104edo
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//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 family|magic temperament]]. The sharp major third val, <104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]]. 104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|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** ||
Original HTML content:
<html><head><title>104edo</title></head><body><em>104edo</em> 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 <a class="wiki_link" href="/Magic%20family">magic temperament</a>. The sharp major third val, <104 165 242|, tempers out 2048/2025 and supports <a class="wiki_link" href="/Diaschismic%20family">diaschismic temperament</a>.<br />
<br />
104edo with the flat third is especially notable as an excellent tuning for <a class="wiki_link" href="/Magic%20family">magic temperament</a>, providing the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for 11-limit magic and the 13-limit magic extension <a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy">necromancy</a>. 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 <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>.<br />
<br />
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.<br />
<br />
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.<br />
<br />
<strong>17-limit Regular Temperaments</strong><br />
<table class="wiki_table">
<tr>
<th>Degree<br />
</th>
<th>Cents<br />
</th>
</tr>
<tr>
<td><strong>2</strong><br />
</td>
<td><strong>23.08</strong><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>34.615<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>46.15<br />
</td>
</tr>
<tr>
<td><strong>5</strong><br />
</td>
<td><strong>57.69</strong><br />
</td>
</tr>
<tr>
<td><strong>7</strong><br />
</td>
<td><strong>80.77</strong><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>92.31<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>103.85<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>115.385<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>126.92<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>138.46<br />
</td>
</tr>
<tr>
<td><strong>13</strong><br />
</td>
<td><strong>150</strong><br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>161.54<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>173.08<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>184.615<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>196.15<br />
</td>
</tr>
<tr>
<td><strong>18</strong><br />
</td>
<td><strong>207.69</strong><br />
</td>
</tr>
<tr>
<td><strong>20</strong><br />
</td>
<td><strong>230.77</strong><br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>242.31<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>253.85<br />
</td>
</tr>
<tr>
<td><strong>23</strong><br />
</td>
<td><strong>265.385</strong><br />
</td>
</tr>
<tr>
<td><strong>25</strong><br />
</td>
<td><strong>288.46</strong><br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>300<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>311.54<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>323.08<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>334.615<br />
</td>
</tr>
<tr>
<td><strong>30</strong><br />
</td>
<td><strong>346.15</strong><br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>357.69<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>369.23<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>380.77<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>392.31<br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>403.85<br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>415.385<br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>438.46<br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>450<br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>461.54<br />
</td>
</tr>
<tr>
<td><strong>41</strong><br />
</td>
<td><strong>473.08</strong><br />
</td>
</tr>
<tr>
<td><strong>43</strong><br />
</td>
<td><strong>496.15</strong><br />
</td>
</tr>
<tr>
<td><strong>45</strong><br />
</td>
<td><strong>519.23</strong><br />
</td>
</tr>
<tr>
<td>46<br />
</td>
<td>530.77<br />
</td>
</tr>
<tr>
<td>47<br />
</td>
<td>542.31<br />
</td>
</tr>
<tr>
<td><strong>48</strong><br />
</td>
<td><strong>553.85</strong><br />
</td>
</tr>
<tr>
<td>50<br />
</td>
<td>576.92<br />
</td>
</tr>
<tr>
<td>51<br />
</td>
<td>588.45<br />
</td>
</tr>
<tr>
<td>52<br />
</td>
<td>600<br />
</td>
</tr>
<tr>
<td>53<br />
</td>
<td>611.54<br />
</td>
</tr>
<tr>
<td><strong>54</strong><br />
</td>
<td><strong>623.08</strong><br />
</td>
</tr>
<tr>
<td>56<br />
</td>
<td>646.15<br />
</td>
</tr>
<tr>
<td>57<br />
</td>
<td>657.69<br />
</td>
</tr>
<tr>
<td>58<br />
</td>
<td>669.23<br />
</td>
</tr>
<tr>
<td>59<br />
</td>
<td>680.77<br />
</td>
</tr>
<tr>
<td><strong>61</strong><br />
</td>
<td><strong>703.85</strong><br />
</td>
</tr>
<tr>
<td>63<br />
</td>
<td>726.92<br />
</td>
</tr>
<tr>
<td>64<br />
</td>
<td>738.46<br />
</td>
</tr>
<tr>
<td>65<br />
</td>
<td>750<br />
</td>
</tr>
<tr>
<td><strong>66</strong><br />
</td>
<td><strong>761.54</strong><br />
</td>
</tr>
<tr>
<td>67<br />
</td>
<td>773.08<br />
</td>
</tr>
<tr>
<td><strong>68</strong><br />
</td>
<td><strong>784.615</strong><br />
</td>
</tr>
<tr>
<td>69<br />
</td>
<td>796.15<br />
</td>
</tr>
<tr>
<td>70<br />
</td>
<td>807.69<br />
</td>
</tr>
<tr>
<td>71<br />
</td>
<td>819.23<br />
</td>
</tr>
<tr>
<td>72<br />
</td>
<td>830.77<br />
</td>
</tr>
<tr>
<td>73<br />
</td>
<td>842.31<br />
</td>
</tr>
<tr>
<td><strong>74</strong><br />
</td>
<td><strong>853.85</strong><br />
</td>
</tr>
<tr>
<td>75<br />
</td>
<td>865.385<br />
</td>
</tr>
<tr>
<td>76<br />
</td>
<td>876.92<br />
</td>
</tr>
<tr>
<td>77<br />
</td>
<td>888.46<br />
</td>
</tr>
<tr>
<td>78<br />
</td>
<td>900<br />
</td>
</tr>
<tr>
<td>79<br />
</td>
<td>911.54<br />
</td>
</tr>
<tr>
<td><strong>81</strong><br />
</td>
<td><strong>934.615</strong><br />
</td>
</tr>
<tr>
<td>82<br />
</td>
<td>946.15<br />
</td>
</tr>
<tr>
<td>83<br />
</td>
<td>957.69<br />
</td>
</tr>
<tr>
<td><strong>84</strong><br />
</td>
<td><strong>969.23</strong><br />
</td>
</tr>
<tr>
<td>86<br />
</td>
<td>992.31<br />
</td>
</tr>
<tr>
<td>87<br />
</td>
<td>1003.85<br />
</td>
</tr>
<tr>
<td>88<br />
</td>
<td>1015.385<br />
</td>
</tr>
<tr>
<td>89<br />
</td>
<td>1026.92<br />
</td>
</tr>
<tr>
<td>90<br />
</td>
<td>1038.46<br />
</td>
</tr>
<tr>
<td><strong>91</strong><br />
</td>
<td><strong>1050</strong><br />
</td>
</tr>
<tr>
<td>92<br />
</td>
<td>1061.54<br />
</td>
</tr>
<tr>
<td>93<br />
</td>
<td>1073.08<br />
</td>
</tr>
<tr>
<td>95<br />
</td>
<td>1096.15<br />
</td>
</tr>
<tr>
<td>96<br />
</td>
<td>1107.69<br />
</td>
</tr>
<tr>
<td><strong>97</strong><br />
</td>
<td><strong>1119.23</strong><br />
</td>
</tr>
<tr>
<td>99<br />
</td>
<td>1142.31<br />
</td>
</tr>
<tr>
<td><strong>100</strong><br />
</td>
<td><strong>1153.85</strong><br />
</td>
</tr>
<tr>
<td>101<br />
</td>
<td>1165.385<br />
</td>
</tr>
<tr>
<td><strong>102</strong><br />
</td>
<td><strong>1176.92</strong><br />
</td>
</tr>
</table>
</body></html>