27edo

← 26edo 27edo 28edo →
Prime factorization	33
Step size	44.4444 ¢
Fifth	16\27 (711.111 ¢)
Semitones (A1:m2)	4:1 (177.8 ¢ : 44.44 ¢)
Consistency limit	9
Distinct consistency limit	7

27 equal divisions of the octave (abbreviated 27edo or 27ed2), also called 27-tone equal temperament (27tet) or 27 equal temperament (27et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 27 equal parts of about 44.4 ¢ each. Each step represents a frequency ratio of 2^1/27, or the 27th root of 2.

Theory

Assuming pure octaves, 27edo divides the octave in 27 equal parts each exactly 444⁄9 cents in size. Its fifth and harmonic seventh are both sharp by 9 ¢, and the major third is the same 400-cent major third as 12edo, sharp by 13.7 ¢. The result is that 6/5, 7/5, and especially 7/6 are all tuned more accurately than this. It can be considered the superpythagorean counterpart of 19edo, as its 5th is audibly indistinguishable from 1/3-septimal-comma superpyth in the same way that 19edo is audibly indistinguishable from 1/3-syntonic-comma meantone: Three fourths (C-Eb) in 19edo reach a near-perfect 6/5 and the same distance in 27edo reaches a near-perfect 7/6.

Though 27edo's 7-limit tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both consistently and distinctly—that is, everything in the 7-odd-limit tonality diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates 19/10, 19/12, and 19/14, so 0 – 7 – 13 – 25 does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the BPS scale in 43edt, although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.

Its step of 44.4 ¢, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very high harmonic entropy. In other words, there is a general perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

The chromatic semitone of 27edo, at 178 ¢, is equal to a submajor second in size, meaning 27edo is a candidate for extraclassical tonality due to its sharp major third of 444 cents.

Odd harmonics

Approximation of odd harmonics in 27edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+9.2	+13.7	+9.0	+18.3	-18.0	+3.9	-21.6	-16.1	+13.6	+18.1	-6.1
Error	Relative (%)	+20.6	+30.8	+20.1	+41.2	-40.5	+8.8	-48.6	-36.1	+30.6	+40.7	-13.6
Steps (reduced)		43 (16)	63 (9)	76 (22)	86 (5)	93 (12)	100 (19)	105 (24)	110 (2)	115 (7)	119 (11)	122 (14)

As a tuning of other temperaments

27edo, with its 400 ¢ major third, tempers out the lesser diesis, 128/125, and the septimal comma, 64/63, and hence 126/125 as well. These it shares with 12edo, making some relationships familiar, and they both support the augene temperament. It shares with 22edo tempering out the sensamagic comma 245/243 as well as 64/63, so that they both support the superpyth temperament, with four quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4. The sharp 9/7 befits a generator for sensi, which 27edo also supports, but a much better tuning is found in 46edo. Another notable temperament 27edo supports is myna, which divides the category of thirds into five different intervals: subminor, minor, neutral, major, and supermajor, representing 7/6, 6/5, 11/9~16/13, 5/4, and 9/7, respectively.

Subsets and supersets

Since 27 factors into primes as 3³, 27edo contains 3edo and 9edo as subsets. Multiplying it by 3 gives 81edo, which is a good meantone tuning.

