81edo

← 80edo

81edo

82edo →

Prime factorization	34
Step size	14.8148 ¢
Fifth	47\81 (696.296 ¢)
Semitones (A1:m2)	5:8 (74.07 ¢ : 118.5 ¢)
Dual sharp fifth	48\81 (711.111 ¢) (→ 16\27)
Dual flat fifth	47\81 (696.296 ¢)
Dual major 2nd	14\81 (207.407 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

By Tom Winspear, utilizing the Accidents shown below. Left: Chain of 4ths/5ths , Right: Chromatic view. Black font represents the '6 accidentals deep' notation that covers the chromatic scale with enharmonics only across EF & BC. White text displays deep enharmonics in the ambiguous infrared & ultraviolet area of the colour notation.

81edo is notable as a tuning for meantone and related temperaments and is the optimal patent val for a number of them. In particular it is the optimal patent val for 5-limit meantone, 7-limit meantone, 11-limit meanpop, 13-limit meanpop, and the rank three temperament erato. The electronic music pioneer Daphne Oram was interested in 81edo. As a step in the Golden meantone series of EDOs, 81 EDO marks the point at which the series ceases to display audible changes to meantone temperament, and is also the EDO with the lowest average and most evenly spread Just-error across the scale (though 31 EDO does have the best harmonic 7th).

Besides meantone, 81edo is a tuning for the cobalt temperament, since it contains 27 as a divisor. It also tunes the unnamed 15 & 51 temperament which divides the octave into 3 equal parts, and is a member of the augmented-cloudy equivalence continuum. 81bd val is a tuning for the septimal porcupine temperament.

In the higher limits, it is a strong tuning for the 2.5.17.19 subgroup, and also can be used to map 19/17 to the meantone major second resulting from stacking of two panent val fifths (13\81).

Odd harmonics

Approximation of odd harmonics in 81edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-5.66	-1.13	-5.86	+3.50	-3.17	+3.92	-6.79	-1.25	-1.22	+3.29	-6.05
	Relative (%)	-38.2	-7.6	-39.6	+23.6	-21.4	+26.4	-45.8	-8.4	-8.2	+22.2	-40.9
Steps (reduced)		128 (47)	188 (26)	227 (65)	257 (14)	280 (37)	300 (57)	316 (73)	331 (7)	344 (20)	356 (32)	366 (42)

