24edo

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24edo

25edo →

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

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

Approximation of odd harmonics in 24 EDO

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 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 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 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.

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.

For the full list, along with some chord progression information, see full article on 24 Edo intervals, and also 24edo Chord Names and Ups and Downs Notation#Chords and Chord Progressions.

Ups and down notation

Degree	Cents	Approximate Ratios	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

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:

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 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 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

Alternate quartertone accidentals

External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif

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= quarter-tone sharp or Jump

††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp

External image: http://www.dolmetsch.com/onequarterflat.gif

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= quarter-tone flat or Drop

External image: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png

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= three quarter-tones flat or drop-flat

For example, the scale 0-5-10-15-20 is written as C-D

External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif

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(or E

External image: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png

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) F G

External image: http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif

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(or A

External image: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png

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) Bb.

Pros: Very easy to distinguish accidentals from one another

Cons: Not practical, tends to clutter a score

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 24-EDO in the Revo flavor of Sagittal:

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.

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.

More good chords in 24-tET:

• 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.

Commas

24edo tempers out the following commas. This assumes 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	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

Rank-2 temperaments

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

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.

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

• Microhex3 Microhex4 and Microhex5 by Shaahin Mohajeri
• Quarterpicnic by Chris Vaisvil
• Quarter Tone Prelude For Two Harps by Nathan BeDell
• Fretless Chrome 1 and Fretless Chrome 2 by Chris Vaisvil
• Lament by Jake Freivald. In the freivaldneutral24 scale.
• Mo - Happy - Happy play by Jake Freivald in Neutral[7] (2.3.11 mohajira), 24et tuning
• Autumn Winds, Easter Time at Nine, Waters of Persia by William Lynch in mohajira, 24et tuning.
• Serena, by Mason Green (intro and coda in 24edo, the rest is in 12edo)
• Autumn Girl, by Mason Green
• "Prométhée enchaîné" by Fromental Halévy (considered the first mainstream western orchestral composition to use quarter tones.)
• "3 Hommages" by Georg Friedrich Haas
• "Folly of a Drunk" by Dawson Berry (mp3) (score)
• List of quartertone pieces on Wikipedia

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.)

External links

• quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory Permalink
• About 24-EDO by Shaahin Mohajeri Permalink
• Notation and Chord Names for 24-EDO by William Lynch
• The place of QUARTERTONES in Today's Xenharmonics by Ivor Darreg