107edo: Difference between revisions

← 106edo 107edo 108edo →
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

← Older edit

VisualWikitext

Latest revision as of 08:57, 27 September 2025

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, and harmonics 3, 5, and 7 are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 subgroup can be used. For the full 7-limit, it has four possible mappings: ⟨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 [28 -22 3⟩ 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 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.

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 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.

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 (%)	+40.9	-44.6	-38.7	-18.2	-15.9	+5.3	-3.7	-35.9	+47.2	+2.2	-2.1
Steps (reduced)		170 (63)	248 (34)	300 (86)	339 (18)	370 (49)	396 (75)	418 (97)	437 (9)	455 (27)	470 (42)	484 (56)

Octave stretch

107edo’s approximations of 3/1, 5/1, 7/1, 13/1, 17/1 and 19/1 are all improved by 1ed175/174, a stretched-octave version of 107edo. The trade-off is a slightly worse 2/1 and 11/1.

There are also several nearby Zeta peak index (ZPI) tunings which can be used to improve some harmonics at the expense of others: 622zpi, 623zpi, 624zpi, 625zpi, 626zpi, 627zpi, 628zpi and 629zpi.

The details of each of those ZPI tunings are visible in User:Contribution’s gallery of Zeta Peak Indexes (1 - 10 000). Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.

