107edo

Prime factorization

107 (prime)

Step size

11.215¢

Fifth

63\107 (706.542¢)

Semitones (A1:m2)

13:6 (145.8¢ : 67.29¢)

Dual sharp fifth

63\107 (706.542¢)

Dual flat fifth

62\107 (695.327¢)

Dual major 2nd

18\107 (201.869¢)

Consistency limit

3

Distinct consistency limit

3

107 equal divisions of the octave (abbreviated 107edo or 107ed2), also called 107-tone equal temperament (107tet) or 107 equal temperament (107et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 107 equal parts of about 11.2 ¢ each. Each step represents a frequency ratio of 2^1/107, or the 107th root of 2.

Theory

107edo is inconsistent to the 5-odd-limit and higher limits, with four mappings possible for the 7-limit: ⟨107 170 248 300] (patent val), ⟨107 169 248 300] (107b), ⟨107 170 249 300] (107c), and ⟨107 170 249 301] (107cd).

Using the patent val, it tempers out 3125/3072 (magic comma) and 33554432000/31381059609 in the 5-limit; 1029/1024, 2240/2187, and 3125/3087 in the 7-limit; 100/99, 1232/1215, and 1331/1323 in the 11-limit.

Using the 107b val, it tempers out 81/80 (syntonic comma) and [-61 -1 27⟩; in the 5-limit; 2401/2400, 2430/2401, and 234375/229376 in the 7-limit; 385/384, 1350/1331, 1375/1372, and 1944/1925 in the 11-limit.

Using the 107c val, it tempers out 1638400/1594323 (immunity comma) and 1990656/1953125 (valentine comma) in the 5-limit; 126/125, 1029/1024, and 307328/295245 in the 7-limit; 121/120, 176/175, 441/440, and 184877/177147 in the 11-limit.

Using the 107cd val, it tempers out 1728/1715, 4000/3969, and 28672/28125 in the 7-limit; 121/120, 896/891, 1375/1372, and 3168/3125 in the 11-limit.

Subsets and supersets

107edo is the 28th prime edo.

Odd harmonics

Approximation of odd harmonics in 107edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+4.59	-5.01	-4.34	-2.04	-1.79	+0.59	-0.42	-4.02	+5.29	+0.25	-0.24
Error	relative (%)	+41	-45	-39	-18	-16	+5	-4	-36	+47	+2	-2
Steps (reduced)		170 (63)	248 (34)	300 (86)	339 (18)	370 (49)	396 (75)	418 (97)	437 (9)	455 (27)	470 (42)	484 (56)

Intervals

"

