102edo

← 101edo

102edo

103edo →

Prime factorization	2 × 3 × 17
Step size	11.7647¢
Fifth	60\102 (705.882¢) (→10\17)
Semitones (A1:m2)	12:6 (141.2¢ : 70.59¢)
Dual sharp fifth	60\102 (705.882¢) (→10\17)
Dual flat fifth	59\102 (694.118¢)
Dual major 2nd	17\102 (200¢) (→1\6)
Consistency limit	5
Distinct consistency limit	5

102 equal divisions of the octave (abbreviated 102edo or 102ed2), also called 102-tone equal temperament (102tet) or 102 equal temperament (102et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102 equal parts of about 11.8 ¢ each. Each step represents a frequency ratio of 2^1/102, or the 102nd root of 2.

In the 5-limit it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as 34edo. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the optimal patent val for 13-limit echidnic temperament, and the rank five temperament tempering out 91/90.

Harmonics

Approximation of odd harmonics in 102edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+3.93	+1.92	-4.12	-3.91	+1.62	-5.23	+5.85	+0.93	-3.40	-0.19	-4.74
	relative (%)	+33	+16	-35	-33	+14	-44	+50	+8	-29	-2	-40
Steps (reduced)		162 (60)	237 (33)	286 (82)	323 (17)	353 (47)	377 (71)	399 (93)	417 (9)	433 (25)	448 (40)	461 (53)

