24edo
← 23edo | 24edo | 25edo → |
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.
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
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 |
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 | −2 | −11⁄2 | −1 | −1⁄2 | 0 | +1⁄2 | +1 | +11⁄2 | +2 |
---|---|---|---|---|---|---|---|---|---|
Symbol |
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:
Further discussion of 24edo notation
- 24edo interval names and harmonies
- 24edo Chord Names
- Ups and Downs Notation#Chords and Chord Progressions.
JI approximation
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.
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 |
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
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:
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:
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.
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
Further reading
- Ellis, Don. Quarter Tones: A Text with Musical Examples, Exercises and Etudes. 1975.
- Sword, Ron. Icosikaitetraphonic Scales for Guitar: Theory and Scales for Twenty-four Equal Divisions of the Octave. 2009. (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.)
See also
- Equal multiplications of MIDI-resolution units
External links
- quarter-tone / 24-edo / 24-ed2 Permalink on Tonalsoft Encyclopedia
- About 24-EDO Permalink by Shaahin Mohajeri
- Notation and Chord Names for 24-EDO by William Lynch
- The place of QUARTERTONES in Today's Xenharmonics by Ivor Darreg
- Tonalsoft Encyclopedia | quarter-tone / 24-edo / 24-ed2
Notes
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.