24edo
| ← 23edo | 24edo | 25edo → |
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
| Odd harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -2.0 | +13.7 | -18.8 | -3.9 | -1.3 | +9.5 | +11.7 | -5.0 | +2.5 | -20.8 | +21.7 | -22.6 | -5.9 | +20.4 | +5.0 |
| relative (%) | -4 | +27 | -38 | -8 | -3 | +19 | +23 | -10 | +5 | -42 | +43 | -45 | -12 | +41 | +10 | |
| Steps (reduced) | 38 (14) | 56 (8) | 67 (19) | 76 (4) | 83 (11) | 89 (17) | 94 (22) | 98 (2) | 102 (6) | 105 (9) | 109 (13) | 111 (15) | 114 (18) | 117 (21) | 119 (23) | |
The 5-limit approximations in 24edo are the same as those in 12edo, therefore 24edo 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 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like 36et, 72et, 84et or 156et. However, it should be noted that 24edo, like 22edo, does temper out the quartisma, linking the otherwise sub-par 7-limit harmonies with those of the 11-limit.
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 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 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.
24edo is the 6th highly composite EDO.
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 | Solfege | ||
|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | P1 | unison | C | Do |
| 1 | 50 | 33/32, 34/33 | ^P1, vm2 | up-unison, downminor 2nd | ^C, vDb | Da/Ru |
| 2 | 100 | 17/16, 18/17 | A1, m2 | aug unison, minor 2nd | C#, Db | Ro |
| 3 | 150 | 12/11 | ~2 | mid 2nd | vD | Ra |
| 4 | 200 | 9/8 | M2 | major 2nd | D | Re |
| 5 | 250 | 22/19 | ^M2, vm3 | upmajor 2nd, downminor 3rd | ^D, vEb | Ri/Mu |
| 6 | 300 | 19/16 | m3 | minor 3rd | Eb | Mo |
| 7 | 350 | 11/9, 27/22 | ~3 | mid 3rd | vE | Ma |
| 8 | 400 | 24/19 | M3 | major 3rd | E | Me |
| 9 | 450 | 22/17 | ^M3, v4 | upmajor 3rd, down-4th | ^E, vF | Mi/Fu |
| 10 | 500 | 4/3 | P4 | fourth | F | Fo |
| 11 | 550 | 11/8 | ^4, ~4 | up-4th, mid-4th | ^F | Fa/Su |
| 12 | 600 | 17/12 | A4, d5 | aug 4th, dim 5th | F#, Gb | Fe/So |
| 13 | 650 | 16/11 | v5, ~5 | down-5th, mid-5th | vG | Fi/Sa |
| 14 | 700 | 3/2 | P5 | fifth | G | Se |
| 15 | 750 | 17/11 | ^5, vm6 | up-fifth, downminor 6th | ^G, vAb | Si/Lu |
| 16 | 800 | 19/12 | m6 | minor 6th | Ab | Lo |
| 17 | 850 | 18/11, 44/27 | ~6 | mid 6th | vA | La |
| 18 | 900 | 32/19 | M6 | major 6th | A | Le |
| 19 | 950 | 19/11 | ^M6, vm7 | upmajor 6th, downminor 7th | ^A, vBb | Li/Tu |
| 20 | 1000 | 16/9 | m7 | minor 7th | Bb | To |
| 21 | 1050 | 11/6 | ~7 | mid 7th | vB | Ta |
| 22 | 1100 | 17/9, 32/17 | M7 | major 7th | B | Te |
| 23 | 1150 | 33/17, 64/33 | ^M7, vP8 | upmajor 7th, down-8ve | ^B, vC | Ti/Du |
| 24 | 1200 | 2/1 | P8 | perfect 8ve | C | 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.
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:
| The twelve new intervals in 24edo | some nearby JI intervals | |||||
|---|---|---|---|---|---|---|
| cents | pions | 7mus (hex) | 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 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
or ^ = quarter-tone sharp or "Jump" or "up"
or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
or v = quarter-tone flat or "Drop" or "down"
or bv or vb = three-quarter-tone sharp 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
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.
Commas
24edo 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 |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
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 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 | Hemidim |
| 6 | 1\24 | Hemisemiaug |
| 8 | 1\24 | Semidim |
| 12 | 1\24 | Catler |
Scales / modes
See 24edo modes.
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 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)
- 0-5-14 (Infra Triad or downminor triad)
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 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.
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.
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.
Music
- Lament in the freivaldneutral24 scale.
- Mo - Happy - Happy play in Neutral[7] (2.3.11 mohajira), 24et tuning
- Serena (intro and coda in 24edo, the rest is in 12edo)
- Autumn Girl
- "3 Hommages"
- "Prométhée enchaîné" (considered the first mainstream western orchestral composition to use quarter tones.)
- Autumn Winds, Easter Time at Nine, Waters of Persia in mohajira, 24et tuning.
See also:
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.)