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

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	+6	-8	+20	+27	-14	-34	-13	-6	-49	+31
Steps (reduced)		147 (54)	216 (30)	261 (75)	295 (16)	322 (43)	344 (65)	363 (84)	380 (8)	395 (23)	408 (36)	421 (49)

93 = 3 * 31, and 93 is a contorted 31 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 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.

Intervals

Steps	Cents	Ups and downs notation (dual flat fifth 54\93)	Ups and downs notation (dual sharp fifth 55\93)	Approximate ratios
0	0	D	D	1/1
1	12.9032	↑D, ↓↓E♭♭	↑D, ↓3E♭
2	25.8065	↑↑D, ↓E♭♭	↑↑D, ↓↓E♭	65/64, 66/65
3	38.7097	↑3D, E♭♭	↑3D, ↓E♭	49/48, 50/49
4	51.6129	↑4D, ↓5E♭	↑4D, E♭	33/32
5	64.5161	↑5D, ↓4E♭	↑5D, ↓12E	26/25, 80/77
6	77.4194	D♯, ↓3E♭	↑6D, ↓11E
7	90.3226	↑D♯, ↓↓E♭	↑7D, ↓10E
8	103.226	↑↑D♯, ↓E♭	↑8D, ↓9E	35/33, 52/49
9	116.129	↑3D♯, E♭	↑9D, ↓8E	15/14, 16/15
10	129.032	↑4D♯, ↓5E	↑10D, ↓7E	14/13
11	141.935	↑5D♯, ↓4E	↑11D, ↓6E	13/12
12	154.839	D𝄪, ↓3E	↑12D, ↓5E	35/32
13	167.742	↑D𝄪, ↓↓E	D♯, ↓4E	11/10
14	180.645	↑↑D𝄪, ↓E	↑D♯, ↓3E
15	193.548	E	↑↑D♯, ↓↓E	28/25
16	206.452	↑E, ↓↓F♭	↑3D♯, ↓E
17	219.355	↑↑E, ↓F♭	E	25/22
18	232.258	↑3E, F♭	↑E, ↓3F	8/7
19	245.161	↑4E, ↓5F	↑↑E, ↓↓F	15/13
20	258.065	↑5E, ↓4F	↑3E, ↓F	64/55, 65/56
21	270.968	E♯, ↓3F	F	7/6, 75/64
22	283.871	↑E♯, ↓↓F	↑F, ↓3G♭	33/28
23	296.774	↑↑E♯, ↓F	↑↑F, ↓↓G♭	77/65
24	309.677	F	↑3F, ↓G♭
25	322.581	↑F, ↓↓G♭♭	↑4F, G♭	77/64
26	335.484	↑↑F, ↓G♭♭	↑5F, ↓12G	40/33
27	348.387	↑3F, G♭♭	↑6F, ↓11G	49/40, 60/49
28	361.29	↑4F, ↓5G♭	↑7F, ↓10G	16/13
29	374.194	↑5F, ↓4G♭	↑8F, ↓9G	26/21
30	387.097	F♯, ↓3G♭	↑9F, ↓8G	5/4
31	400	↑F♯, ↓↓G♭	↑10F, ↓7G	44/35, 49/39
32	412.903	↑↑F♯, ↓G♭	↑11F, ↓6G	14/11, 33/26
33	425.806	↑3F♯, G♭	↑12F, ↓5G	32/25
34	438.71	↑4F♯, ↓5G	F♯, ↓4G
35	451.613	↑5F♯, ↓4G	↑F♯, ↓3G	13/10
36	464.516	F𝄪, ↓3G	↑↑F♯, ↓↓G	64/49
37	477.419	↑F𝄪, ↓↓G	↑3F♯, ↓G	33/25
38	490.323	↑↑F𝄪, ↓G	G	65/49
39	503.226	G	↑G, ↓3A♭	75/56
40	516.129	↑G, ↓↓A♭♭	↑↑G, ↓↓A♭	35/26, 66/49
41	529.032	↑↑G, ↓A♭♭	↑3G, ↓A♭	65/48
42	541.935	↑3G, A♭♭	↑4G, A♭	48/35
43	554.839	↑4G, ↓5A♭	↑5G, ↓12A	11/8
44	567.742	↑5G, ↓4A♭	↑6G, ↓11A
45	580.645	G♯, ↓3A♭	↑7G, ↓10A	7/5
46	593.548	↑G♯, ↓↓A♭	↑8G, ↓9A
