50 [-1 1 -2 0 0 0 1 34.28 Sogugu Large septendecimal sixth tone
17 136/135 [3 -3 -1 0 0 0 1 12.78 Sogu Diatisma, fiventeen comma
17 170/169 [1 0 1 0 0 -2 1 10.21 Sothuthuyo Major naiadma
17 221/220 [-2 0 -1 0 -1 1 1 7.85 Sotholugu Minor naiadma
17 256/255 [8 -1 -1 0 0 0 -1 6.78 Sugu Charisma, septendecimal kleisma
17 289/288 [-5 -2 0 0 0 0 2 6.00 Soso Semitonisma
17 1225/1224 [-3 -2 2 2 0 0 -1 1.41 Subizoyo Noellisma
19 77/76 [-2 0 0 1 1 0 0 -1 22.63 Nulozo Small undevicesimal ninth tone
19 76/75 [2 -1 -2 0 0 0 0 1 22.93 Nogugu Large undevicesimal ninth tone
19 96/95 [5 1 -1 0 0 0 0 -1 18.13 Nugu 19th Partial chroma
19 133/132 [-2 -1 0 1 -1 0 0 1 13.07 Noluzo Minithirdma
19 153/152 [-3 2 0 0 0 0 1 -1 11.35 Nuso Ganassisma
19 171/170 [-1 2 -1 0 0 0 -1 1 10.15 Nosugu Malcolmisma
19 209/208 [-4 0 0 0 1 -1 0 1 8.30 Nothulo Yama comma
19 324/323 [2 4 0 0 0 0 -1 -1 5.35 Nusu Photisma
19 361/360 [-3 -2 -1 0 0 0 0 2 4.80 Nonogu Go comma
19 5776/5775 [4 -1 -2 -1 -1 0 0 2 0.30 Nonolurugugu Neovish comma

### Rank-2 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 8ve
Generator Name
1 1\24
1 5\24 Semaphore, godzilla, bridgetown
1 7\24 Mohajira (patent val), neutrominant (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 Hemidim
6 1\24 Hemisemiaug
8 1\24 Semidim
12 1\24 Catler

## Scales and modes

See: 24edo scales and List of MOS scales in 24edo.

## Tetrachords

See 24edo tetrachords.

## 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 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 two:

Fundamental triads of 24edo
JI Chord Edosteps Notes of C Chord Written name Spoken name
6:7:9, 26:30:39 0 – 5 – 14 C – E – G Cvm
Cm(3), Cmin(3)
C subminor
C minor semiflat-three
10:12:15 0 – 6 – 14 C – E♭ – G Cm, Cmin C minor
18:22:27, 22:27:33 0-7-14 C – E – G C~, Cneu C neutral
4:5:6 0 – 8 – 14 C – E – G C, Cmaj C, C major
14:18:21, 10:13:15 0 – 9 – 14 C – E – G C^
C(3), Cmaj(3)
C supermajor
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.

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.

More good chords in 24edo:

• 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 (Godzilla) pentatonic scale (0 – 5 – 10 – 14 – 19 – 24), that is close to equi-pentatonic and also close to several Indonesian slendro scales. 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:

Fundamental tetrads of 24edo
Degrees of 24edo Chord spelling Notes of C chord Written name Spoken name Audio example
0 – 5 – 14 – 19 1 – vb3 – 5 – vb7 C – E – G – B smin7
min7(3, 7)
Subminor seven
Minor seven semiflat-three semiflat-seven
0 – 6 – 14 – 20 1 – b3 – 5 – b7 C – E♭ – G – B♭ m7, min7 Minor seven
0 – 7 – 14 – 21 1 – v3 – 5 – v7 C – E – G – B n7, neu7 Neutral seven
0 – 8 – 14 – 22 1 – b3 – 5 – b7 C – E – G – B maj7 Major seven
0 – 8 – 14 – 22 1 – b3 – 5 – b7 C – E – G – B smaj7
maj7(3, 7)
Supermajor seven
Major seven semisharp-three semisharp-seven
0 – 8 – 14 – 20 1 – 3 – 5 – b7 C – E – G – B♭ 7, dom7 Dominant seven
0 – 8 – 14 – 19 1 – 3 – 5 – vb7 C – E – G – B h7
7(7)
Harmonic seven
Dominant 7 semiflat-seven
0 – 5 – 14 – 20 1 – vb3 – 5 – b7 C – E – G – B♭ min7(3) Arto
Minor seven semiflat-three
0 – 9 – 14 – 19 1 – ^3 – 5 – vb7 C – E – G – B h7(3)
7(3, 7)
Tendo
Harmonic seven semisharp-three
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.

## 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.

Furthermore, in the same fashion, every sequence of intervals available in 12edo are reachable by equal contrary motion in 24edo.

 Every sequence of 12edo intervals are reachable by strict contrary motion in 24edo. >

## 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.

Guitars with 24 frets per octave are also an option and some guitar makers, such as Ron Sword's Metatonal Music, can make custom instruments and perform re-fretting, with an example below:

However, 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.

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.

24edo can also be played on the Lumatone: see Lumatone mapping for 24edo

## Music

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