93edo

← 92edo

93edo

94edo →

Prime factorization	3 × 31
Step size	12.9032¢
Fifth	54\93 (696.774¢) (→18\31)
Semitones (A1:m2)	6:9 (77.42¢ : 116.1¢)
Dual sharp fifth	55\93 (709.677¢)
Dual flat fifth	54\93 (696.774¢) (→18\31)
Dual major 2nd	16\93 (206.452¢)
Consistency limit	7
Distinct consistency limit	7

93 equal divisions of the octave (abbreviated 93edo or 93ed2), also called 93-tone equal temperament (93tet) or 93 equal temperament (93et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 93 equal parts of about 12.9 ¢ each. Each step represents a frequency ratio of 2^1/93, or the 93rd root of 2.

Theory

93 = 3 × 31, and 93edo is a contorted 31edo through the 7-limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11- and 13-limit trimean (43 & 50) temperament. It is the 13th no-3s zeta peak edo. The bd val is close to the 9-odd limit minimax tuning for superpyth.

Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710 ¢, 103.226 ¢, and 296.774 ¢ respectively, octave-reduced), it also allows one to give a clearer harmonic identity to 31edo's excellent approximation of 13:17:19.

Odd harmonics

Approximation of odd harmonics in 93edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-5.18	+0.78	-1.08	+2.54	+3.52	-1.82	-4.40	-1.73	-0.74	-6.26	+3.98
	Relative (%)	-40.2	+6.1	-8.4	+19.7	+27.3	-14.1	-34.1	-13.4	-5.7	-48.6	+30.9
Steps (reduced)		147 (54)	216 (30)	261 (75)	295 (16)	322 (43)	344 (65)	363 (84)	380 (8)	395 (23)	408 (36)	421 (49)