47	606.452	↑↑G♯, ↓A♭	↑9G, ↓8A
48	619.355	↑3G♯, A♭	↑10G, ↓7A	10/7
49	632.258	↑4G♯, ↓5A	↑11G, ↓6A	75/52
50	645.161	↑5G♯, ↓4A	↑12G, ↓5A	16/11
51	658.065	G𝄪, ↓3A	G♯, ↓4A	35/24
52	670.968	↑G𝄪, ↓↓A	↑G♯, ↓3A	65/44
53	683.871	↑↑G𝄪, ↓A	↑↑G♯, ↓↓A	49/33, 52/35, 77/52
54	696.774	A	↑3G♯, ↓A
55	709.677	↑A, ↓↓B♭♭	A
56	722.581	↑↑A, ↓B♭♭	↑A, ↓3B♭	50/33
57	735.484	↑3A, B♭♭	↑↑A, ↓↓B♭	49/32, 75/49
58	748.387	↑4A, ↓5B♭	↑3A, ↓B♭	20/13, 77/50
59	761.29	↑5A, ↓4B♭	↑4A, B♭
60	774.194	A♯, ↓3B♭	↑5A, ↓12B	25/16
61	787.097	↑A♯, ↓↓B♭	↑6A, ↓11B	11/7, 52/33
62	800	↑↑A♯, ↓B♭	↑7A, ↓10B	35/22, 78/49
63	812.903	↑3A♯, B♭	↑8A, ↓9B	8/5
64	825.806	↑4A♯, ↓5B	↑9A, ↓8B	21/13
65	838.71	↑5A♯, ↓4B	↑10A, ↓7B	13/8
66	851.613	A𝄪, ↓3B	↑11A, ↓6B	49/30, 80/49
67	864.516	↑A𝄪, ↓↓B	↑12A, ↓5B	33/20
68	877.419	↑↑A𝄪, ↓B	A♯, ↓4B
69	890.323	B	↑A♯, ↓3B
70	903.226	↑B, ↓↓C♭	↑↑A♯, ↓↓B
71	916.129	↑↑B, ↓C♭	↑3A♯, ↓B	56/33
72	929.032	↑3B, C♭	B	12/7
73	941.935	↑4B, ↓5C	↑B, ↓3C	55/32
74	954.839	↑5B, ↓4C	↑↑B, ↓↓C	26/15
75	967.742	B♯, ↓3C	↑3B, ↓C	7/4
76	980.645	↑B♯, ↓↓C	C	44/25
77	993.548	↑↑B♯, ↓C	↑C, ↓3D♭
78	1006.45	C	↑↑C, ↓↓D♭	25/14
79	1019.35	↑C, ↓↓D♭♭	↑3C, ↓D♭
80	1032.26	↑↑C, ↓D♭♭	↑4C, D♭	20/11
81	1045.16	↑3C, D♭♭	↑5C, ↓12D	64/35
82	1058.06	↑4C, ↓5D♭	↑6C, ↓11D	24/13
83	1070.97	↑5C, ↓4D♭	↑7C, ↓10D	13/7
84	1083.87	C♯, ↓3D♭	↑8C, ↓9D	15/8, 28/15
85	1096.77	↑C♯, ↓↓D♭	↑9C, ↓8D	49/26, 66/35
86	1109.68	↑↑C♯, ↓D♭	↑10C, ↓7D
87	1122.58	↑3C♯, D♭	↑11C, ↓6D
88	1135.48	↑4C♯, ↓5D	↑12C, ↓5D	25/13, 77/40
89	1148.39	↑5C♯, ↓4D	C♯, ↓4D	64/33
90	1161.29	C𝄪, ↓3D	↑C♯, ↓3D	49/25
91	1174.19	↑C𝄪, ↓↓D	↑↑C♯, ↓↓D	65/33
92	1187.1	↑↑C𝄪, ↓D	↑3C♯, ↓D
93	1200	D	D	2/1

Scales

Meantone Chromatic

• 116.129
• 193.548
• 309.677
• 387.097
• 503.226
• 580.645
• 696.774
• 812.903
• 890.323
• 1006.452
• 1083.871
• 1200.000

Superpyth Chromatic

• 51.613
• 219.355
• 270.968
• 438.710
• 490.323
• 658.065
• 709.677
• 761.290
• 929.032
• 980.645
• 1148.387
• 1200.000

Superpyth Shailaja

• 270.968
• 709.677
• 761.290
• 980.645
• 1200.000

Superpyth Subminor Hexatonic

• 219.355
• 270.968
• 490.323
• 709.677
• 980.645
• 1200.000

Superpyth Subminor Pentatonic

• 270.968
• 490.323
• 709.677
• 980.645
• 1200.000

See Also

• 93edo and stretched hemififths