Subsets and supersets

107edo is the 28th prime edo, following 103edo and before 109edo. 214edo, which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 62\107)	Ups and downs notation (Dual sharp fifth 63\107)
0	0	1/1	D	D
1	11.2		^D, ^^E♭♭♭	^D, v⁵E♭
2	22.4		^^D, v³E♭♭	^^D, v⁴E♭
3	33.6		^³D, vvE♭♭	^³D, v³E♭
4	44.9	39/38, 41/40	vvD♯, vE♭♭	^⁴D, vvE♭
5	56.1	31/30, 32/31	vD♯, E♭♭	^⁵D, vE♭
6	67.3		D♯, ^E♭♭	^⁶D, E♭
7	78.5	22/21, 23/22, 45/43	^D♯, ^^E♭♭	v⁶D♯, ^E♭
8	89.7		^^D♯, v³E♭	v⁵D♯, ^^E♭
9	100.9		^³D♯, vvE♭	v⁴D♯, ^³E♭
10	112.1	16/15	vvD𝄪, vE♭	v³D♯, ^⁴E♭
11	123.4	44/41	vD𝄪, E♭	vvD♯, ^⁵E♭
12	134.6	40/37	D𝄪, ^E♭	vD♯, ^⁶E♭
13	145.8	37/34	^D𝄪, ^^E♭	D♯, v⁶E
14	157	23/21	^^D𝄪, v³E	^D♯, v⁵E
15	168.2	32/29, 43/39	^³D𝄪, vvE	^^D♯, v⁴E
16	179.4	41/37	vvD♯𝄪, vE	^³D♯, v³E
17	190.7	29/26	E	^⁴D♯, vvE
18	201.9		^E, ^^F♭♭	^⁵D♯, vE
19	213.1	26/23, 43/38	^^E, v³F♭	E
20	224.3	33/29	^³E, vvF♭	^E, v⁵F
21	235.5		vvE♯, vF♭	^^E, v⁴F
22	246.7	15/13	vE♯, F♭	^³E, v³F
23	257.9		E♯, ^F♭	^⁴E, vvF
24	269.2		^E♯, ^^F♭	^⁵E, vF
25	280.4	20/17	^^E♯, v³F	F
26	291.6	45/38	^³E♯, vvF	^F, v⁵G♭
27	302.8	31/26	vvE𝄪, vF	^^F, v⁴G♭
28	314		F	^³F, v³G♭
29	325.2	41/34	^F, ^^G♭♭♭	^⁴F, vvG♭
30	336.4	17/14	^^F, v³G♭♭	^⁵F, vG♭
31	347.7		^³F, vvG♭♭	^⁶F, G♭
32	358.9	16/13	vvF♯, vG♭♭	v⁶F♯, ^G♭
33	370.1	26/21	vF♯, G♭♭	v⁵F♯, ^^G♭
34	381.3		F♯, ^G♭♭	v⁴F♯, ^³G♭
35	392.5		^F♯, ^^G♭♭	v³F♯, ^⁴G♭
36	403.7	24/19	^^F♯, v³G♭	vvF♯, ^⁵G♭
37	415	33/26	^³F♯, vvG♭	vF♯, ^⁶G♭
38	426.2		vvF𝄪, vG♭	F♯, v⁶G
39	437.4		vF𝄪, G♭	^F♯, v⁵G
40	448.6	22/17	F𝄪, ^G♭	^^F♯, v⁴G
41	459.8	30/23, 43/33	^F𝄪, ^^G♭	^³F♯, v³G
42	471	21/16	^^F𝄪, v³G	^⁴F♯, vvG
43	482.2	37/28, 41/31	^³F𝄪, vvG	^⁵F♯, vG
44	493.5		vvF♯𝄪, vG	G
45	504.7		G	^G, v⁵A♭
46	515.9	31/23	^G, ^^A♭♭♭	^^G, v⁴A♭
47	527.1	42/31	^^G, v³A♭♭	^³G, v³A♭
48	538.3	15/11	^³G, vvA♭♭	^⁴G, vvA♭
49	549.5	11/8	vvG♯, vA♭♭	^⁵G, vA♭
50	560.7	29/21	vG♯, A♭♭	^⁶G, A♭
51	572	32/23	G♯, ^A♭♭	v⁶G♯, ^A♭
52	583.2	7/5	^G♯, ^^A♭♭	v⁵G♯, ^^A♭
53	594.4	31/22	^^G♯, v³A♭	v⁴G♯, ^³A♭
54	605.6	44/31	^³G♯, vvA♭	v³G♯, ^⁴A♭
55	616.8	10/7	vvG𝄪, vA♭	vvG♯, ^⁵A♭
56	628	23/16	vG𝄪, A♭	vG♯, ^⁶A♭
57	639.3	42/29	G𝄪, ^A♭	G♯, v⁶A
58	650.5	16/11	^G𝄪, ^^A♭	^G♯, v⁵A
59	661.7	22/15, 41/28	^^G𝄪, v³A	^^G♯, v⁴A
60	672.9	31/21	^³G𝄪, vvA	^³G♯, v³A
61	684.1	43/29, 46/31	vvG♯𝄪, vA	^⁴G♯, vvA
62	695.3		A	^⁵G♯, vA
63	706.5		^A, ^^B♭♭♭	A
64	717.8		^^A, v³B♭♭	^A, v⁵B♭
65	729	32/21	^³A, vvB♭♭	^^A, v⁴B♭
66	740.2	23/15	vvA♯, vB♭♭	^³A, v³B♭
67	751.4	17/11	vA♯, B♭♭	^⁴A, vvB♭
68	762.6	45/29	A♯, ^B♭♭	^⁵A, vB♭
69	773.8		^A♯, ^^B♭♭	^⁶A, B♭
70	785		^^A♯, v³B♭	v⁶A♯, ^B♭
71	796.3	19/12	^³A♯, vvB♭	v⁵A♯, ^^B♭
72	807.5		vvA𝄪, vB♭	v⁴A♯, ^³B♭
73	818.7		vA𝄪, B♭	v³A♯, ^⁴B♭
74	829.9	21/13	A𝄪, ^B♭	vvA♯, ^⁵B♭
75	841.1	13/8	^A𝄪, ^^B♭	vA♯, ^⁶B♭
76	852.3		^^A𝄪, v³B	A♯, v⁶B
77	863.6	28/17	^³A𝄪, vvB	^A♯, v⁵B
78	874.8		vvA♯𝄪, vB	^^A♯, v⁴B
79	886		B	^³A♯, v³B
80	897.2		^B, ^^C♭♭	^⁴A♯, vvB
81	908.4		^^B, v³C♭	^⁵A♯, vB
82	919.6	17/10	^³B, vvC♭	B
83	930.8		vvB♯, vC♭	^B, v⁵C
84	942.1		vB♯, C♭	^^B, v⁴C
85	953.3	26/15	B♯, ^C♭	^³B, v³C
86	964.5		^B♯, ^^C♭	^⁴B, vvC
87	975.7		^^B♯, v³C	^⁵B, vC
88	986.9	23/13	^³B♯, vvC	C
89	998.1		vvB𝄪, vC	^C, v⁵D♭
90	1009.3	43/24	C	^^C, v⁴D♭
91	1020.6		^C, ^^D♭♭♭	^³C, v³D♭
92	1031.8	29/16	^^C, v³D♭♭	^⁴C, vvD♭
93	1043	42/23	^³C, vvD♭♭	^⁵C, vD♭
94	1054.2		vvC♯, vD♭♭	^⁶C, D♭
95	1065.4	37/20	vC♯, D♭♭	v⁶C♯, ^D♭
96	1076.6	41/22	C♯, ^D♭♭	v⁵C♯, ^^D♭
97	1087.9	15/8	^C♯, ^^D♭♭	v⁴C♯, ^³D♭
98	1099.1		^^C♯, v³D♭	v³C♯, ^⁴D♭
99	1110.3		^³C♯, vvD♭	vvC♯, ^⁵D♭
100	1121.5	21/11, 44/23	vvC𝄪, vD♭	vC♯, ^⁶D♭
101	1132.7		vC𝄪, D♭	C♯, v⁶D
102	1143.9	31/16	C𝄪, ^D♭	^C♯, v⁵D
103	1155.1		^C𝄪, ^^D♭	^^C♯, v⁴D
104	1166.4		^^C𝄪, v³D	^³C♯, v³D
105	1177.6		^³C𝄪, vvD	^⁴C♯, vvD
106	1188.8		vvC♯𝄪, vD	^⁵C♯, vD
107	1200	2/1	D	D

