103edo: Difference between revisions

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== Scales ==
== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}
=== 13-limit temperaments ===
=== 13-limit temperaments ===
{| class="wikitable"
{| class="wikitable"

Revision as of 21:11, 30 May 2023

← 102edo 103edo 104edo →
Prime factorization 103 (prime)
Step size 11.6505 ¢ 
Fifth 60\103 (699.029 ¢)
Semitones (A1:m2) 8:9 (93.2 ¢ : 104.9 ¢)
Consistency limit 7
Distinct consistency limit 7

The 103 equal divisions of the octave (103edo), or the 103(-tone) equal temperament (103tet, 103et) when viewed from a regular temperament perspective, is the equal division of the octave into 103 steps of about 11.7 cents each.

Theory

103edo is a good miracle tuning, especially for the 7- and 13-limit and benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.

103edo is the 27th prime edo.

Prime harmonics

Approximation of prime harmonics in 103edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -2.93 -1.85 -1.84 -3.75 -1.69 -0.10 +5.40 +0.85 -4.33 -3.29
Relative (%) +0.0 -25.1 -15.9 -15.8 -32.1 -14.5 -0.9 +46.3 +7.3 -37.2 -28.2
Steps
(reduced) 		103
(0) 		163
(60) 		239
(33) 		289
(83) 		356
(47) 		381
(72) 		421
(9) 		438
(26) 		466
(54) 		500
(88) 		510
(98)

Intervals

Degree Cents Approximate Ratios
1 11.650 81/80, 126/125
2 23.301 65/64, 66/65, 78/77
3 34.951 49/48, 50/49, 64/63
4 46.602 33/32, 35/34, 36/35
5 58.252 27/26, 34/33
6 69.903 25/24, 26/25, 28/27
7 81.553 21/20, 22/21
8 93.204 18/17
9 104.854 17/16
10 116.505 15/14, 16/15
11 128.155 14/13
12 139.806 13/12
13 151.456 12/11
14 163.107 11/10
15 174.757 72/65
16 186.408 10/9
17 198.058 9/8
18 209.708
19 221.359 17/15, 25/22
20 233.010 8/7
21 244.660 15/13
22 256.311
23 267.961 7/6
24 279.712 20/17
25 291.262 13/11
26 303.013 25/21
27 314.563 6/5
28 326.214 63/52, 65/54
29 337.864 17/14, 39/32
30 349.615 11/9, 27/22
31 361.165 16/13, 21/17
32 372.816 26/21, 81/65
33 384.466 5/4
34 396.117 44/35
35 407.767 33/26
36 419.417 14/11
37 431.068 9/7
38 442.708 22/17
39 454.369 13/10
40 466.019 17/13, 21/16
41 477.670
42 489.320 65/49
43 500.971 4/3
44 512.621 27/20
45 524.272 65/48
46 535.922 15/11
47 547.573 11/8
48 559.223 18/13
49 570.874 25/18
50 582.524 7/5
51 594.175 24/17
… … …

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-163 103⟩ [⟨103 166]] +0.923 0.924 7.92
2.3.5 78732/78125, 34171875/33554432 [⟨103 166 239]] +0.881 0.757 6.49
2.3.5.7 225/224, 1029/1024, 78732/78125 [⟨103 166 239 289]] +0.824 0.663 5.68
2.3.5.7.11 225/224, 243/242, 385/384, 43923/43750 [⟨103 166 239 289 356]] +0.876 0.602 5.16
2.3.5.7.11.13 225/224, 243/242, 351/350, 385/384, 847/845 [⟨103 166 239 289 356 381]] +0.806 0.571 4.90
2.3.5.7.11.13.17 225/224, 243/242, 273/272, 351/350, 375/374, 847/845 [⟨103 166 239 289 356 381 421]] +0.694 0.595 5.10

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 3\103 34.951 1990656/1953125 Gammic (5-limit)
1 5\103 58.252 27/26 Hemisecordite
1 9\103 104.854 17/16 Septendesemi
1 10\103 116.505 15/14~16/15 Miracle / benediction
1 16\103 186.408 10/9 Mintone
1 20\103 233.010 8/7 Slendric
1 21\103 244.660 15/13 Subsemifourth
1 26\103 303.013 25/21 Quinmite
1 31\103 361.165 16/13 Phicordial
1 37\103 431.06 77/60 Lockerbie
1 38\103 442.708 162/125 Sensipent / sensei
1 39\103 454.369 13/10 Fibo
1 40\103 466.019 55/42 Hemiseptisix
1 42\103 489.320 65/49 Catafourth
1 45\103 524.272 65/48 Widefourth
1 47\103 547.573 11/8 Heinz
1 48\103 559.223 242/175 Tritriple
1 50\103 582.524 7/5 Neptune

