27edo

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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 (27edo), or 27(-tone) equal temperament (27tet, 27et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 27 equally large steps. Each step represents a frequency ratio of the 27th root of 2, or 44.4 cents.

Theory

If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444… cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics 3, 5, and 7 sharply. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614-edo, which corresponds to a step size of 44.3023 cents.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13+2/3 cents. 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, resulting in three of them reaching a near perfect minor third/major sixth in both, with 19edo reaching a near-6/5 and 27edo reaching a near-7/6.

27edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also). These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with 22edo tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support the superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.

Though the 7-limit tuning of 27edo 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 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 temperament. It also approximates 19/10, 19/12, and 19/14, so 0-7-13-25 does quite well as a 10:12:14:19; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest harmonic entropy possible and thus is, in theory, most dissonant, assuming the relatively common values of a = 2 and s = 1%. This property is shared with all edos between around 24 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 27-note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 edosteps, just one edostep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.

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
	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)

Intervals

#	Cents	Approximate Ratios*	Ups and Downs Notation			6L 1s notation		Solfege
0	0.00	1/1	P1	perfect unison	D	perfect unison	C	do
1	44.44	28/27, 36/35, 39/38, 49/48, 50/49, 81/80	^1, m2	up unison, minor 2nd	^D, Eb	aug 1sn, double-dim 2nd	C#, Dbbb	di
2	88.89	16/15, 21/20, 25/24, 19/18, 20/19	^^1, ^m2	dup unison, upminor 2nd	^^D, ^Eb	double-aug 1sn, dim 2nd	Cx, Dbb	ra
3	133.33	15/14, 14/13, 13/12	vA1, ~2	downaug 1sn, mid 2nd	vD#, vvE	minor 2nd	Db	ru
4	177.78	10/9	A1, vM2	aug 1sn, downmajor 2nd	D#, vE	major 2nd	D	reh
5	222.22	8/7, 9/8	M2	major 2nd	E	aug 2nd, double-dim 3rd	D#, Ebbb	re
6	266.67	7/6	m3	minor 3rd	F	double-aug 2nd, dim 3rd	Dx, Ebb	ma
7	311.11	6/5, 19/16	^m3	upminor 3rd	Gb	minor 3rd	Eb	me
8	355.56	16/13	~3	mid 3rd	^Gb	major 3rd	E	mu
9	400.00	5/4, 24/19	vM3	downmajor 3rd	vF#	aug 3rd, double-dim 4th	E#, Fbbb	mi
10	444.44	9/7, 13/10	M3	major 3rd	F#	double-aug 3rd, dim 4th	Ex, Fbb	mo
11	488.89	4/3	P4	perfect 4th	G	minor 4th	Ex#, Fb	fa
12	533.33	27/20, 48/35, 19/14, 26/19	^4	up 4th	Ab	major 4th	F	fih
13	577.78	7/5, 18/13	~4, ^d5	mid 4th, updim 5th	^^G, ^Ab	aug 4th, double-dim 5th	F#, Gbbb	fi
14	622.22	10/7, 13/9	vA4, ~5	downaug 4th, mid 5th	vG#, vvA	double-aug 4th, dim 5th	Fx, Gbb	se
15	666.67	40/27, 35/24, 19/13, 28/19	v5	down fifth	G#	minor 5th	Fx#, Gb	sih
16	711.11	3/2	P5	perfect 5th	A	major 5th	G	so/sol
17	755.56	14/9, 20/13	m6	minor 6th	Bb	aug 5th, double-dim 6th	G#, Abbb	lo
18	800.00	8/5, 19/12	^m6	upminor 6th	^Bb	double-aug 5th, dim 6th	Gx, Abb	le
19	844.44	13/8	~6	mid 6th	vA#	minor 6th	Ab	lu
20	888.89	5/3, 32/19	vM6	downmajor 6th	A#	major 6th	A	la
21	933.33	12/7	M6	major 6th	B	aug 6th, double-dim 7th	A#, Bbbb	li
22	977.78	7/4, 16/9	m7	minor 7th	C	double-aug 6th, dim 7th	Ax, Bbb	ta
23	1022.22	9/5	^m7	upminor 7th	Db	minor 7th	Bb	te
24	1066.67	28/15, 13/7, 24/13	~7	mid 7th	^Db	major 7th	B	tu
25	1111.11	15/8, 40/21, 48/25, 19/10, 36/19	vM7	downmajor 7th	vC#	aug 7th, double-dim 8ve	B#, Cbb	ti
26	1155.56	27/14, 35/18, 96/49, 49/25, 160/81	M7	major 7th	C#	double-aug 7th, dim 8ve	Bx, Cb	da
27	1200.00	2/1	P8	8ve	D	8ve	C	do

* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.