Intervals

#	Cents	Interval regions	Approximate ratios^{[note 1]}	Ups and downs notation
0	0.0	Unison	1/1	D
1	44.4	Diesis	28/27, 36/35, 39/38, 49/48, 50/49, 81/80	^D, E♭
2	88.9	Minor second	16/15, 21/20, 25/24, 19/18, 20/19	^^D, ^E♭
3	133.3	Neutral second	15/14, 14/13, 13/12	vD♯, vvE
4	177.8	Small major second	10/9	D♯, vE
5	222.2	Large major second	8/7, 9/8	E
6	266.7	Subminor third	7/6	F
7	311.1	Minor third	6/5, 19/16	^F, G♭
8	355.6	Neutral third	16/13	^^F, ^G♭
9	400.0	Major third	5/4, 24/19	vF♯, vvG
10	444.4	Supermajor third	9/7, 13/10	F♯, vG
11	488.9	Perfect fourth	4/3	G
12	533.3	Superfourth	19/14, 26/19, 27/20, 48/35	^G, A♭
13	577.8	Small tritone	7/5, 18/13	^^G, ^A♭
14	622.2	Large tritone	10/7, 13/9	vG♯, vvA
15	666.7	Subfifth	19/13, 28/19, 35/24, 40/27	G♯, vA
16	711.1	Perfect fifth	3/2	A
17	755.6	Subminor sixth	14/9, 20/13	^A, B♭
18	800.0	Minor sixth	8/5, 19/12	^^A, ^B♭
19	844.4	Neutral sixth	13/8	vA♯, vvB
20	888.9	Major sixth	5/3, 32/19	A♯, vB
21	933.3	Supermajor sixth	12/7	B
22	977.8	Harmonic seventh	7/4, 16/9	C
23	1022.2	Large minor seventh	9/5	^C, D♭
24	1066.7	Neutral seventh	13/7, 24/13, 28/15	^^C, ^D♭
25	1111.1	Major seventh	15/8, 19/10, 36/19, 40/21, 48/25	vC♯, vvD
26	1155.6	Supermajor seventh	27/14, 35/18, 49/25, 96/49, 160/81	C♯, vD
27	1200.0	Octave	2/1	D

↑ As a 2.3.5.7.13.19-subgroup temperament, inconsistent intervals in italic.

Proposed interval names and solfèges

Table of proposed interval names and solfèges
#	Cents	Ups and downs notation (EUs: v⁴A1 and vm2)			Solfèges
0	0.0	P1	perfect unison	D	da	do
1	44.4	^1, m2	up unison, minor 2nd	^D, Eb	fra	di
2	88.9	^^1, ^m2	dup unison, upminor 2nd	^^D, ^Eb	fru	ra
3	133.3	vA1, ~2	downaug 1sn, mid 2nd	vD#, vvE	ri	ru
4	177.8	A1, vM2	aug 1sn, downmajor 2nd	D#, vE	ro	reh
5	222.2	M2	major 2nd	E	ra	re
6	266.7	m3	minor 3rd	F	na	ma
7	311.1	^m3	upminor 3rd	Gb	nu	me
8	355.6	~3	mid 3rd	^Gb	mi	mu
9	400.0	vM3	downmajor 3rd	vF#	mo	mi
10	444.4	M3	major 3rd	F#	ma	mo
11	488.9	P4	perfect 4th	G	fa	fa
12	533.3	^4	up 4th	Ab	fu/sha	fih
13	577.8	~4, ^d5	mid 4th, updim 5th	^^G, ^Ab	fi/shu	fi
14	622.2	vA4, ~5	downaug 4th, mid 5th	vG#, vvA	po/si	se
15	666.7	v5	down fifth	G#	pa/so	sih
16	711.1	P5	perfect 5th	A	sa	so/sol
17	755.6	m6	minor 6th	Bb	fla	lo
18	800.0	^m6	upminor 6th	^Bb	flu	le
19	844.4	~6	mid 6th	vA#	li	lu
20	888.9	vM6	downmajor 6th	A#	lo	la
21	933.3	M6	major 6th	B	la	li
22	977.8	m7	minor 7th	C	tha	ta
23	1022.2	^m7	upminor 7th	Db	thu	te
24	1066.7	~7	mid 7th	^Db	ti	tu
25	1111.1	vM7	downmajor 7th	vC#	to	ti
26	1155.6	M7	major 7th	C#	ta	da
27	1200.0	P8	8ve	D	da	do

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality	Color name	Monzo format	Examples
minor	zo	[a, b, 0, 1⟩	7/6, 7/4
minor	fourthward wa	[a, b⟩, b < −1	32/27, 16/9
upminor	gu	[a, b, −1⟩	6/5, 9/5
mid	tho	[a, b, 0, 0, 0, 1⟩	13/12, 13/8
mid	thu	[a, b, 0, 0, 0, −1⟩	16/13, 24/13
downmajor	yo	[a, b, 1⟩	5/4, 5/3
major	fifthward wa	[a, b⟩, b > 1	9/8, 27/16
major	ru	[a, b, 0, −1⟩	9/7, 12/7