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 54\93)	Ups and downs notation (Dual sharp fifth 55\93)
0	0	1/1	D	D
1	12.9		^D, vvE♭♭	^D, v3E♭
2	25.8		^^D, vE♭♭	^^D, vvE♭
3	38.7		^3D, E♭♭	^3D, vE♭
4	51.6	33/32, 34/33, 35/34	vvD♯, ^E♭♭	^4D, E♭
5	64.5		vD♯, ^^E♭♭	^5D, ^E♭
6	77.4	23/22	D♯, v3E♭	^6D, ^^E♭
7	90.3	20/19, 39/37	^D♯, vvE♭	v6D♯, ^3E♭
8	103.2	17/16, 35/33	^^D♯, vE♭	v5D♯, ^4E♭
9	116.1	31/29	^3D♯, E♭	v4D♯, ^5E♭
10	129	14/13, 41/38	vvD𝄪, ^E♭	v3D♯, ^6E♭
11	141.9	25/23, 38/35	vD𝄪, ^^E♭	vvD♯, v6E
12	154.8	35/32	D𝄪, v3E	vD♯, v5E
13	167.7	32/29	^D𝄪, vvE	D♯, v4E
14	180.6		^^D𝄪, vE	^D♯, v3E
15	193.5	19/17, 28/25	E	^^D♯, vvE
16	206.5		^E, vvF♭	^3D♯, vE
17	219.4	17/15, 25/22, 42/37	^^E, vF♭	E
18	232.3	8/7	^3E, F♭	^E, v3F
19	245.2	15/13, 38/33	vvE♯, ^F♭	^^E, vvF
20	258.1	29/25	vE♯, ^^F♭	^3E, vF
21	271		E♯, v3F	F
22	283.9	20/17, 33/28	^E♯, vvF	^F, v3G♭
23	296.8	19/16	^^E♯, vF	^^F, vvG♭
24	309.7		F	^3F, vG♭
25	322.6	41/34	^F, vvG♭♭	^4F, G♭
26	335.5	17/14, 40/33	^^F, vG♭♭	^5F, ^G♭
27	348.4		^3F, G♭♭	^6F, ^^G♭
28	361.3	16/13, 37/30	vvF♯, ^G♭♭	v6F♯, ^3G♭
29	374.2	31/25, 41/33	vF♯, ^^G♭♭	v5F♯, ^4G♭
30	387.1	5/4	F♯, v3G♭	v4F♯, ^5G♭
31	400	29/23	^F♯, vvG♭	v3F♯, ^6G♭
32	412.9	33/26	^^F♯, vG♭	vvF♯, v6G
33	425.8	32/25	^3F♯, G♭	vF♯, v5G
34	438.7	40/31	vvF𝄪, ^G♭	F♯, v4G
35	451.6	13/10	vF𝄪, ^^G♭	^F♯, v3G
36	464.5	17/13	F𝄪, v3G	^^F♯, vvG
37	477.4	25/19, 29/22	^F𝄪, vvG	^3F♯, vG
38	490.3		^^F𝄪, vG	G
39	503.2		G	^G, v3A♭
40	516.1	31/23, 35/26	^G, vvA♭♭	^^G, vvA♭
41	529	19/14	^^G, vA♭♭	^3G, vA♭
42	541.9	26/19, 41/30	^3G, A♭♭	^4G, A♭
43	554.8	40/29	vvG♯, ^A♭♭	^5G, ^A♭
44	567.7	43/31	vG♯, ^^A♭♭	^6G, ^^A♭
45	580.6	7/5	G♯, v3A♭	v6G♯, ^3A♭
46	593.5	31/22	^G♯, vvA♭	v5G♯, ^4A♭
47	606.5		^^G♯, vA♭	v4G♯, ^5A♭
48	619.4	10/7	^3G♯, A♭	v3G♯, ^6A♭
49	632.3		vvG𝄪, ^A♭	vvG♯, v6A
50	645.2	29/20	vG𝄪, ^^A♭	vG♯, v5A
51	658.1	19/13, 41/28	G𝄪, v3A	G♯, v4A
52	671	28/19	^G𝄪, vvA	^G♯, v3A
53	683.9	43/29	^^G𝄪, vA	^^G♯, vvA
54	696.8		A	^3G♯, vA
55	709.7		^A, vvB♭♭	A
56	722.6	38/25	^^A, vB♭♭	^A, v3B♭
57	735.5	26/17	^3A, B♭♭	^^A, vvB♭
58	748.4	20/13, 37/24	vvA♯, ^B♭♭	^3A, vB♭
59	761.3	31/20	vA♯, ^^B♭♭	^4A, B♭
60	774.2	25/16	A♯, v3B♭	^5A, ^B♭
61	787.1	41/26	^A♯, vvB♭	^6A, ^^B♭
62	800		^^A♯, vB♭	v6A♯, ^3B♭
63	812.9	8/5	^3A♯, B♭	v5A♯, ^4B♭
64	825.8		vvA𝄪, ^B♭	v4A♯, ^5B♭
65	838.7	13/8	vA𝄪, ^^B♭	v3A♯, ^6B♭
66	851.6		A𝄪, v3B	vvA♯, v6B
67	864.5	28/17, 33/20	^A𝄪, vvB	vA♯, v5B
68	877.4		^^A𝄪, vB	A♯, v4B
69	890.3		B	^A♯, v3B
70	903.2	32/19	^B, vvC♭	^^A♯, vvB
71	916.1	17/10	^^B, vC♭	^3A♯, vB
72	929	41/24	^3B, C♭	B
73	941.9		vvB♯, ^C♭	^B, v3C
74	954.8	26/15, 33/19	vB♯, ^^C♭	^^B, vvC
75	967.7	7/4	B♯, v3C	^3B, vC
76	980.6	30/17, 37/21	^B♯, vvC	C
77	993.5		^^B♯, vC	^C, v3D♭
78	1006.5	25/14, 34/19	C	^^C, vvD♭
79	1019.4		^C, vvD♭♭	^3C, vD♭
80	1032.3	29/16	^^C, vD♭♭	^4C, D♭
81	1045.2		^3C, D♭♭	^5C, ^D♭
82	1058.1	35/19	vvC♯, ^D♭♭	^6C, ^^D♭
83	1071	13/7	vC♯, ^^D♭♭	v6C♯, ^3D♭
84	1083.9	43/23	C♯, v3D♭	v5C♯, ^4D♭
85	1096.8	32/17	^C♯, vvD♭	v4C♯, ^5D♭
86	1109.7	19/10	^^C♯, vD♭	v3C♯, ^6D♭
87	1122.6		^3C♯, D♭	vvC♯, v6D
88	1135.5		vvC𝄪, ^D♭	vC♯, v5D
89	1148.4	33/17	vC𝄪, ^^D♭	C♯, v4D
90	1161.3	43/22	C𝄪, v3D	^C♯, v3D
91	1174.2		^C𝄪, vvD	^^C♯, vvD
92	1187.1		^^C𝄪, vD	^3C♯, vD
93	1200	2/1	D	D

Scales

• Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
• Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)
• Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
• Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)

See also

• 93edo and stretched hemififths