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
	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
	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
	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 Eb G	Cm	C minor
gu	10:12:15	0-7-16	C ^Eb G	C^m	C upminor
ilo	18:22:27	0-8-16	C vvE G	C~	C mid
yo	4:5:6	0-9-16	C vE 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.

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 27edo. Prime harmonics are in bold; inconsistent intervals are in italic.

15-odd-limit intervals by direct approximation (even if inconsistent)

Interval, complement	Error (abs, ¢)	Error (rel, %)
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

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

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				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]]	-3.70	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.

Rank-2 temperaments

• List of 27edo rank two temperaments by badness
• List of edo-distinct 27e rank two temperaments

Periods per 8ve	Generator	Temperaments	MOS Scales
1	1\27	Quartonic/quarto
1	2\27	Octacot/octocat	1L_12s, 13L_1s
1	4\27	Tetracot/modus/wollemia	1L_5s, 6L_1s, 7L_6s, 7L_13s
1	5\27	Machine/kumonga	1L_4s, 5L_1s, 5L_6s, 11L_5s
1	7\27	Myna/coleto/minah/oolong	4L_3s, 4L_7s, 4L_11s, 4L_15s, 4L_19s
1	8\27	Beatles/ringo	3L_4s, 7L_3s, 10L_7s
1	10\27	Sensi/sensis	3L_2s, 3L_5s, 8L_3s, 8L_11s
1	11\27	Superpyth	5L_2s, 5L_7s, 5L_12s, 5L_17s
1	13\27	Fervor	2L_3s, 2L_5s, 2L_7s, 2L_9s, 2L_11s, etc ... 2L_23s
3	1\27	Semiaug/hemiaug
3	2\27	Augmented/Augene/ogene	3L_3s, 3L_6s, 3L_9s, 12L_3s
3	4\27	Oodako/terrain	3L_3s, 6L_3s, 6L_9s, 6L_15s
9	1\27	Terrible version of Ennealimmal /niner	9L_9s

Commas

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

Prime Limit	Ratio	Monzo	Cents	Color name	Name(s)
5	128/125	[7 0 -3⟩	41.06	Trigu	Diesis, augmented comma
5	20000/19683	[5 -9 4⟩	27.66	Saquadyo	Minimal diesis, Tetracot comma
5	78732/78125	[2 9 -7⟩	13.40	Sepgu	Medium semicomma, 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, Archytas' comma, Leipziger Komma
7	50421/50000	[-4 1 -5 5⟩	14.52	Quinzogu	Trimyna
7	245/243	[0 -5 1 2⟩	14.19	Zozoyo	Sensamagic
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Septimal semicomma, Starling comma
7	4000/3969	[5 -4 3 -2⟩	13.47	Rurutriyo	Octagar
7	1728/1715	[6 3 -1 -3⟩	13.07	Triru-agu	Orwellisma, Orwell comma
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	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	896/891	[7 -4 0 1 -1⟩	9.69	Saluzo	Pentacircle
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap

Music

See also: Category:27edo tracks

Beheld

• Thick vibe (2023)

Brendan Byrnes

• Sunspots

Igliashon Jones

• Sad Like Winter Leaves^{[dead link]}
• Superpythagorean Waltz
• Stuttering Anticipation (2021)

Peter Kosmorsky

• miniature prelude and fugue

Claudi Meneghin

• Chorale in 27edo for Organ

Nae Ayy

• What Happens Next

Dustin Schallert

• Tetracot Perc-Sitar^{[dead link]} (on SoundCloud)^{[dead link]}
• Tetracot Jam^{[dead link]} (on SoundCloud)^{[dead link]}
• Tetracot Pump^{[dead link]} (on SoundCloud)^{[dead link]}
• 27-EDO Guitar 1^{[dead link]}

Gene Ward Smith

• Music For Your Ears play – the central portion is in 27edo, the rest in 46edo.

Joel Taylor

• Galticeran Sonatina^{[dead link]}

Chris Vaisvil

• Chicago Pile-1

Xotla

• Funkrotonal

See also

• Lumatone mapping for 27edo