All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:

Color of the 3rd	JI chord	Notes as edosteps	Notes of C chord	Written name	Spoken name
zo	6:7:9	0–6–16	C–E♭–G	Cm	C minor
gu	10:12:15	0–7–16	C–F♭–G, C–E⁠ ⁠–G	C^m	C upminor
ilo	18:22:27	0–8–16	C–E⁠ ⁠–G	C~	C mid
yo	4:5:6	0–9–16	C–D♯–G, C–E⁠ ⁠–G	Cv	C downmajor or C down
ru	14:18:21	0–10–16	C–E–G	C	C major or C

For a more complete list, see Ups and downs notation #Chords and chord progressions. See also the 22edo page.

Notation

Circle of fifths in 27edo
Cents	Extended Pythagorean notation		Quartertone notation
0.0	C		A⁠ ⁠
711.1	G		E⁠ ⁠
222.2	D		B⁠ ⁠	F
933.3	A		C
444.4	E		G
1155.6	B		D
666.7	F♯		A
177.8	C♯		E
888.9	G♯		B
400.0	D♯		F⁠ ⁠
1111.1	A♯		C⁠ ⁠
622.2	E♯		G⁠ ⁠
133.3	B♯	F𝄫	D⁠ ⁠
844.4	F𝄪	C𝄫	A⁠ ⁠
355.6	C𝄪	G𝄫	E⁠ ⁠
1066.7	G𝄪	D𝄫	B⁠ ⁠
577.8	D𝄪	A𝄫	F⁠ ⁠
88.9	A𝄪	E𝄫	C⁠ ⁠
800.0	E𝄪	B𝄫	G⁠ ⁠
311.1	B𝄪	F♭	D⁠ ⁠
1022.2	C♭		A⁠ ⁠
533.3	G♭		E⁠ ⁠
44.4	D♭		B⁠ ⁠
755.6	A♭		F⁠ ⁠
266.7	E♭		C⁠ ⁠
977.8	B♭		G⁠ ⁠
488.9	F		D⁠ ⁠
0.0	C		A⁠ ⁠

Extended Pythagorean notation

27edo being a superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C–E is a 9/7 supermajor third and so the note located 5/4 above C must be notated as D♯. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D–F is a 7/6 subminor third and so the note located 6/5 above D must be notated as G♭. The diminished 2nd is a descending interval, thus A♯ is higher than B♭. Though here very exaggerated, this should be familiar to those working with the Pythagorean scale (see 53edo), and also to many classically trained violinists.

Quartertone notation

Using standard chain-of-fifths notation, a sharp (an augmented unison) raises a note by 4 edosteps, just one edostep beneath the following nominal, and the flat conversely lowers. The sharp is quite wide at about 178¢, sounding like a narrow major 2nd. C to C♯ describes the approximate 10/9 and 11/10 interval. An accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats. The half-sharp is notated as a quartertone, but at about 89¢ it sounds more like a narrow semitone. The gamut from C to D is C, D♭, C⁠ ⁠, D⁠ ⁠, C♯, and D, with many ascending intervals appearing to be descending on the staff.

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset	0	1	2	3	4	5	6	7	8	9
Sharp symbol										
Flat symbol										

Kite's ups and downs notation

27edo can also be notated with Kite's ups and downs, spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

Step offset	0	1	2	3	4	5	6	7	8	9
Sharp symbol
Flat symbol

Sagittal notation

This notation is a subset of the notation for 54edo.

Evo and Revo flavors

Alternative Evo flavor

Evo-SZ flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

6L 1s (archeotonic) notation

The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as LLLLLLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 edostep in this instance, is denoted by ♯ and ♭.

