24edo

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24edo

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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, 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. 24edo is also good at the 13th harmonic, which makes it a good 2.3.5.11.13 system. Specifically, intervals of 13/5 are particularly well approximated. And of course, 24edo shares its 17 and 19 tunings with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.

While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically: a supermajor chord is available at [0 9 14] and a subminor chord at [0 5 14], though they're more ultramajor and inframinor.

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 being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). 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.

Intervals

Degree	Cents	Approximate ratios	Ups and downs notation (EUs: vvA1 and d2)			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, 19/18	A1, m2	aug unison, minor 2nd	D#, Eb	aug unison, minor 2nd	A1, m2	D#, Eb	Ro
3	150	13/12, 12/11, 11/10	~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	15/13, 22/19	^M2, vm3	upmajor 2nd, downminor 3rd	^E, vF	supermajor 2nd, subminor 3rd	SM2, sm3	SE, sF	Ri/Mu
6	300	6/5, 13/11, 19/16	m3	minor 3rd	F	minor 3rd	m3	F	Mo
7	350	11/9, 16/13, 27/22, 39/32	~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	13/10, 17/13, 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, 15/11	^4, ~4	up-4th, mid-4th	^G	uber 4th/neutral 4th	U4/N4	UG	Fa/Su
12	600	17/12, 24/17, 45/32, 64/45	A4, d5	aug 4th, dim 5th	G#, Ab	aug 4th, dim 5th	A4, d5	G#/Ab	Fe/So
13	650	16/11, 22/15	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, 20/13	^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	13/8, 18/11, 44/27, 64/39	~6	mid 6th	vB	neutral 6th	N6	UBb, uB	La
18	900	5/3, 22/13, 32/19	M6	major 6th	B	major 6th	M6	B	Le
19	950	19/11, 26/15	^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, 20/11	~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.

Notation

Ups and downs notation

Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

Semitones	−2	−1+1⁄2	−1	−1⁄2	0	+1⁄2	+1	+1+1⁄2	+2
Symbol

Stein–Zimmermann accidentals

Semitones	−2	−1+1⁄2	−1	−1⁄2	0	+1⁄2	+1	+1+1⁄2	+2
Symbol

	A "semisharp" or "half-sharp" 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" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left.
	A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.

Pros: familiar, intuitive, and fairly easy to learn.

Cons: can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.

Persian quartertone accidentals

English Wikipedia has articles on:

Koron (music)

Sori (music)

	Koron = quarter-tone flat
	Sori = quarter-tone sharp

Pros: easy to read.

Cons: hard to write on a computer, does not fit with standard notation well.

Sagittal notation

This notation uses the same sagittal sequence as edos 17, 31, and 38, is a subset of the notations for edos 48 and 72, and is a superset of the notations for edos 12, 8, and 6.

Evo flavor

Revo flavor

Evo-SZ flavor

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

Pros and cons

Revo 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:

Interval and chord naming

Combining ups and downs with 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 interval and chord names

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

Further discussion of interval and chord naming

Main article: 24edo/Interval names and harmonies

• 24edo Chord Names
• Ups and downs notation#Chords and Chord Progressions.

Approximation to JI

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 23.8 and 24.2

Min. size	Max. size	Wart notation	Map
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]

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	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	Diminished comma, greater diesis
5	(12 digits)	[18 -4 -5⟩	60.61	Saquingu	Passion comma
5	128/125	[7 0 -3⟩	41.06	Trigu	Augmented comma, lesser diesis
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	Semaphoresma, 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
11	(14 digits)	[-1 -11 -1 0 6⟩	0.089	Satribilo-agu	Parimo
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	351/350	[-1 3 -2 -1 0 1⟩	4.94	Thorugugu	Ratwolfsma
13	352/351	[5 -3 0 0 1 -1⟩	4.93	Thulo	Minor minthma
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	76/75	[2 -1 -2 0 0 0 0 1⟩	22.93	Nogugu	Large undevicesimal ninth tone
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
19	5776/5775	[4 -1 -2 -1 -1 0 0 2⟩	0.30	Nonolurugugu	Neovish comma