Subsets and supersets

Since 81 is equal to 3⁴, a perfect power of 3, 81edo contains subset edos 1, 3, 9, 27.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 47\81)	Ups and downs notation (Dual sharp fifth 48\81)
0	0	1/1	D	D
1	14.8		^D, vvE♭♭	^D, vvE♭
2	29.6		^^D, vE♭♭	^^D, vE♭
3	44.4	38/37, 39/38, 40/39	vvD♯, E♭♭	^3D, E♭
4	59.3	29/28, 30/29, 31/30	vD♯, ^E♭♭	^4D, ^E♭
5	74.1	23/22, 24/23	D♯, ^^E♭♭	^5D, ^^E♭
6	88.9	20/19, 39/37	^D♯, vvE♭	^6D, ^3E♭
7	103.7	17/16, 35/33	^^D♯, vE♭	v5D♯, ^4E♭
8	118.5	15/14, 31/29	vvD𝄪, E♭	v4D♯, ^5E♭
9	133.3	40/37	vD𝄪, ^E♭	v3D♯, v6E
10	148.1	12/11, 37/34	D𝄪, ^^E♭	vvD♯, v5E
11	163	11/10, 34/31	^D𝄪, vvE	vD♯, v4E
12	177.8	31/28	^^D𝄪, vE	D♯, v3E
13	192.6	19/17	E	^D♯, vvE
14	207.4	35/31	^E, vvF♭	^^D♯, vE
15	222.2	25/22, 33/29	^^E, vF♭	E
16	237	39/34	vvE♯, F♭	^E, vvF
17	251.9	22/19, 37/32	vE♯, ^F♭	^^E, vF
18	266.7	7/6	E♯, ^^F♭	F
19	281.5	20/17, 33/28	^E♯, vvF	^F, vvG♭
20	296.3	19/16	^^E♯, vF	^^F, vG♭
21	311.1		F	^3F, G♭
22	325.9	29/24, 35/29	^F, vvG♭♭	^4F, ^G♭
23	340.7	28/23, 39/32	^^F, vG♭♭	^5F, ^^G♭
24	355.6	38/31	vvF♯, G♭♭	^6F, ^3G♭
25	370.4	31/25	vF♯, ^G♭♭	v5F♯, ^4G♭
26	385.2	5/4	F♯, ^^G♭♭	v4F♯, ^5G♭
27	400	29/23, 39/31	^F♯, vvG♭	v3F♯, v6G
28	414.8	14/11	^^F♯, vG♭	vvF♯, v5G
29	429.6	32/25	vvF𝄪, G♭	vF♯, v4G
30	444.4	22/17, 31/24, 40/31	vF𝄪, ^G♭	F♯, v3G
31	459.3	30/23	F𝄪, ^^G♭	^F♯, vvG
32	474.1	25/19	^F𝄪, vvG	^^F♯, vG
33	488.9		^^F𝄪, vG	G
34	503.7		G	^G, vvA♭
35	518.5	31/23	^G, vvA♭♭	^^G, vA♭
36	533.3	34/25	^^G, vA♭♭	^3G, A♭
37	548.1	11/8	vvG♯, A♭♭	^4G, ^A♭
38	563		vG♯, ^A♭♭	^5G, ^^A♭
39	577.8		G♯, ^^A♭♭	^6G, ^3A♭
40	592.6	31/22	^G♯, vvA♭	v5G♯, ^4A♭
41	607.4		^^G♯, vA♭	v4G♯, ^5A♭
42	622.2	33/23	vvG𝄪, A♭	v3G♯, v6A
43	637		vG𝄪, ^A♭	vvG♯, v5A
44	651.9	16/11, 35/24	G𝄪, ^^A♭	vG♯, v4A
45	666.7	25/17	^G𝄪, vvA	G♯, v3A
46	681.5	37/25	^^G𝄪, vA	^G♯, vvA
47	696.3		A	^^G♯, vA
48	711.1		^A, vvB♭♭	A
49	725.9	35/23, 38/25	^^A, vB♭♭	^A, vvB♭
50	740.7	23/15	vvA♯, B♭♭	^^A, vB♭
51	755.6	17/11, 31/20	vA♯, ^B♭♭	^3A, B♭
52	770.4	25/16, 39/25	A♯, ^^B♭♭	^4A, ^B♭
53	785.2	11/7	^A♯, vvB♭	^5A, ^^B♭
54	800		^^A♯, vB♭	^6A, ^3B♭
55	814.8	8/5	vvA𝄪, B♭	v5A♯, ^4B♭
56	829.6		vA𝄪, ^B♭	v4A♯, ^5B♭
57	844.4	31/19	A𝄪, ^^B♭	v3A♯, v6B
58	859.3	23/14	^A𝄪, vvB	vvA♯, v5B
59	874.1		^^A𝄪, vB	vA♯, v4B
60	888.9		B	A♯, v3B
61	903.7	32/19	^B, vvC♭	^A♯, vvB
62	918.5	17/10	^^B, vC♭	^^A♯, vB
63	933.3	12/7	vvB♯, C♭	B
64	948.1	19/11	vB♯, ^C♭	^B, vvC
65	963		B♯, ^^C♭	^^B, vC
66	977.8		^B♯, vvC	C
67	992.6	39/22	^^B♯, vC	^C, vvD♭
68	1007.4	34/19	C	^^C, vD♭
69	1022.2		^C, vvD♭♭	^3C, D♭
70	1037	20/11, 31/17	^^C, vD♭♭	^4C, ^D♭
71	1051.9	11/6	vvC♯, D♭♭	^5C, ^^D♭
72	1066.7	37/20	vC♯, ^D♭♭	^6C, ^3D♭
73	1081.5	28/15	C♯, ^^D♭♭	v5C♯, ^4D♭
74	1096.3	32/17	^C♯, vvD♭	v4C♯, ^5D♭
75	1111.1	19/10	^^C♯, vD♭	v3C♯, v6D
76	1125.9	23/12	vvC𝄪, D♭	vvC♯, v5D
77	1140.7	29/15	vC𝄪, ^D♭	vC♯, v4D
78	1155.6	37/19, 39/20	C𝄪, ^^D♭	C♯, v3D
79	1170.4		^C𝄪, vvD	^C♯, vvD
80	1185.2		^^C𝄪, vD	^^C♯, vD
81	1200	2/1	D	D

Notation

Tom Winspear's notation

81 EDO Accidentals created and used by Tom Winspear, based on those provided in Scala though with a logic correction. The innermost accidentals represent one EDOstep, followed by two, then the bracket representing three. Conventional sharp/doublesharp/flat/doubleflat accidentals are reached in steps of five and the pattern repeats itself on them. The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.

Regular temperament properties

Commas

• 5-limit commas: 81/80, [-48 1 20⟩
• 7-limit commas: 81/80, 126/125, [-24 1 0 8⟩
• 11-limit commas: 81/80, 126/125, 385/384, 12005/11979
• 13-limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591

Scales

• Leantone6
• Leantone7
• Leantone13