Scales

13-limit temperaments

Marvel and Benediction Hemisecordite
Degree cents Difference from 72edo Degree cents Difference from 62edo
1 11.6505 -5.016¢ 2 23.301 3.946¢
3 34.9515 1.618¢ 3 34.9515 -3.758¢
4 46.602 -3.398¢ 5 58.252 0.188¢
6 69.903 3.236¢ 7 81.553 4.134¢
7 81.553 -1.78¢ 8 93.204 -3.57¢
9 104.854 4.854¢ 10 116.505 0.376¢
10 116.5045 -0.162¢ 12 139.806 4.322¢
11 128.155 -5.178¢ 13 151.456 -3.382¢
13 151.456¢ 1.456¢ 15 174.757 0.563¢
14 163.107¢ -3.56¢ 17 198.058 4.51¢
16 186.408 3.074¢ 18 209.709 -3.1945¢
17 198.058 -1.942¢ 20 233.01 0.751¢
19 221.359 4.693¢ 22 256.311 4.698¢
20 233.01 -0.324¢ 23 267.961 -3.007¢
21 244.66 -5.34¢ 25 291.262 0.94¢
23 267.961 1.2945¢ 27 314.563 4.886¢
24 279.612 -3.722¢ 28 326.214 -2.819¢
26 302.913 2.913¢ 30 349.515 1.1275¢
27 314.563 -2.104¢ 32 372.8155 5.074¢
29 337.864 4.531¢ 33 384.466 -2.631¢
30 349.515 -0.485¢ 35 407.767 1.315¢
31 361.165 -5.502¢ 37 431.068 5.2615¢
33 384.466 1.133¢ 38 442.718 -2.443¢
34 396.1165 -3.8835¢ 40 466.0190 1.503¢
36 419.4175 2.751¢ 42 489.32 5.449¢
37 431.068 -2.265¢ 43 500.971 -2.255¢
39 454.369 4.369¢ 45 524.272 1.691¢
40 466.019 -0.647¢ 47 547.573 5.637¢
41 477.67 -5.663¢ 48 559.223 -2.067¢
43 500.971 0.971¢ 50 582.524 1.879¢
44 512.621¢ -4.045¢ 52 605.825 5.825¢
46 535.922¢ 2.589¢ 53 617.476 -1.879¢
47 547.573¢ -2.427¢ 55 640.777¢ 2.067
49 570.874¢ 4.207¢ 56 652.427 -5.637¢
50 582.524 -0.809¢ 58 675.728 -1.691
52 605.825 5.825¢ 60 699.029 2.255¢
53 617.475 0.809¢ 61 710.68 -5.449¢
54 629.126¢ -4.207¢ 63 733.981 -1.503¢
56 652.427¢ 2.427¢ 65 757.282 2.443¢
57 664.078 -2.589¢ 66 768.932 -5.2615¢
59 687.379 4.045¢ 68 792.233 -1.315¢
60 699.029 -0.971¢ 70 815.534 2.631¢
62 722.33 5.663¢ 71 827.1845 -5.074¢
63 733.981 0.647¢ 73 850.485 -1.1275¢
64 745.631 -4.369¢ 75 873.786 2.819¢
66 768.932 2.265¢ 76 885.437 -4.886¢
67 780.5825 -2.751¢ 78 908.738 -0.94¢
69 803.8835 3.8835¢ 80 932.039 3.007¢
70 815.534 -1.133¢ 81 943.689 -4.698¢
72 838.835 5.501¢ 83 966.99 -0.752¢
73 850.485 0.485¢ 85 990.291 3.1945¢
74 862.136 -4.531¢ 86 1001.942 -4.51¢
76 885.439 2.104¢ 88 1025.243 -0.564¢
77 897.087 -2.913¢ 90 1048.544 3.382¢
79 920.388 3.722¢ 91 1060.194 -4.322¢
80 932.039 -1.2945¢ 93 1083.495 -0.376¢
82 955.34 5.34¢ 95 1106.796 3.57¢
83 966.99 0.324¢ 96 1118.447¢ -4.134¢
84 978.641 -4.693¢ 98 1141.748 -0.188¢
86 1001.942 1.942¢ 100 1165.0485 3.758¢
87 1013.592 -3.074¢ 101 1176.699 -3.946¢
89 1036.893 3.56¢
90 1048.544 -1.456¢
92 1071.845 5.178¢
93 1083.495 0.162¢
94 1095.146 -4.854¢
96 1118.447 1.78¢
97 1130.097 -3.236¢
99 1153.398 3.398¢
100 1165.0485 -1.618¢
102 1188.3495 5.016¢

Music

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