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
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.
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 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 edosteps, 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.)
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:
24 EDO Interval names and Harmonies
24 EDO or 24 ET divides the octave into 24 equal parts and is also a multiple of twelve, therefore, 24 EDO contains all of the original harmonies found in 12 EDO. This page
seeks to explore the new harmonies available in a 24 Tone system.
Tone Sizes
While 12 EDO contains only two tone sizes: the whole tone at 200 cents, and the semitone at 100 cents,
24 EDO contains five being that it has three additional new tone sizes. Generally, as it divides the octave into 24 parts, it’s a good idea
to approach intervals and tones with this mindset of there being a new wider or narrower version of the previous intervals. These
tone sizes are mainly used in context of scale steps and sometimes modulation but not usually in context of a chord or scale degrees.
Quarter Tone
The quarter tone is the smallest tone size in 24 EDO. At only 50 cents, It is a highly dissonant interval and has a characteristic washy, beating sound to it and is reminiscent of tuning instrument. Melodically it can function similar to the way a semitone does in 12 EDO but it tends to sound really different as it’s such a small interval. Due to the high dissonance, this interval is challenging to make it sound good in a chord within the context of tonal music but can work quite well for composers who wish to explore the dissonance of 24 EDO. Through chord changes, the quarter tone is very effective in creating a sound of a record player going in and out of pitch. It can be a nice effect in smooth jazz progressions or post-modal music to simply move a diatonic chord from 12 EDO up a quarter tone as quarter tone root movement is quite novel in sound. Within scale context, the quarter tone is represented by a lowercase q. An example of quarter tone is C to Ct or enharmonically C to Ddb.
Whole Tones
24 EDO there are not one, but two distinct sizes of larger tones. The wide whole tone usually called “wide tone” at 250 cents, and the natural whole tone, usually called “whole tone” at 200 cents, therefore the natural whole tone is exactly the same interval that appears in 12 EDO as a whole tone. The wide whole tone generally has a more metallic sound than the narrow tone as well as a more moody character compared to the brightness of the natural whole tone. In context of a major chord, the wide tone brings a much colder flavor to the major chord than the
whole tone which enhances the brightness of the major chord. The wide tone from the root clashes heavily with minor chords as the minor third and the wide tone are only a quarter tone apart. Diatonic chords tend to move naturally by wide tone movement such as moving an Am chord to an Gd major chord. The wide tone is fairly unique to 24 EDO as in
it’s too small to be considered a good 7/6 and fits more as being described as a 15/13. The major whole tone is represented by a lowercase w while to wide whole tone is represented by an uppercase W. An example of a whole tone is C to D and a wide tone is C to Dt or enharmonically C to Edb.
Semitones
Like whole tones, there are two distinct sizes of semitones in 24 EDO: The narrow semitone at 100 cents and the wider semitone at 150 cents called “neutral”. While the narrow semitone is exactly the same as the 12 EDO semitone, the neutral tone is unique. The neutral tone is called so because it represents both a narrow whole tone and a wide semitone depending on how it’s used. It is heavily used in persian, turkish, and other forms of eastern music as well as some east asian scales though normally is slightly sharp or flat from 24 ET. The character of the neutral tone resembles the sound of bells, a car horn, and other sounds that are normally considered “non-musical” which can be a valuable asset to those trying to impressionistically compose music to mimic sounds such as trains and car horns. If fact, the dialing tone in the US is fairly close to a 150 cent neutral tone so I guess you could call it a “dial tone” hehe.
An example of a narrow semitone is C to C# or enharmonically, C to Db. An example of a wide semitone is C to Dd.
Seconds
Within a theory context, the above tone sizes will normally be referred to as seconds of the following names from great to small:
Infra Second 50c – Minor Second 100c – Neutral Second 150c – Major Second 200c – Ultra Second 250c. Therefore an Infra second is the
same as a Quarter Tone, a Minor second is the same as a semitone, a Neutral second is the same enharmonically as a neutral tone/wide semitone/narrow whole tone, a Major second is the same as a whole tone, and an Ultra second is the same as a Wide tone.