Steps	Cents	Ups and downs notation (dual flat fifth 62\107)	Ups and downs notation (dual sharp fifth 63\107)	Approximate ratios
0	0	D	D	1/1
1	11.215	^D, v⁴E♭♭	^D, v⁵E♭
2	22.4299	^^D, v³E♭♭	^^D, v⁴E♭	65/64
3	33.6449	^³D, vvE♭♭	^³D, v³E♭	50/49, 56/55
4	44.8598	^⁴D, vE♭♭	^⁴D, vvE♭	77/75
5	56.0748	^⁵D, E♭♭	^⁵D, vE♭	33/32
6	67.2897	D♯, v⁵E♭	^⁶D, E♭	80/77
7	78.5047	^D♯, v⁴E♭	^⁷D, v¹²E	22/21
8	89.7196	^^D♯, v³E♭	^⁸D, v¹¹E
9	100.935	^³D♯, vvE♭	^⁹D, v¹⁰E
10	112.15	^⁴D♯, vE♭	^¹⁰D, v⁹E	16/15
11	123.364	^⁵D♯, E♭	^¹¹D, v⁸E	15/14
12	134.579	D𝄪, v⁵E	^¹²D, v⁷E	13/12
13	145.794	^D𝄪, v⁴E	D♯, v⁶E
14	157.009	^^D𝄪, v³E	^D♯, v⁵E
15	168.224	^³D𝄪, vvE	^^D♯, v⁴E	11/10
16	179.439	^⁴D𝄪, vE	^³D♯, v³E
17	190.654	E	^⁴D♯, vvE
18	201.869	^E, v⁴F♭	^⁵D♯, vE	55/49
19	213.084	^^E, v³F♭	E
20	224.299	^³E, vvF♭	^E, v⁵F
21	235.514	^⁴E, vF♭	^^E, v⁴F	8/7
22	246.729	^⁵E, F♭	^³E, v³F	15/13, 52/45
23	257.944	E♯, v⁵F	^⁴E, vvF	65/56
24	269.159	^E♯, v⁴F	^⁵E, vF
25	280.374	^^E♯, v³F	F
26	291.589	^³E♯, vvF	^F, v⁵G♭	13/11, 77/65
27	302.804	^⁴E♯, vF	^^F, v⁴G♭
28	314.019	F	^³F, v³G♭
29	325.234	^F, v⁴G♭♭	^⁴F, vvG♭
30	336.449	^^F, v³G♭♭	^⁵F, vG♭	63/52
31	347.664	^³F, vvG♭♭	^⁶F, G♭	49/40
32	358.879	^⁴F, vG♭♭	^⁷F, v¹²G	16/13
33	370.093	^⁵F, G♭♭	^⁸F, v¹¹G	26/21
34	381.308	F♯, v⁵G♭	^⁹F, v¹⁰G
35	392.523	^F♯, v⁴G♭	^¹⁰F, v⁹G
36	403.738	^^F♯, v³G♭	^¹¹F, v⁸G
37	414.953	^³F♯, vvG♭	^¹²F, v⁷G	14/11, 33/26
38	426.168	^⁴F♯, vG♭	F♯, v⁶G
39	437.383	^⁵F♯, G♭	^F♯, v⁵G
40	448.598	F𝄪, v⁵G	^^F♯, v⁴G
41	459.813	^F𝄪, v⁴G	^³F♯, v³G
42	471.028	^^F𝄪, v³G	^⁴F♯, vvG	21/16
43	482.243	^³F𝄪, vvG	^⁵F♯, vG
44	493.458	^⁴F𝄪, vG	G	65/49
45	504.673	G	^G, v⁵A♭	75/56
46	515.888	^G, v⁴A♭♭	^^G, v⁴A♭
47	527.103	^^G, v³A♭♭	^³G, v³A♭
48	538.318	^³G, vvA♭♭	^⁴G, vvA♭	15/11
49	549.533	^⁴G, vA♭♭	^⁵G, vA♭	11/8
50	560.748	^⁵G, A♭♭	^⁶G, A♭
51	571.963	G♯, v⁵A♭	^⁷G, v¹²A
52	583.178	^G♯, v⁴A♭	^⁸G, v¹¹A	7/5
53	594.393	^^G♯, v³A♭	^⁹G, v¹⁰A	45/32
54	605.607	^³G♯, vvA♭	^¹⁰G, v⁹A	64/45
55	616.822	^⁴G♯, vA♭	^¹¹G, v⁸A	10/7
56	628.037	^⁵G♯, A♭	^¹²G, v⁷A
57	639.252	G𝄪, v⁵A	G♯, v⁶A
58	650.467	^G𝄪, v⁴A	^G♯, v⁵A	16/11
59	661.682	^^G𝄪, v³A	^^G♯, v⁴A	22/15
60	672.897	^³G𝄪, vvA	^³G♯, v³A	65/44
61	684.112	^⁴G𝄪, vA	^⁴G♯, vvA
62	695.327	A	^⁵G♯, vA