#	Cents	Note	Name	Associated ratio
0	0.0	C	perfect unison	1/1
1	44.4	C#, Dbbb	aug 1sn, triple-dim 2nd	40/39, 45/44, 55/54, 81/80
2	88.9	Cx, Dbb	double-aug 1sn, double-dim 2nd	16/15, 25/24
3	133.3	Db	dim 2nd	12/11, 13/12
4	177.8	D	perfect 2nd	10/9, 11/10
5	222.2	D#, Ebbb	aug 2nd, double-dim 3rd	9/8
6	266.7	Dx, Ebb	double-aug 2nd, dim 3rd	15/13
7	311.1	Eb	minor 3rd	6/5
8	355.6	E	major 3rd	11/9, 16/13
9	400.0	E#, Fbbb	aug 3rd, double-dim 4th	5/4
10	444.4	Ex, Fbb	double-aug 3rd, dim 4th	13/10
11	488.9	Ex#, Fb	minor 4th	4/3
12	533.3	F	major 4th	15/11, 27/20
13	577.8	F#, Gbbb	aug 4th, double-dim 5th	11/8, 18/13
14	622.2	Fx, Gbb	double-aug 4th, dim 5th	13/9, 16/11
15	666.7	Fx#, Gb	minor 5th	22/15, 40/27
16	711.1	G	major 5th	3/2
17	755.6	G#, Abbb	aug 5th, double-dim 6th	20/13
18	800.0	Gx, Abb	double-aug 5th, dim 6th	8/5
19	844.4	Ab	minor 6th	13/8, 18/11
20	888.9	A	major 6th	5/3
21	933.3	A#, Bbbb	aug 6th, double-dim 7th	26/15
22	977.8	Ax, Bbb	double-aug 6th, dim 7th	16/9
23	1022.2	Bb	perfect 7th	9/5, 20/11
24	1066.7	B	aug 7th	11/6, 24/13
25	1111.1	B#, Cbb	double-aug 7th, double-dim 8ve	15/8, 48/25
26	1155.6	Bx, Cb	triple-aug 7th, dim 8ve	39/20, 88/45, 108/55, 160/81
27	1200.0	C	8ve	2/1

Approximation to JI

alt : Your browser has no SVG support. — Selected 19-limit intervals approximated in 27edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 27edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 27edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/6, 12/7	0.204	0.5
15/11, 22/15	3.617	8.1
13/8, 16/13	3.917	8.8
5/3, 6/5	4.530	10.2
9/5, 10/9	4.626	10.4
7/5, 10/7	4.734	10.7
13/7, 14/13	5.035	11.3
13/12, 24/13	5.239	11.8
11/9, 18/11	8.148	18.3
7/4, 8/7	8.952	20.1
3/2, 4/3	9.156	20.6
9/7, 14/9	9.360	21.1
13/10, 20/13	9.770	22.0
11/10, 20/11	12.774	28.7
5/4, 8/5	13.686	30.8
15/14, 28/15	13.891	31.3
13/9, 18/13	14.395	32.4
11/6, 12/11	17.304	38.9
11/7, 14/11	17.508	39.4
11/8, 16/11	17.985	40.5
9/8, 16/9	18.312	41.2
15/13, 26/15	18.926	42.6
15/8, 16/15	21.602	48.6
13/11, 22/13	21.901	49.3

15-odd-limit intervals in 27edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/6, 12/7	0.204	0.5
13/8, 16/13	3.917	8.8
5/3, 6/5	4.530	10.2
9/5, 10/9	4.626	10.4
7/5, 10/7	4.734	10.7
13/7, 14/13	5.035	11.3
13/12, 24/13	5.239	11.8
7/4, 8/7	8.952	20.1
3/2, 4/3	9.156	20.6
9/7, 14/9	9.360	21.1
13/10, 20/13	9.770	22.0
5/4, 8/5	13.686	30.8
15/14, 28/15	13.891	31.3
13/9, 18/13	14.395	32.4
11/8, 16/11	17.985	40.5
9/8, 16/9	18.312	41.2
15/13, 26/15	18.926	42.6
13/11, 22/13	21.901	49.3
15/8, 16/15	22.842	51.4
11/7, 14/11	26.936	60.6
11/6, 12/11	27.141	61.1
11/10, 20/11	31.671	71.3
11/9, 18/11	36.297	81.7
15/11, 22/15	40.827	91.9