Rank-2 temperaments

• List of 24et rank two temperaments by badness
• List of edo-distinct 24et rank two temperaments

Periods per 8ve	Generator	Name
1	1\24	Hemiripple (24)
1	5\24	Godzilla (24) / baragon (24) / varan (24)
1	7\24	Mohajira (24) / neutrominant (24d) / migration (24d)
1	11\24	Cohemiripple (24), freivald (24)
2	1\24	Shrutar (24)
2	5\24	Sruti (24), anguirus (24), decimal (24c)
3	1\24	Hemiaug (24)
3	3\24	Triforce (24)
4	1\24	Hemidim (24)
6	1\24	Hemisemiaug (24)
8	1\24	Semidim (24)
12	1\24	Catler

Important MOSes include:

• Semaphore 4L1s 55455 (generator: 5\24)
• Semaphore 5L4s 414141414 (generator: 5\24)
• Mohajira 3L4s 3434343 (generator: 7\24)
• Mohajira 7L3s 3313313313 (generator: 7\24)

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 inframinor 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 ultramajor 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 barbados major (10:13:15) and barbodos minor (26:30:39) triads 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)	Inframinor 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)	Ultramajor 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.

Guitar

Adam Hoey Xen (on YouTube) has used a "neutral thirds tuning" of F#-At-C#-Et-G#-Bt on a standard guitar to play in quartertones.

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:

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.

Harp, Harpsichord, and Piano

Scordatura tuning of 12edo instruments

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. Iceface tuning is one type of scordatura piano (or other keyboard instrument) tuning. A more complex type of scordatura tuning was required for a performance of Charles Ives' 4th Symphony which calls for a quarter-tone piano, but for which no quarter-tone piano was available, as described by Thomas Broadhead in this video. For this composition the gamut of notes needed would not be met using a simple transformation such as Iceface.

Although no recording using the above tuning is currently legally freely available, Paweł Mykietyn has used a similar idea with harp and harpsichord. A score video of this is available as Klave for Microtonal Harpsichord and Chamber Orchestra (Score-Video) (2004, performed by Elżbieta Chojnacka with Marek Moś conducting the AUKSO chamber orchestra of the city of Tychy, uploaded by Quinone Bob with permission from Paweł Mykietyn); the video starts with slides explaining the scordatura tuning of each manual of the Revival harpsichord (with each manual having a differrent scordatura tuning), followed by the scordatura tuning of the harp.

Quarter-tone instruments

A very small number of quarter-tone pianos have been built — here are a couple of videos of these instruments being tested/played experimentally (to demonstrate their capabilities rather than to play specific compositions that would qualify for the 24edo Music section):

Quarter-tone grand piano, Czech Museum of Music (this piano is essentially two stacked grand pianos, and as such is massive, in order to avoid sacrificing strings per note)

• Demonstration short video by Nahre Sol (2024)

Quarter-tone upright piano, Academy of Music in Prague (Czech Republic) (this piano apparently sacrificed number of strings per note in order to be able to fit into a reasonable amount of space)

• Demonstration video by Steve Cohn (2011)

Electronic Keyboards

24edo can also be played on the Lumatone, with better ergonomics than the quarter-tone pianos noted above: see Lumatone mapping for 24edo

Flute

Likewise, some flutes have been built by Eva Kingma — here is a video exploring the capabilities of these, intermixed with regular 12edo playing:

Quarter-tone flute, made by Eva Kingma

• Visit to the workshop of Eva Kingma, followed by test by Manuel Luis Cochofel (2010) (demonstration of fingering starts at 06:56)

Music

English Wikipedia has an article on:

List of quarter tone pieces

Main article: 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
  • 48edo (2mu tuning)
  • 96edo (3mu tuning)
  • 192edo (4mu tuning)
  • 384edo (5mu tuning)
  • 768edo (6mu tuning)
  • 1536edo (7mu tuning)
  • 3072edo (8mu tuning)
  • 6144edo (9mu tuning)
  • 12288edo (10mu tuning)
  • 24576edo (11mu tuning)
  • 49152edo (12mu tuning)
  • 98304edo (13mu tuning)
  • 196608edo (14mu tuning)

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