Thirds
Like seconds, 24 EDO contains five sizes of thirds which are in order: Infra Third 250c – Minor Third 300c – Neutral third 350c – Major Third 400c – Ultra Third 450c. Obviously, the Infra Third is enharmonically the same as the ultra second but appear differently on the staff and function differently. In a chord such as C E G Dt or 000 – 400 – 700 – 250, 250c is functioning as an ultra ninth but in a chord such as 000 – 250 – 700 – 950, we could say that it’s probably functioning as an infra third. The infra third is thought to represent 15/13, the neutral third is 11/9 and the ultra third is 13/10.
Fourths
24 EDO contains five distinct sizes of fourths: Diminished fourth 400c – Narrow fourth 450c – Perfect fourth 500c – Wide fourth 550c – Augmented fourth 600c. The wide fourth is a great representation of the eleventh harmonic 11/8, while the narrow fourth is enharmonically the same as an ultra third. 11/8 is a fantastic addition to major triads while 13/10 can sound good with minor triads.
Fifths
24 EDO contains four distinct sizes of fifths: Diminished Fifth 600 – Narrow fifth 650c – Perfect fifth 700c – Wide fifth 750c – Augmented fifth 800c. The sound of the narrow and wide fifths are very cool and extremely different from 12 ET sounds. The narrow fifth, which is considered to be the most dissonant interval in 24 ET next to the ultra seventh, is a fantastic representation of the 11th subharmonic 16/11. The narrow fifth sometimes called “wolf fifth” can be used to create dynamic texture and voice leading but has a rough character that can be challenging to incorporate into the music well. The wide fifth is a good representation of 17/11.
Sixths
24 EDO contains five distinct sizes of sixths: Infra Sixth 750c – Minor Sixth 800c – Neutral Sixth 850c – Major Sixth 900c – Ultra Sixth – 950c. The Infra sixth is enharmonically the same as the wide fifth. The neutral sixth is a decent approximation of 13/8, the thirteenth harmonic, however not quite as good as how well 24 approximates 11/8; The interval may more closely approximate 18/11. The Neutral sixth is though to be the sweetest sounding neutral interval in the tuning.
The Ultra sixth approximates the interval 26/15 very closely but also can be considered a rather poor 7/4, yet can functionally harmonize in the same manner.
Sevenths
24 EDO contains five distinct sizes of sevenths: Infra Seventh 950c – Minor Seventh 1000c – Neutral Seventh 1050c – Major Seventh 1100c – Ultra Seventh – 1150c. The Infra seventh is enharmonically the same as the Ultra Sixth as it represents 26/15 or sometimes 7/4 but rather poorly. The Neutral seventh is a fantastic 11/6 and the Ultra Seventh represents a good 35/18 and has a very rough character as it is the inversion of the infra second.
Special Enharmonics
24 EDO contains certain enharmonics that are good to keep in mind, the list is as follows:
An ultra second and infra third are both 250 cents.
An ultra third and narrow fourth are both 450 cents.
A wide fifth and infra sixth are both 750 cents.
An ultra sixth and infra seventh are both 950 cents.
Interval Class Categories
Neutral
Neutral intervals basically are right between the major and minor version of an interval in 12 EDO. For example, the neutral third is between the major and minor third. The name also suggests that the interval can function as either depending on how it’s used. In addition, neutral intervals contain very special color to them that makes them unique.
Ultra/Infra
Ultra and Infra are used to describe intervals that are borderline between two classes. For example, an Ultra third borderlines a perfect fourth and can also be called a narrow fourth. Only seconds, thirds, sixths, and sevenths have ultra and infra classes, and the names are never used to describe fourths, fifths, or octaves.
Narrow/Wide
Narrow and wide are used to describe intervals that are between perfect and diminished/augmented. While they aren’t normally used to describe anything but fourths, fifths, and sometimes octaves, they can be used to describe extreme altered intervals such as 350 cents being a narrow diminished fourth.
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:
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
Taken from the xenharmonic wiki to preserve some knowledge
Music by me in this tuning