104edo
104edo divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, <104 165 241|, tempers out 3125/3072, and supports magic temperament. The sharp major third val, <104 165 242|, tempers out 2048/2025 and supports diaschismic temperament.
104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the optimal patent val for 11-limit magic and the 13-limit magic extension necromancy. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.
17-limit Regular Temperaments
Degree | Cents |
---|---|
2 | 23.08 |
3 | 34.615 |
4 | 46.15 |
5 | 57.69 |
7 | 80.77 |
8 | 92.31 |
9 | 103.85 |
10 | 115.385 |
11 | 126.92 |
12 | 138.46 |
13 | 150 |
14 | 161.54 |
15 | 173.08 |
16 | 184.615 |
17 | 196.15 |
18 | 207.69 |
20 | 230.77 |
21 | 242.31 |
22 | 253.85 |
23 | 265.385 |
25 | 288.46 |
26 | 300 |
27 | 311.54 |
28 | 323.08 |
29 | 334.615 |
30 | 346.15 |
31 | 357.69 |
32 | 369.23 |
33 | 380.77 |
34 | 392.31 |
35 | 403.85 |
36 | 415.385 |
38 | 438.46 |
39 | 450 |
40 | 461.54 |
41 | 473.08 |
43 | 496.15 |
45 | 519.23 |
46 | 530.77 |
47 | 542.31 |
48 | 553.85 |
50 | 576.92 |
51 | 588.45 |
52 | 600 |
53 | 611.54 |
54 | 623.08 |
56 | 646.15 |
57 | 657.69 |
58 | 669.23 |
59 | 680.77 |
61 | 703.85 |
63 | 726.92 |
64 | 738.46 |
65 | 750 |
66 | 761.54 |
67 | 773.08 |
68 | 784.615 |
69 | 796.15 |
70 | 807.69 |
71 | 819.23 |
72 | 830.77 |
73 | 842.31 |
74 | 853.85 |
75 | 865.385 |
76 | 876.92 |
77 | 888.46 |
78 | 900 |
79 | 911.54 |
81 | 934.615 |
82 | 946.15 |
83 | 957.69 |
84 | 969.23 |
86 | 992.31 |
87 | 1003.85 |
88 | 1015.385 |
89 | 1026.92 |
90 | 1038.46 |
91 | 1050 |
92 | 1061.54 |
93 | 1073.08 |
95 | 1096.15 |
96 | 1107.69 |
97 | 1119.23 |
99 | 1142.31 |
100 | 1153.85 |
101 | 1165.385 |
102 | 1176.92 |