104edo: Difference between revisions

Revision as of 11:22, 8 August 2016

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Original Wikitext content:

//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, &lt;104 165 241|, tempers out 3125/3072, and supports <a class="wiki_link" href="/Magic%20family">magic temperament</a>. The sharp major third val, &lt;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&amp;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>

104edo: Difference between revisions

Revision as of 11:22, 8 August 2016

IMPORTED REVISION FROM WIKISPACES

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