15-odd-limit intervals by 27e val mapping
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/6, 12/7	0.204	0.5
15/11, 22/15	3.617	8.1
13/8, 16/13	3.917	8.8
5/3, 6/5	4.530	10.2
9/5, 10/9	4.626	10.4
7/5, 10/7	4.734	10.7
13/7, 14/13	5.035	11.3
13/12, 24/13	5.239	11.8
11/9, 18/11	8.148	18.3
7/4, 8/7	8.952	20.1
3/2, 4/3	9.156	20.6
9/7, 14/9	9.360	21.1
13/10, 20/13	9.770	22.0
11/10, 20/11	12.774	28.7
5/4, 8/5	13.686	30.8
15/14, 28/15	13.891	31.3
13/9, 18/13	14.395	32.4
11/6, 12/11	17.304	38.9
11/7, 14/11	17.508	39.4
9/8, 16/9	18.312	41.2
15/13, 26/15	18.926	42.6
13/11, 22/13	22.543	50.7
15/8, 16/15	22.842	51.4
11/8, 16/11	26.460	59.5

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[43 -27⟩	[⟨27 43]]	−2.89	2.88	6.50
2.3.5	128/125, 20000/19683	[⟨27 43 63]]	−3.88	2.74	6.19
2.3.5.7	64/63, 126/125, 245/243	[⟨27 43 63 76]]	−3.71	2.39	5.40
2.3.5.7.13	64/63, 91/90, 126/125, 169/168	[⟨27 43 63 76 100]]	−3.18	2.39	5.39
2.3.5.7.13.19	64/63, 76/75, 91/90, 126/125, 169/168	[⟨27 43 63 76 100 115]]	−3.18	2.18	4.92

27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next equal temperaments doing better in those subgroups are 31, 31, and 46, respectively.
27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next equal temperament that does better in this subgroup is 53.

Uniform maps

13-limit uniform maps between 26.8 and 27.2
Min. size	Max. size	Wart notation	Map
26.7385	26.8145	27bcdf	⟨27 42 62 75 93 99]
26.8145	26.8887	27cdf	⟨27 43 62 75 93 99]
26.8887	26.8936	27cd	⟨27 43 62 75 93 100]
26.8936	26.9173	27c	⟨27 43 62 76 93 100]
26.9173	27.0276	27	⟨27 43 63 76 93 100]
27.0276	27.1589	27e	⟨27 43 63 76 94 100]
27.1589	27.2498	27eff	⟨27 43 63 76 94 101]

Rank-2 temperaments

Periods per 8ve	Generator	Temperaments	Mos scales
1	1\27	Quartonic / quarto (27e) / quartz (27)
1	2\27	Octacot / octocat (27e)	1L 12s, 13L 1s
1	4\27	Tetracot (27e) / modus (27e) / wollemia (27e)	1L 5s, 6L 1s, 7L 6s, 7L 13s
1	5\27	Machine (27) Kumonga (27e)	1L 4s, 5L 1s, 5L 6s, 11L 5s
1	7\27	Myna (27e) / coleto (27e) / myno (27) Oolong (27e)	4L 3s, 4L 7s, 4L 11s, 4L 15s, 4L 19s
1	8\27	Beatles (27e) / ringo (27e) / beetle (27)	3L 4s, 7L 3s, 10L 7s
1	10\27	Sensi	3L 2s, 3L 5s, 8L 3s, 8L 11s
1	11\27	Superpyth (27e)	5L 2s, 5L 7s, 5L 12s, 5L 17s
1	13\27	Fervor (27e)	2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, etc. … 2L 23s
3	1\27	Hemiaug (27e)
3	2\27	Augene (27e) / eugene (27)	3L 3s, 3L 6s, 3L 9s, 12L 3s
3	4\27	Oodako (27e) Terrain	3L 3s, 6L 3s, 6L 9s, 6L 15s
9	1\27	Niner (27e)	9L 9s

In addition, 27edo can be used as a detempering target for ennealimmal.

Commas

27et tempers out the following commas. (Note: This assumes the patent val, ⟨27 43 63 76 93 100].)

