# 103edo

 ← 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

103 equal divisions of the octave (103edo), or 103-tone equal temperament (103tet), 103 equal temperament (103et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 103 equal parts of about 11.7 ¢ each.

## Theory

103edo is a good miracle tuning, especially for the 7-, and for 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.

In 103edo, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

### 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 -25 -16 -16 -32 -15 -1 +46 +7 -37 -28
Steps
(reduced)
103
(0)
163
(60)
239
(33)
289
(83)
356
(47)
381
(72)
421
(9)
438
(26)
466
(54)
500
(88)
510
(98)

### Subsets and supersets

103edo is the 27th prime edo.

## 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 8ve
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 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¢