Intervals

Steps	Cents	Ups and downs notation (dual flat fifth 59\102)	Ups and downs notation (dual sharp fifth 60\102)	Approximate ratios
0	0	D	D	1/1
1	11.7647	↑D, E♭♭♭	↑D, ↓5E♭
2	23.5294	↑↑D, ↓4E♭♭	↑↑D, ↓4E♭	64/63, 65/64, 78/77
3	35.2941	↑3D, ↓3E♭♭	↑3D, ↓3E♭
4	47.0588	↑4D, ↓↓E♭♭	↑4D, ↓↓E♭	40/39
5	58.8235	D♯, ↓E♭♭	↑5D, ↓E♭
6	70.5882	↑D♯, E♭♭	↑6D, E♭	25/24, 80/77
7	82.3529	↑↑D♯, ↓4E♭	↑7D, ↓11E	21/20, 22/21
8	94.1176	↑3D♯, ↓3E♭	↑8D, ↓10E
9	105.882	↑4D♯, ↓↓E♭	↑9D, ↓9E	52/49
10	117.647	D𝄪, ↓E♭	↑10D, ↓8E
11	129.412	↑D𝄪, E♭	↑11D, ↓7E	14/13
12	141.176	↑↑D𝄪, ↓4E	D♯, ↓6E
13	152.941	↑3D𝄪, ↓3E	↑D♯, ↓5E	12/11, 35/32
14	164.706	↑4D𝄪, ↓↓E	↑↑D♯, ↓4E	11/10
15	176.471	D♯𝄪, ↓E	↑3D♯, ↓3E
16	188.235	E	↑4D♯, ↓↓E	39/35
17	200	↑E, F♭♭	↑5D♯, ↓E
18	211.765	↑↑E, ↓4F♭	E	44/39
19	223.529	↑3E, ↓3F♭	↑E, ↓5F	25/22
20	235.294	↑4E, ↓↓F♭	↑↑E, ↓4F	8/7, 55/48, 63/55
21	247.059	E♯, ↓F♭	↑3E, ↓3F
22	258.824	↑E♯, F♭	↑4E, ↓↓F	64/55, 65/56
23	270.588	↑↑E♯, ↓4F	↑5E, ↓F
24	282.353	↑3E♯, ↓3F	F
25	294.118	↑4E♯, ↓↓F	↑F, ↓5G♭	77/65
26	305.882	E𝄪, ↓F	↑↑F, ↓4G♭	25/21
27	317.647	F	↑3F, ↓3G♭	6/5, 77/64
28	329.412	↑F, G♭♭♭	↑4F, ↓↓G♭	40/33
29	341.176	↑↑F, ↓4G♭♭	↑5F, ↓G♭	39/32
30	352.941	↑3F, ↓3G♭♭	↑6F, G♭
31	364.706	↑4F, ↓↓G♭♭	↑7F, ↓11G
32	376.471	F♯, ↓G♭♭	↑8F, ↓10G
33	388.235	↑F♯, G♭♭	↑9F, ↓9G	5/4
34	400	↑↑F♯, ↓4G♭	↑10F, ↓8G	44/35, 63/50
35	411.765	↑3F♯, ↓3G♭	↑11F, ↓7G	80/63
36	423.529	↑4F♯, ↓↓G♭	F♯, ↓6G	32/25
37	435.294	F𝄪, ↓G♭	↑F♯, ↓5G
38	447.059	↑F𝄪, G♭	↑↑F♯, ↓4G
39	458.824	↑↑F𝄪, ↓4G	↑3F♯, ↓3G
40	470.588	↑3F𝄪, ↓3G	↑4F♯, ↓↓G	21/16, 55/42, 72/55
41	482.353	↑4F𝄪, ↓↓G	↑5F♯, ↓G	33/25
42	494.118	F♯𝄪, ↓G	G	4/3
43	505.882	G	↑G, ↓5A♭
44	517.647	↑G, A♭♭♭	↑↑G, ↓4A♭	35/26
45	529.412	↑↑G, ↓4A♭♭	↑3G, ↓3A♭
46	541.176	↑3G, ↓3A♭♭	↑4G, ↓↓A♭	15/11
47	552.941	↑4G, ↓↓A♭♭	↑5G, ↓A♭	11/8
48	564.706	G♯, ↓A♭♭	↑6G, A♭	25/18
49	576.471	↑G♯, A♭♭	↑7G, ↓11A	39/28
50	588.235	↑↑G♯, ↓4A♭	↑8G, ↓10A
51	600	↑3G♯, ↓3A♭	↑9G, ↓9A
52	611.765	↑4G♯, ↓↓A♭	↑10G, ↓8A
53	623.529	G𝄪, ↓A♭	↑11G, ↓7A	56/39, 63/44
54	635.294	↑G𝄪, A♭	G♯, ↓6A	36/25
55	647.059	↑↑G𝄪, ↓4A	↑G♯, ↓5A	16/11
56	658.824	↑3G𝄪, ↓3A	↑↑G♯, ↓4A	22/15
57	670.588	↑4G𝄪, ↓↓A	↑3G♯, ↓3A
58	682.353	G♯𝄪, ↓A	↑4G♯, ↓↓A	52/35, 77/52
59	694.118	A	↑5G♯, ↓A
60	705.882	↑A, B♭♭♭	A	3/2
61	717.647	↑↑A, ↓4B♭♭	↑A, ↓5B♭	50/33
62	729.412	↑3A, ↓3B♭♭	↑↑A, ↓4B♭	32/21, 55/36
63	741.176	↑4A, ↓↓B♭♭	↑3A, ↓3B♭
64	752.941	A♯, ↓B♭♭	↑4A, ↓↓B♭	65/42
65	764.706	↑A♯, B♭♭	↑5A, ↓B♭
66	776.471	↑↑A♯, ↓4B♭	↑6A, B♭	25/16
67	788.235	↑3A♯, ↓3B♭	↑7A, ↓11B	63/40
68	800	↑4A♯, ↓↓B♭	↑8A, ↓10B	35/22
69	811.765	A𝄪, ↓B♭	↑9A, ↓9B	8/5
70	823.529	↑A𝄪, B♭	↑10A, ↓8B
71	835.294	↑↑A𝄪, ↓4B	↑11A, ↓7B
72	847.059	↑3A𝄪, ↓3B	A♯, ↓6B
73	858.824	↑4A𝄪, ↓↓B	↑A♯, ↓5B	64/39
74	870.588	A♯𝄪, ↓B	↑↑A♯, ↓4B	33/20
75	882.353	B	↑3A♯, ↓3B	5/3
76	894.118	↑B, C♭♭	↑4A♯, ↓↓B	42/25
77	905.882	↑↑B, ↓4C♭	↑5A♯, ↓B
78	917.647	↑3B, ↓3C♭	B
79	929.412	↑4B, ↓↓C♭	↑B, ↓5C
80	941.176	B♯, ↓C♭	↑↑B, ↓4C	55/32
81	952.941	↑B♯, C♭	↑3B, ↓3C
82	964.706	↑↑B♯, ↓4C	↑4B, ↓↓C	7/4
83	976.471	↑3B♯, ↓3C	↑5B, ↓C	44/25
84	988.235	↑4B♯, ↓↓C	C	39/22
85	1000	B𝄪, ↓C	↑C, ↓5D♭
86	1011.76	C	↑↑C, ↓4D♭	70/39
87	1023.53	↑C, D♭♭♭	↑3C, ↓3D♭
88	1035.29	↑↑C, ↓4D♭♭	↑4C, ↓↓D♭	20/11
89	1047.06	↑3C, ↓3D♭♭	↑5C, ↓D♭	11/6, 64/35
90	1058.82	↑4C, ↓↓D♭♭	↑6C, D♭
91	1070.59	C♯, ↓D♭♭	↑7C, ↓11D	13/7
92	1082.35	↑C♯, D♭♭	↑8C, ↓10D
93	1094.12	↑↑C♯, ↓4D♭	↑9C, ↓9D	49/26
94	1105.88	↑3C♯, ↓3D♭	↑10C, ↓8D
95	1117.65	↑4C♯, ↓↓D♭	↑11C, ↓7D	21/11, 40/21
96	1129.41	C𝄪, ↓D♭	C♯, ↓6D	48/25, 77/40
97	1141.18	↑C𝄪, D♭	↑C♯, ↓5D
98	1152.94	↑↑C𝄪, ↓4D	↑↑C♯, ↓4D	39/20
99	1164.71	↑3C𝄪, ↓3D	↑3C♯, ↓3D
100	1176.47	↑4C𝄪, ↓↓D	↑4C♯, ↓↓D	63/32, 77/39
101	1188.24	C♯𝄪, ↓D	↑5C♯, ↓D
102	1200	D	D	2/1

13-limit Echidnic

Degree	Cents	Difference from 46edo
2	23.529	-2.5575¢
4	47.059	-5.115¢
7	82.353	4.092¢
9	105.882	1.5345¢
11	129.412	-1.023¢
13	152.941	8.184¢
16	188.235	5.627¢
18	211.765	3.069¢
20	235.294	0.511¢
22	258.824	-2.046¢
24	282.353	-4.604¢
27	317.647	4.604¢
29	341.176	2.046¢
31	364.706	-0.5115¢
33	388.235	-3.069¢
35	411.765	-5.627¢
38	447.059	3.581¢
40	470.588	1.023¢
42	494.117	-1.5345¢
44	517.647	-4.092¢
47	552.941	5.115¢
49	576.471	2.5575¢