83edo

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83edo

84edo →

Prime factorization	83 (prime)
Step size	14.4578¢
Fifth	49\83 (708.434¢)
Semitones (A1:m2)	11:4 (159¢ : 57.83¢)
Dual sharp fifth	49\83 (708.434¢)
Dual flat fifth	48\83 (693.976¢)
Dual major 2nd	14\83 (202.41¢)
Consistency limit	7
Distinct consistency limit	7

83 equal divisions of the octave (abbreviated 83edo or 83ed2), also called 83-tone equal temperament (83tet) or 83 equal temperament (83et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 83 equal parts of about 14.5 ¢ each. Each step represents a frequency ratio of 2^1/83, or the 83rd root of 2.

Theory

Approximation of odd harmonics in 83edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+6.48	+4.05	-0.15	-1.50	-1.92	-1.97	-3.93	-3.75	+6.10	+6.33	-6.59
	relative (%)	+45	+28	-1	-10	-13	-14	-27	-26	+42	+44	-46
Steps (reduced)		132 (49)	193 (27)	233 (67)	263 (14)	287 (38)	307 (58)	324 (75)	339 (7)	353 (21)	365 (33)	375 (43)

The 3/1 is 6.5 cents sharp and the 5/1 is 4 cents sharp, with 7, 11, and 13 more accurate but a little flat. It tempers out 15625/15552 in the 5-limit and 686/675, 4000/3969 and 6144/6125 in the 7-limit, and provides the optimal patent val for the 7-limit 27&56 temperament with wedgie ⟨⟨5 18 17 17 13 -11]]. In the 11-limit it tempers out 121/120, 176/175 and 385/384, and in the 13-limit 91/90, 169/168 and 196/195, and it provides the optimal patent val for the 11-limit 22&61 temperament and the 13-limit 15&83 temperament. 83edo is the 23rd prime EDO.

Every odd harmonic between the 7th and the 17th is tuned flatly. As a consequence, this tuning provides a good approximation of the 7:9:11:13:15:17 hexad, and especially of the 9:11:13 triad.