63	706.542	^A, v⁴B♭♭	A
64	717.757	^^A, v³B♭♭	^A, v⁵B♭
65	728.972	^³A, vvB♭♭	^^A, v⁴B♭	32/21
66	740.187	^⁴A, vB♭♭	^³A, v³B♭	75/49
67	751.402	^⁵A, B♭♭	^⁴A, vvB♭	77/50
68	762.617	A♯, v⁵B♭	^⁵A, vB♭
69	773.832	^A♯, v⁴B♭	^⁶A, B♭
70	785.047	^^A♯, v³B♭	^⁷A, v¹²B	11/7, 52/33
71	796.262	^³A♯, vvB♭	^⁸A, v¹¹B
72	807.477	^⁴A♯, vB♭	^⁹A, v¹⁰B
73	818.692	^⁵A♯, B♭	^¹⁰A, v⁹B
74	829.907	A𝄪, v⁵B	^¹¹A, v⁸B	21/13
75	841.121	^A𝄪, v⁴B	^¹²A, v⁷B	13/8
76	852.336	^^A𝄪, v³B	A♯, v⁶B	80/49
77	863.551	^³A𝄪, vvB	^A♯, v⁵B
78	874.766	^⁴A𝄪, vB	^^A♯, v⁴B
79	885.981	B	^³A♯, v³B
80	897.196	^B, v⁴C♭	^⁴A♯, vvB
81	908.411	^^B, v³C♭	^⁵A♯, vB	22/13
82	919.626	^³B, vvC♭	B	75/44
83	930.841	^⁴B, vC♭	^B, v⁵C
84	942.056	^⁵B, C♭	^^B, v⁴C
85	953.271	B♯, v⁵C	^³B, v³C	26/15, 45/26
86	964.486	^B♯, v⁴C	^⁴B, vvC	7/4
87	975.701	^^B♯, v³C	^⁵B, vC
88	986.916	^³B♯, vvC	C
89	998.131	^⁴B♯, vC	^C, v⁵D♭
90	1009.35	C	^^C, v⁴D♭
91	1020.56	^C, v⁴D♭♭	^³C, v³D♭
92	1031.78	^^C, v³D♭♭	^⁴C, vvD♭	20/11
93	1042.99	^³C, vvD♭♭	^⁵C, vD♭
94	1054.21	^⁴C, vD♭♭	^⁶C, D♭
95	1065.42	^⁵C, D♭♭	^⁷C, v¹²D	24/13
96	1076.64	C♯, v⁵D♭	^⁸C, v¹¹D	28/15
97	1087.85	^C♯, v⁴D♭	^⁹C, v¹⁰D	15/8
98	1099.07	^^C♯, v³D♭	^¹⁰C, v⁹D
99	1110.28	^³C♯, vvD♭	^¹¹C, v⁸D
100	1121.5	^⁴C♯, vD♭	^¹²C, v⁷D	21/11
101	1132.71	^⁵C♯, D♭	C♯, v⁶D	77/40
102	1143.93	C𝄪, v⁵D	^C♯, v⁵D	64/33
103	1155.14	^C𝄪, v⁴D	^^C♯, v⁴D
104	1166.36	^^C𝄪, v³D	^³C♯, v³D	49/25, 55/28
105	1177.57	^³C𝄪, vvD	^⁴C♯, vvD
106	1188.79	^⁴C𝄪, vD	^⁵C♯, vD
107	1200	D	D	2/1

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.9	[339 -107⟩	⟨107 339]	+0.322	0.322	2.87
2.9.5	9765625/9565938, [-34 10 1⟩	⟨107 339 248]	+0.933	0.904	8.06
2.9.5.7	225/224, 84035/82944, [14 -6 7 -4⟩	⟨107 339 248 300]	+1.087	0.827	7.37
2.9.5.7.11	225/224, 441/440, 26411/26244, 161280/161051	⟨107 339 248 300 370]	+0.973	0.774	6.90
2.9.5.7.11.13	225/224, 325/324, 441/440, 847/845, 24500/24167	⟨107 339 248 300 370 396]	+0.783	0.823	7.33
2.9.5.7.11.13.17	170/169, 225/224, 325/324, 441/440, 847/845, 2000/1989	⟨107 339 248 300 370 396 437]	+0.812	0.765	6.82

107edo

Contents

Theory

Subsets and supersets

Odd harmonics

Intervals

Regular temperament properties

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