Prime limit	Ratio^{[note 1]}	Monzo	Cents	Color name	Name
5	128/125	[7 0 -3⟩	41.06	Trigu	Augmented comma, lesser diesis
5	20000/19683	[5 -9 4⟩	27.66	Saquadyo	Tetracot comma, minimal diesis
5	78732/78125	[2 9 -7⟩	13.40	Sepgu	Sensipent comma
5	(26 digits)	[1 -27 18⟩	0.86	Satritribiyo	Ennealimma
7	686/675	[1 -3 -2 3⟩	27.99	Trizo-agugu	Senga
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal comma
7	50421/50000	[-4 1 -5 5⟩	14.52	Quinzogu	Trimyna comma
7	245/243	[0 -5 1 2⟩	14.19	Zozoyo	Sensamagic comma
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Starling comma
7	4000/3969	[5 -4 3 -2⟩	13.47	Rurutriyo	Octagar comma
7	1728/1715	[6 3 -1 -3⟩	13.07	Triru-agu	Orwellisma
7	(16 digits)	[-11 -9 0 9⟩	1.84	Tritrizo	Septimal ennealimma
7	(12 digits)	[-6 -8 2 5⟩	1.12	Quinzo-ayoyo	Wizma
7	2401/2400	[-5 -1 -2 4⟩	0.72	Bizozogu	Breedsma
7	4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
7	(12 digits)	[-4 6 -6 3⟩	0.33	Trizogugu	Landscape comma
11	55/54	[-1 -3 1 0 1⟩	31.77	Loyo	Telepathma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	896/891	[7 -4 0 1 -1⟩	9.69	Saluzo	Pentacircle comma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
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	325/324	[-2 -4 2 0 0 1⟩	5.34	Thoyoyo	Marveltwin comma
13	351/350	[-1 3 -2 -1 0 1⟩	4.94	Thorugugu	Ratwolfsma
13	31213/31104	[-7 -5 0 4 0 1⟩	6.06	Thoquadzo	Praveensma
17	85/84	[-2 -1 1 -1 0 0 1⟩	20.49	Soruyo	Monk comma
17	154/153	[1 -2 0 1 1 0 -1⟩	11.28	Sulozo	Augustma
19	77/76	[2 -1 -2 0 0 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

↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Octave stretch or compression

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for octave compression. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 ¢. More generally, narrowing the steps to between 44.2 and 44.35 ¢ would be better in theory; 43edt, 70ed6, 90ed10, and 97ed12 are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, 4.11, and 2.55 ¢, respectively. 106zpi is another possible choice.

Scales

MOS scales

Main article: List of MOS scales in 27edo

Superpyth pentic – Superpyth[5] 2L 3s (gen = 11\27): 5 5 6 5 6
Superpyth diatonic – Superpyth[7] 5L 2s (gen = 11\27): 5 5 1 5 5 5 1
Superpyth chromatic – Superpyth[12] 5L 7s (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1
Superpyth enharmonic – Superpyth[17] 5L 12s (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1
Augene[6] 3L 3s (period = 9\27, gen = 2\27): 7 2 7 2 7 2
Augene[9] 3L 6s (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2
Augene[12] 3L 9s (period = 9\27, gen = 2\27): 3 2 2 2 3 2 2 2 3 2 2 2
Augene[15] 12L 3s (period = 9\27, gen = 2\27): 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2
Beatles[7] 3L 4s (gen = 8\27): 3 5 3 5 3 5 3
Beatles[10] 7L 3s (gen = 8\27): 3 3 2 3 3 2 3 3 2 3
Beatles[17] 10L 7s (gen = 8\27): 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 2 1
Machine[5] 1L 4s (gen = 5\27): 5 5 5 5 7
Machine[6] 5L 1s (gen = 5\27): 5 5 5 5 5 2
Machine[11] 5L 6s (gen = 5\27): 2 3 2 3 2 3 2 3 2 3 2
Machine[16] 11L 5s (gen = 5\27): 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
Myna[7] 4L 3s (gen = 7\27): 6 1 6 1 6 1 6
Myna[11] 4L 7s (gen = 7\27): 5 1 1 5 1 1 5 1 1 5 1
Myna[15] 4L 11s (gen = 7\27): 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1
Myna[19] 4L 15s (gen = 7\27): 3 1 1 1 1 3 1 1 1 1 3 1 1 1 1 3 1 1 1
Octacot[13] 1L 12s (gen = 2\27): 2 2 2 2 2 2 2 2 2 2 2 2 3
Octacot[14] 13L 1s (gen = 2\27): 2 2 2 2 2 2 2 2 2 2 2 2 2 1
Sensi[5] 3L 2s (gen = 10\27): 7 3 7 3 7
Sensi[8] 3L 5s (gen = 10\27): 3 4 3 3 4 3 3 4
Sensi[11] 8L 3s (gen = 10\27): 3 3 1 3 3 3 1 3 3 3 1
Tetracot[6] 1L 5s (gen = 4\27): 4 4 4 4 4 7
Tetracot[7] 6L 1s (gen = 4\27): 4 4 4 4 4 4 3
Tetracot[13] 7L 6s (gen = 4\27): 3 1 3 1 3 1 3 1 3 1 3 1 3
Tetracot[20] 7L 13s (gen = 4\27): 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1