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

@@ Line 1: / Line 1: @@
-'''107edo''' is the [[EDO|equal division of the octave]] into 107 parts of 11.214953271 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: &lt;107 170 248 300| (patent val), &lt;107 169 248 300| (107b), &lt;107 170 249 300| (107c), and &lt;107 170 249 301| (107cd). Using the patent val, it tempers out the [[Magic family|small diesis]], 3125/3072 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 the [[syntonic comma]], 81/80 and |-61 -1 27&gt; 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 the immunity comma, 1638400/1594323 and the valentine comma, 1990656/1953125 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.
+{{Infobox ET}}
+{{ED intro}}
-It is the 28th [[prime edo]].
+== Theory ==
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, and [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 [[subgroup]] can be used. For the full 7-limit, it has four possible [[mapping]]s: {{val| 107 170 248 300 }} ([[patent val]]), {{val| 107 '''169''' 248 300 }} (107b), {{val| 107 170 '''249''' 300 }} (107c), and {{val| 107 170 '''249''' '''301''' }} (107cd).
-[[Category:Edo]]
+Using the patent val, it tempers out 3125/3072 ([[magic comma]]) and {{monzo| 28 -22 3 }} 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.
-[[Category:Prime EDO]]
+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.
+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 107b val, it tempers out 81/80 ([[syntonic comma]]) and {{monzo| -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.
+=== Odd harmonics ===
+{{Harmonics in equal|107}}
+=== Octave stretch ===
+edo’s approximations of 3/1, 5/1, 7/1, 13/1, 17/1 and 19/1 are all improved by [[AS|1ed175/174]], a [[Octave stretch|stretched-octave]] version of 107edo. The trade-off is a slightly worse 2/1 and 11/1.
+There are also several nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve some harmonics at the expense of others: 622zpi, 623zpi, 624zpi, 625zpi, 626zpi, 627zpi, 628zpi and 629zpi.
+The details of each of those ZPI tunings are visible in [[User:Contribution]]’s gallery of [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)|Zeta Peak Indexes (1 - 10 000)]]. Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.
+=== Subsets and supersets ===
+edo is the 28th [[prime edo]], following [[103edo]] and before [[109edo]]. [[214edo]], which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.
+== Intervals ==
+{{Interval table}}
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.9
+| {{monzo| 339 -107 }}
+| {{mapping| 107 339 }}
+| +0.322
+| 0.322
+| 2.87
+|-
+| 2.9.5
+| 9765625/9565938, {{monzo| -34 10 1 }}
+| {{mapping| 107 339 248 }}
+| +0.933
+| 0.904
+| 8.06
+|-
+| 2.9.5.7
+| 225/224, 84035/82944, {{monzo| 14 -6 7 -4 }}
+| {{mapping| 107 339 248 300 }}
+| +1.087
+| 0.827
+| 7.37
+|-
+| 2.9.5.7.11
+| 225/224, 441/440, 26411/26244, 161280/161051
+| {{mapping| 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
+| {{mapping| 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
+| {{mapping| 107 339 248 300 370 396 437 }}
+| +0.812
+| 0.765
+| 6.82
+|}

107edo: Difference between revisions

Latest revision as of 08:57, 27 September 2025

Contents

Theory

Odd harmonics

Octave stretch

Subsets and supersets

Intervals

Regular temperament properties

Navigation menu

107edo: Difference between revisions

Latest revision as of 08:57, 27 September 2025

Theory

Odd harmonics

Octave stretch

Subsets and supersets

Intervals

Regular temperament properties

Navigation menu

Search