Intervals

Steps	Cents	Ups and downs notation (dual flat fifth 48\83)	Ups and downs notation (dual sharp fifth 49\83)	Approximate ratios
0	0	D	D	1/1
1	14.4578	↑D, E♭♭♭	↑D, ↓3E♭
2	28.9157	↑↑D, ↓3E♭♭	↑↑D, ↓↓E♭	56/55, 65/64, 66/65
3	43.3735	↑3D, ↓↓E♭♭	↑3D, ↓E♭	40/39
4	57.8313	D♯, ↓E♭♭	↑4D, E♭	33/32
5	72.2892	↑D♯, E♭♭	↑5D, ↓10E	25/24
6	86.747	↑↑D♯, ↓3E♭	↑6D, ↓9E	21/20
7	101.205	↑3D♯, ↓↓E♭	↑7D, ↓8E	35/33, 52/49, 55/52
8	115.663	D𝄪, ↓E♭	↑8D, ↓7E
9	130.12	↑D𝄪, E♭	↑9D, ↓6E	14/13
10	144.578	↑↑D𝄪, ↓3E	↑10D, ↓5E
11	159.036	↑3D𝄪, ↓↓E	D♯, ↓4E	35/32
12	173.494	D♯𝄪, ↓E	↑D♯, ↓3E
13	187.952	E	↑↑D♯, ↓↓E	39/35, 49/44
14	202.41	↑E, F♭♭	↑3D♯, ↓E	55/49
15	216.867	↑↑E, ↓3F♭	E
16	231.325	↑3E, ↓↓F♭	↑E, ↓3F	8/7, 55/48
17	245.783	E♯, ↓F♭	↑↑E, ↓↓F
18	260.241	↑E♯, F♭	↑3E, ↓F	64/55, 65/56
19	274.699	↑↑E♯, ↓3F	F
20	289.157	↑3E♯, ↓↓F	↑F, ↓3G♭	13/11, 33/28, 77/65
21	303.614	E𝄪, ↓F	↑↑F, ↓↓G♭	25/21
22	318.072	F	↑3F, ↓G♭	6/5, 77/64
23	332.53	↑F, G♭♭♭	↑4F, G♭	40/33
24	346.988	↑↑F, ↓3G♭♭	↑5F, ↓10G	39/32, 49/40
25	361.446	↑3F, ↓↓G♭♭	↑6F, ↓9G	16/13
26	375.904	F♯, ↓G♭♭	↑7F, ↓8G
27	390.361	↑F♯, G♭♭	↑8F, ↓7G	5/4, 49/39
28	404.819	↑↑F♯, ↓3G♭	↑9F, ↓6G	63/50
29	419.277	↑3F♯, ↓↓G♭	↑10F, ↓5G	14/11
30	433.735	F𝄪, ↓G♭	F♯, ↓4G	50/39
31	448.193	↑F𝄪, G♭	↑F♯, ↓3G
32	462.651	↑↑F𝄪, ↓3G	↑↑F♯, ↓↓G	55/42, 64/49
33	477.108	↑3F𝄪, ↓↓G	↑3F♯, ↓G	33/25
34	491.566	F♯𝄪, ↓G	G	65/49
35	506.024	G	↑G, ↓3A♭
36	520.482	↑G, A♭♭♭	↑↑G, ↓↓A♭	65/48, 66/49
37	534.94	↑↑G, ↓3A♭♭	↑3G, ↓A♭
38	549.398	↑3G, ↓↓A♭♭	↑4G, A♭	11/8, 48/35
39	563.855	G♯, ↓A♭♭	↑5G, ↓10A	25/18
40	578.313	↑G♯, A♭♭	↑6G, ↓9A	7/5, 39/28
41	592.771	↑↑G♯, ↓3A♭	↑7G, ↓8A	55/39
42	607.229	↑3G♯, ↓↓A♭	↑8G, ↓7A	78/55
43	621.687	G𝄪, ↓A♭	↑9G, ↓6A	10/7, 56/39
44	636.145	↑G𝄪, A♭	↑10G, ↓5A	36/25
45	650.602	↑↑G𝄪, ↓3A	G♯, ↓4A	16/11, 35/24
46	665.06	↑3G𝄪, ↓↓A	↑G♯, ↓3A
47	679.518	G♯𝄪, ↓A	↑↑G♯, ↓↓A	49/33, 65/44, 77/52
48	693.976	A	↑3G♯, ↓A
49	708.434	↑A, B♭♭♭	A
50	722.892	↑↑A, ↓3B♭♭	↑A, ↓3B♭	50/33
51	737.349	↑3A, ↓↓B♭♭	↑↑A, ↓↓B♭	49/32
52	751.807	A♯, ↓B♭♭	↑3A, ↓B♭	65/42
53	766.265	↑A♯, B♭♭	↑4A, B♭	39/25
54	780.723	↑↑A♯, ↓3B♭	↑5A, ↓10B	11/7
55	795.181	↑3A♯, ↓↓B♭	↑6A, ↓9B
56	809.639	A𝄪, ↓B♭	↑7A, ↓8B	8/5, 78/49
57	824.096	↑A𝄪, B♭	↑8A, ↓7B
58	838.554	↑↑A𝄪, ↓3B	↑9A, ↓6B	13/8
59	853.012	↑3A𝄪, ↓↓B	↑10A, ↓5B	64/39, 80/49
60	867.47	A♯𝄪, ↓B	A♯, ↓4B	33/20
61	881.928	B	↑A♯, ↓3B	5/3
62	896.386	↑B, C♭♭	↑↑A♯, ↓↓B	42/25
63	910.843	↑↑B, ↓3C♭	↑3A♯, ↓B	22/13, 56/33
64	925.301	↑3B, ↓↓C♭	B
65	939.759	B♯, ↓C♭	↑B, ↓3C	55/32
66	954.217	↑B♯, C♭	↑↑B, ↓↓C
67	968.675	↑↑B♯, ↓3C	↑3B, ↓C	7/4
68	983.133	↑3B♯, ↓↓C	C
69	997.59	B𝄪, ↓C	↑C, ↓3D♭
70	1012.05	C	↑↑C, ↓↓D♭	70/39
71	1026.51	↑C, D♭♭♭	↑3C, ↓D♭
72	1040.96	↑↑C, ↓3D♭♭	↑4C, D♭	64/35
73	1055.42	↑3C, ↓↓D♭♭	↑5C, ↓10D
74	1069.88	C♯, ↓D♭♭	↑6C, ↓9D	13/7
75	1084.34	↑C♯, D♭♭	↑7C, ↓8D
76	1098.8	↑↑C♯, ↓3D♭	↑8C, ↓7D	49/26, 66/35
77	1113.25	↑3C♯, ↓↓D♭	↑9C, ↓6D	40/21
78	1127.71	C𝄪, ↓D♭	↑10C, ↓5D	48/25
79	1142.17	↑C𝄪, D♭	C♯, ↓4D	64/33
80	1156.63	↑↑C𝄪, ↓3D	↑C♯, ↓3D	39/20
81	1171.08	↑3C𝄪, ↓↓D	↑↑C♯, ↓↓D	55/28, 65/33
82	1185.54	C♯𝄪, ↓D	↑3C♯, ↓D
83	1200	D	D	2/1