JI chords

These are those overtone scales 27edo approximates with neat-looking horograms, which preserves their mapping well when rotated:

5afdo (rotated): 6 5 5 4 7
6afdo: 6 5 5 4 4 3
7afdo (rotated): 3 3 5 5 4 4 3
9afdo (rotated): 3 3 3 2 3 2 4 4 3
15afdo (rotated): 2 2 2 2 2 1 2 1 2 1 2 1 3 2 2
18afdo: 2 2 2 2 2 1 2 1 2 1 2 1 1 1 2 1 1 1
21afdo: 2 2 1 2 1 2 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1

These are other JI chords that 27edo approximates well:

12afdo without 17/12

(11 tones)
JI - 12:13:14:15:16:18:19:20:21:22:23:24
Included edosteps - 0, 3, 6, 9, 11, 16, 18, 20, 22, 24, 25, 27

an over-13 chord

(9 tones)
JI - 13:14:16:18:19:20:21:23:24:26
Included edosteps - 0, 3, 8, 13, 15, 17, 19, 22, 24, 27

an over-14 chord

(9 tones)
JI - 14:16:18:19:20:21:23:24:26:28
Included edosteps - 0, 5, 10, 12, 14, 16, 19, 21, 24, 27

Other scales

Pinetone

5-limit / pental / Pinetone major pentatonic: 5 4 7 4 7
5-limit / pental / Pinetone minor pentatonic: 7 4 5 7 4
Pinetone diatonic: 4 3 4 5 4 3 4
Pinetone major-harmonic octatonic: 4 3 4 2 3 4 3 4
Pinetone minor-harmonic octatonic: 4 3 2 4 3 4 4 3
Pinetone diminished octatonic / Porcusmine: 3 4 2 4 3 4 3 4
Pinetone harmonic diminished: 3 4 2 5 2 4 3 4
Pinetone chromatic / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3

Superpyth

Superpyth melodic minor – Superpyth 2|4 #6 #7 or 5|1 b3: 5 1 5 5 5 5 1
Superpyth harmonic minor – Superpyth 2|4 #7: 5 1 5 5 1 9 1
Superpyth harmonic major – Superpyth 5|1 b6: 5 5 1 5 1 9 1
Superpyth double harmonic major – Superpyth 5|1 b2 b6: 1 9 1 5 1 9 1

Tonality diamonds

5-odd limit tonality diamond: 7 2 2 5 2 2 7
7-odd limit tonality diamond: 5 1 1 2 2 2 1 2 2 2 1 1 5
9-odd limit tonality diamond: 4 1 1 1 2 1 1 2 1 2 1 1 2 1 1 1 4

