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{{Infobox ET}}
{{Infobox ET}}
'''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: <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 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> 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.
'''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: {{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). 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 {{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. 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.


It is the 28th [[prime edo]].
It is the 28th [[prime edo]].


==Theory==
Since 107edo has a step of 11.214953271 cents, it also allows one to use its MOS scales as circulating temperaments.  
Since 107edo has a step of 11.214953271 cents, it also allows one to use its MOS scales as circulating temperaments.  
{| class="wikitable"
{| class="wikitable"
Line 268: Line 269:
|85
|85
|22L 63s
|22L 63s
|}
{{Harmonics in equal|107}}
==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]] (%)
|-
|2.3
|{{monzo|170 -107}}
|{{val|107 170}}
| -1.4471
| 1.4453
| 12.89
|-
|2.3.5
|3125/3072, {{monzo|18 -23 8}}
|{{val|107 170 248}}
| -0.2497
| 2.0685
| 18.44
|-
|2.3.5.7
|2240/2187, 1029/1024, 3125/3072
|{{val|107 170 248 300}}
| +0.1987
| 1.9529
| 17.41
|-
|2.3.5.7.11
|100/99, 1232/1215, 1375/1372, 1029/1024
|{{val|107 170 248 300 370}}
| +0.2622
| 1.7513
| 15.62
|-
|2.3.5.7.11.13
|100/99, 196/195, 275/273, 1232/1215, 1029/1024
|{{val|107 170 248 300 370 396}}
| +0.1917
| 1.6065
| 14.32
|-
|2.3.5.7.11.13.17
|100/99, 196/195, 136/135, 275/273, 1232/1215, 1547/1536
|{{val|107 170 248 300 370 396}}
| +0.3048
| 1.5129
| 13.49
|}
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Prime EDO]]

Revision as of 12:53, 12 April 2023

← 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

107edo is the 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: ⟨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 the 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⟩; 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.

It is the 28th prime edo.

Theory

Since 107edo has a step of 11.214953271 cents, it also allows one to use its MOS scales as circulating temperaments.

Circulating temperaments in 107edo
Tones Pattern L:s
5 2L 3s 22:21
6 5L 1s 18:17
7 2L 5s 16:15
8 3L 5s 14:13
9 8L 1s 12:11
10 7L 3s 11:10
11 8L 3s 10:9
12 11L 1s 9:8
13 3L 10s
14 9L 5s 8:7
15 2L 13s
16 11L 5s 7:6
17 5L 12s
18 17L 1s 6:5
19 12L 7s
20 7L 13s
21 2L 19s
22 19L 3s 5:4
23 15L 8s
24 11L 13s
25 7L 18s
26 3L 23s
27 26L 1s 4:3
28 23L 5s
29 20L 9s
30 17L 13s
31 14L 17s
32 11L 21s
33 8L 25s
34 5L 29s
35 2L 33s
36 35L 1s 3:2
37 33L 4s
38 31L 7s
39 29L 10s
40 27L 13s
41 25L 16s
42 23L 19s
43 21L 22s
44 19L 25s
45 17L 28s
46 15L 31s
47 13L 34s
48 11L 37s
49 9L 40s
50 7L 43s
51 5L 46s
52 3L 49s
53 1L 52s
54 53L 1s 2:1
55 52L 3s
56 51L 5s
57 50L 7s
58 49L 9s
59 48L 11s
60 47L 13s
61 46L 15s
62 45L 17s
63 44L 19s
64 43L 21s
65 42L 23s
66 41L 25s
67 40L 27s
68 39L 29s
69 38L 31s
70 37L 33s
71 36L 35s
72 35L 37s
73 34L 39s
74 33L 41s
75 32L 43s
76 31L 45s
77 30L 47s
78 29L 49s
79 28L 51s
80 27L 53s
81 26L 55s
82 25L 57s
83 24L 59s
84 23L 61s
85 22L 63s


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
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)

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [170 -107⟩ ⟨107 170] -1.4471 1.4453 12.89
2.3.5 3125/3072, [18 -23 8⟩ ⟨107 170 248] -0.2497 2.0685 18.44
2.3.5.7 2240/2187, 1029/1024, 3125/3072 ⟨107 170 248 300] +0.1987 1.9529 17.41
2.3.5.7.11 100/99, 1232/1215, 1375/1372, 1029/1024 ⟨107 170 248 300 370] +0.2622 1.7513 15.62
2.3.5.7.11.13 100/99, 196/195, 275/273, 1232/1215, 1029/1024 ⟨107 170 248 300 370 396] +0.1917 1.6065 14.32
2.3.5.7.11.13.17 100/99, 196/195, 136/135, 275/273, 1232/1215, 1547/1536 ⟨107 170 248 300 370 396] +0.3048 1.5129 13.49
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