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 superpine (43 & 50) temperament. It is the 13th no-3s zeta peak edo.

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	12.903		^D, vvE♭♭	^D, v3E♭
2	25.806		^^D, vE♭♭	^^D, vvE♭
3	38.71		^3D, E♭♭	^3D, vE♭
4	51.613	33/32, 34/33, 35/34	^4D, v5E♭	^4D, E♭
5	64.516		^5D, v4E♭	^5D, v12E
6	77.419	23/22	D♯, v3E♭	^6D, v11E
7	90.323	20/19, 39/37	^D♯, vvE♭	^7D, v10E
8	103.226	17/16, 35/33	^^D♯, vE♭	^8D, v9E
9	116.129	31/29	^3D♯, E♭	^9D, v8E
10	129.032	14/13, 41/38	^4D♯, v5E	^10D, v7E
11	141.935	25/23, 38/35	^5D♯, v4E	^11D, v6E
12	154.839	35/32	D𝄪, v3E	^12D, v5E
13	167.742	32/29	^D𝄪, vvE	D♯, v4E
14	180.645		^^D𝄪, vE	^D♯, v3E
15	193.548	19/17, 28/25	E	^^D♯, vvE
16	206.452		^E, vvF♭	^3D♯, vE
17	219.355	17/15, 25/22, 42/37	^^E, vF♭	E
18	232.258	8/7	^3E, F♭	^E, v3F
19	245.161	15/13, 38/33	^4E, v5F	^^E, vvF
20	258.065	29/25	^5E, v4F	^3E, vF
21	270.968		E♯, v3F	F
22	283.871	20/17, 33/28	^E♯, vvF	^F, v3G♭
23	296.774	19/16	^^E♯, vF	^^F, vvG♭
24	309.677		F	^3F, vG♭
25	322.581	41/34	^F, vvG♭♭	^4F, G♭
26	335.484	17/14, 40/33	^^F, vG♭♭	^5F, v12G
27	348.387		^3F, G♭♭	^6F, v11G
28	361.29	16/13, 37/30	^4F, v5G♭	^7F, v10G
29	374.194	31/25, 41/33	^5F, v4G♭	^8F, v9G
30	387.097	5/4	F♯, v3G♭	^9F, v8G
31	400	29/23	^F♯, vvG♭	^10F, v7G
32	412.903	33/26	^^F♯, vG♭	^11F, v6G
33	425.806	32/25	^3F♯, G♭	^12F, v5G
34	438.71	40/31	^4F♯, v5G	F♯, v4G
35	451.613	13/10	^5F♯, v4G	^F♯, v3G
36	464.516	17/13	F𝄪, v3G	^^F♯, vvG
37	477.419	25/19, 29/22	^F𝄪, vvG	^3F♯, vG
38	490.323		^^F𝄪, vG	G
39	503.226		G	^G, v3A♭
40	516.129	31/23, 35/26	^G, vvA♭♭	^^G, vvA♭
41	529.032	19/14	^^G, vA♭♭	^3G, vA♭
42	541.935	26/19, 41/30	^3G, A♭♭	^4G, A♭
43	554.839	40/29	^4G, v5A♭	^5G, v12A
44	567.742	43/31	^5G, v4A♭	^6G, v11A
45	580.645	7/5	G♯, v3A♭	^7G, v10A
46	593.548	31/22	^G♯, vvA♭	^8G, v9A
47	606.452		^^G♯, vA♭	^9G, v8A
48	619.355	10/7	^3G♯, A♭	^10G, v7A
49	632.258		^4G♯, v5A	^11G, v6A
50	645.161	29/20	^5G♯, v4A	^12G, v5A
51	658.065	19/13, 41/28	G𝄪, v3A	G♯, v4A
52	670.968	28/19	^G𝄪, vvA	^G♯, v3A
53	683.871	43/29	^^G𝄪, vA	^^G♯, vvA
54	696.774		A	^3G♯, vA
55	709.677		^A, vvB♭♭	A
56	722.581	38/25	^^A, vB♭♭	^A, v3B♭
57	735.484	26/17	^3A, B♭♭	^^A, vvB♭
58	748.387	20/13, 37/24	^4A, v5B♭	^3A, vB♭
59	761.29	31/20	^5A, v4B♭	^4A, B♭
60	774.194	25/16	A♯, v3B♭	^5A, v12B
61	787.097	41/26	^A♯, vvB♭	^6A, v11B
62	800		^^A♯, vB♭	^7A, v10B
63	812.903	8/5	^3A♯, B♭	^8A, v9B
64	825.806		^4A♯, v5B	^9A, v8B
65	838.71	13/8	^5A♯, v4B	^10A, v7B
66	851.613		A𝄪, v3B	^11A, v6B
67	864.516	28/17, 33/20	^A𝄪, vvB	^12A, v5B
68	877.419		^^A𝄪, vB	A♯, v4B
69	890.323		B	^A♯, v3B
70	903.226	32/19	^B, vvC♭	^^A♯, vvB
71	916.129	17/10	^^B, vC♭	^3A♯, vB
72	929.032	41/24	^3B, C♭	B
73	941.935		^4B, v5C	^B, v3C
74	954.839	26/15, 33/19	^5B, v4C	^^B, vvC
75	967.742	7/4	B♯, v3C	^3B, vC
76	980.645	30/17, 37/21	^B♯, vvC	C
77	993.548		^^B♯, vC	^C, v3D♭
78	1006.452	25/14, 34/19	C	^^C, vvD♭
79	1019.355		^C, vvD♭♭	^3C, vD♭
80	1032.258	29/16	^^C, vD♭♭	^4C, D♭
81	1045.161		^3C, D♭♭	^5C, v12D
82	1058.065	35/19	^4C, v5D♭	^6C, v11D
83	1070.968	13/7	^5C, v4D♭	^7C, v10D
84	1083.871	43/23	C♯, v3D♭	^8C, v9D
85	1096.774	32/17	^C♯, vvD♭	^9C, v8D
86	1109.677	19/10	^^C♯, vD♭	^10C, v7D
87	1122.581		^3C♯, D♭	^11C, v6D
88	1135.484		^4C♯, v5D	^12C, v5D
89	1148.387	33/17	^5C♯, v4D	C♯, v4D
90	1161.29	43/22	C𝄪, v3D	^C♯, v3D
91	1174.194		^C𝄪, vvD	^^C♯, vvD
92	1187.097		^^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