5-limit scales

5-limit / pental double harmonic hexatonic (Augmented[6] 4M): 2 7 2 7 7 2, 7 7 2 2 7 2
5-limit / pental tetrachordal major: 5 4 2 5 5 4 2
5-limit / pental tetrachordal minor: 5 2 4 5 5 2 4
5-limit / pental melodic minor: 5 2 4 5 4 5 2
5-limit / pental harmonic minor: 5 2 4 5 2 7 2
5-limit / pental harmonic major: 5 4 2 5 2 7 2
5-limit / pental double harmonic major: 2 7 2 5 2 7 2
5-limit / pental double harmonic nonatonic (subset of Augene[12]): 2 5 2 2 5 2 5 2 2, 2 2 5 2 5 2 2 5 2 (Augene[9] 4M)
5-limit / pental double harmonic decatonic (subset of Augene[12]): 2 5 2 2 3 2 2 5 2 2
5-limit / pental double harmonic chromatic: 2 2 3 2 2 3 2 2 2 3 2 2, 2 2 3 2 2 2 3 2 2 3 2 2 (Augene[12] 4M)

Miscellaneous

Blackdye / syntonic dipentatonic (superset of Zarlino): 1 4 2 4 1 4 2 4 1 4
Blackville / 5-limit dipentatonic (superset of Zarlino): 3 2 4 2 3 2 4 2 3 2
enharmonic trichord octave species: 9 2 5 9 2, 2 9 5 2 9
enharmonic tetrachord octave species: 9 1 1 5 9 1 1, 1 9 1 5 1 9 1 (also Superpyth double harmonic major), 1 1 9 5 1 1
Zarlino / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2
"Just" minor (inverse of "just" major): 5 2 4 5 2 5 49
Direct sunlight^{[idiosyncratic term]} (original/default tuning; subset of Sensi[19]): 1 2 8 5 1 9 1
Hypersakura^{[idiosyncratic term]} (original/default tuning; subset of Sensi[19]): 1 10 5 1 10
Gutierrez wisp scale^{[idiosyncratic term]} (scale's period is nonoctave)
Lambeth will-o-wisps' scale^{[idiosyncratic term]} (scale's period is nonoctave)
User:BudjarnLambeth/Augene18 subsets in 97ed12

Instruments

While playing 27edo instruments requires significantly more frets or keys than 12edo, it is still possible to create physical instruments that can play all its notes. Probably the most notable of these is owned by Brendan Byrnes and played on some of his tracks listed in the music section.

However, the frets are very close together and playing high up the neck requires careful use of fingernails for fretting. A skip-fretted guitar would have notes only slightly closer together than 12edo and be easier to play, particularly when tuned in the configuration detailed below.

Skip fretting system 27 2 9

27edo can also be played on the Lumatone, with various layouts discussed here.

Lumatone mapping for 27edo

Music

See also: Category:27edo tracks

Modern renderings

Scott Joplin

Maple Leaf Rag (1899) – arranged with syntonic chroma adjustment for harpsichord and rendered by Claudi Meneghin (2025)

21st century

Abnormality

Boiling (2024)

Nae Ayy

What Happens Next (2021)

Beheld

Thick vibe (2023)

Gregoire Blanc

A microtonal teatime jam (2023)

Brendan Byrnes

Sunspots (2022)
27 EDO Etude (2022)
- on Bandcamp
- on YouTube (with sheet music)
Istril Bloom (2025)

Flora Canou

"Prelude: the Triad Challenge" from TOTMC Suite (2023–2025) – in superpyth, 70ed6 tuning

Bryan Deister

microtonal improvisation in 27edo (2023)
27edo waltz (2025)
27edo improv (2025)
Flies Control My Pain - 27edo (2026)
- [short 1] (using tetracot Lumatone mapping)
- [short 2] (using kumonga Lumatone mapping)

Francium

Happy Birthday in 27edo (2025)
"Router-Pseudoscientist" from TOTMC 2025 (2025) – Spotify | Bandcamp | YouTube
Waltz No. 11 in A flat major (2026)
Plane Sonatina No. 4 (2026)

groundfault

From A New Dusk (2024) – Bandcamp | YouTube
- "Back Stalk"
- "Superior Intermedial" – in part, the rest being in 31edo
- "Revelation of Your Forever"
"Sakura Blade Minivan", from Souvenirs of the Affliction (2025) – Bandcamp | YouTube (23:56–27:58) – in part, the rest being in 35edo