24edo

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

highly composite

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 2^1/24, or the 24th root of 2.

English Wikipedia has an article on:

Quarter tone

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 and 7/6) are almost as inaccurate 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, 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 and 17. Another approach would be to treat 24edo as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.

Prime harmonics

Approximation of prime harmonics in 24edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.0	-2.0	+13.7	-18.8	-1.3	+9.5	-5.0	+2.5	+21.7	+20.4	+5.0
Error	relative (%)	+0	-4	+27	-38	-3	+19	-10	+5	+43	+41	+10
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	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	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	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	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	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	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	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	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

↑ based on treating 24edo as a 2.3.11.17.19 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
minor	gu	(a, b, -1)	6/5, 9/5
mid	ilo	(a, b, 0, 0, 1)	11/9, 11/6
mid	lu	(a, b, 0, 0, -1)	12/11, 18/11
major	yo	(a, b, 1)	5/4, 5/3
major	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	quartertone, 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

	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

Regular temperament properties

Selected 19-limit intervals approximated in 24edo

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^[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	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	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	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

↑ 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 (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 / modes

See 24edo scales.

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, there are three new fundamental options, giving five basic triads over 12edo's two:

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 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:

Degrees of 24edo	Chord spelling	Written name	Spoken name
0–5–14–19	1–vb3–5–vb7	smin7 min7( 3, 7)	Subminor seven Minor seven semiflat-three semiflat-seven
0–6–14–20	1–b3–5–b7	m7, min7	Minor seven
0–7–14–21	1–v3–5–v7	n7, neu7	Neutral seven
0–8–14–22	1–b3–5–b7	maj7	Major seven
0–8–14–22	1–b3–5–b7	smaj7 maj7( 3, 7)	Supermajor seven Major seven semisharp-three semisharp-seven
0–8–14–20	1–3–5–b7	7, dom7	Dominant seven
0–8–14–19	1–3–5–vb7	h7 7( 7)	Harmonic seven Dominant 7 semiflat-seven
0–5–14–20	1–vb3–5–b7	min7( 3)	Arto Minor seven semiflat-three
0–9–14–19	1–^3–5–vb7	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 are also an option and are available from Ron Sword / Metatonal Music, with an example below.

However, while playable, these do make playing ordinary chords and navigating the fretboard significantly more challenging. More common is the "Sazocaster" tuning popularised by Australian band King Gizzard and the Lizard Wizard, which adds quartertones between approximately half the regular frets. Multiple guitar companies have produced variants of this, including Eastwood and Revelation.

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