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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-14 17:06:24 UTC</tt>.<br>
| |
| : The original revision id was <tt>599442450</tt>.<br>
| |
| : The revision comment was: <tt></tt><br>
| |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| |
| <h4>Original Wikitext content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit [[val]]s, 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]].
| |
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|
| 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]]. | | == Theory == |
| | 104edo is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.<!-- Add commas in 2.3.7.11.13.17.19 as well --> It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap| 17 & 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth. |
|
| |
|
| 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.
| | Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup. |
|
| |
|
| 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.
| | If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament. Additionally, it is viable to treat 104edo as dual-5, or as a 2.3.25.7.11.13.17.19 subgroup temperament. |
|
| |
|
| **17-limit Regular Temperaments**
| | 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 also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val. |
| ||~ 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** ||</pre></div>
| |
| <h4>Original HTML content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 <a class="wiki_link" href="/val">val</a>s, 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 />
| |
|
| |
|
| | 104edo 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]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit. |
|
| |
|
| <table class="wiki_table">
| | === Prime harmonics === |
| <tr>
| | {{Harmonics in equal|104}} |
| <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></pre></div>
| | === Octave stretch === |
| | 104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. |
| | |
| | === Subsets and supersets === |
| | Since 104 factors into primes as {{nowrap| 2<sup>3</sup> × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. |
| | |
| | == Regular temperament properties == |
| | {| class="wikitable center-4 center-5 center-6" |
| | |- |
| | ! rowspan="2" | [[Subgroup]] |
| | ! rowspan="2" | [[Comma list]] |
| | ! rowspan="2" | [[Mapping]] |
| | ! rowspan="2" | Optimal<br />8ve stretch (¢) |
| | ! colspan="2" | Tuning error |
| | |- |
| | ! [[TE error|Absolute]] (¢) |
| | ! [[TE simple badness|Relative]] (%) |
| | |- |
| | | 2.3 |
| | | {{monzo| 165 -104 }} |
| | | {{mapping| 104 165 }} |
| | | −0.597 |
| | | 0.596 |
| | | 5.17 |
| | |- |
| | | 2.3.5 |
| | | 2048/2025, {{monzo| 0 22 -15 }} |
| | | {{mapping| 104 165 242 }} (104c) |
| | | −1.258 |
| | | 1.054 |
| | | 9.14 |
| | |- |
| | | 2.3.5.7 |
| | | 126/125, 2048/2025, 117649/116640 |
| | | {{mapping| 104 165 242 292 }} (104c) |
| | | −0.980 |
| | | 1.032 |
| | | 8.95 |
| | |- |
| | | 2.3.5.7.11 |
| | | 126/125, 176/175, 896/891, 14641/14580 |
| | | {{mapping| 104 165 242 292 360 }} (104c) |
| | | −0.930 |
| | | 0.929 |
| | | 8.05 |
| | |- |
| | | 2.3.5.7.11.13 |
| | | 126/125, 176/175, 196/195, 364/363, 2197/2187 |
| | | {{mapping| 104 165 242 292 360 385 }} (104c) |
| | | −0.855 |
| | | 0.864 |
| | | 7.49 |
| | |} |
| | |
| | === Rank-2 temperaments === |
| | {| class="wikitable center-all left-5" |
| | |+ style="font-size: 105%;" | Patent val |
| | |- |
| | ! Periods<br />per 8ve |
| | ! Generator* |
| | ! Cents* |
| | ! Associated<br />ratio* |
| | ! Temperament |
| | |- |
| | | 1 |
| | | 33\104 |
| | | 380.77 |
| | | 5/4 |
| | | [[Magic]] / necromancy / divination |
| | |- |
| | | 1 |
| | | 51\104 |
| | | 588.46 |
| | | 7/5 |
| | | [[Untriton]] |
| | |- |
| | | 4 |
| | | 9\104 |
| | | 103.85 |
| | | 18/17 |
| | | [[Undim]] |
| | |} |
| | |
| | {| class="wikitable center-all left-5" |
| | |+ style="font-size: 105%;" | 104c val |
| | |- |
| | ! Periods<br />per 8ve |
| | ! Generator* |
| | ! Cents* |
| | ! Associated<br />ratio* |
| | ! Temperament |
| | |- |
| | | 1 |
| | | 11\104 |
| | | 126.92 |
| | | 27/25 |
| | | [[Mowgli]] |
| | |- |
| | | 1 |
| | | 21\104 |
| | | 242.31 |
| | | 147/128 |
| | | [[Septiquarter]] |
| | |- |
| | | 1 |
| | | 27\104 |
| | | 311.54 |
| | | 6/5 |
| | | [[Oolong]] |
| | |- |
| | | 1 |
| | | 47\104 |
| | | 542.31 |
| | | 15/11 |
| | | [[Casablanca]] / marrakesh |
| | |- |
| | | 2 |
| | | 21\104 |
| | | 242.31 |
| | | 121/105 |
| | | [[Semiseptiquarter]] |
| | |- |
| | | 2 |
| | | 43\104<br />(9\104) |
| | | 496.15<br />(103.85) |
| | | 4/3<br />(17/16) |
| | | [[Diaschismic]] |
| | |- |
| | | 8 |
| | | 49\104<br />(2\104) |
| | | 565.38<br />(34.62) |
| | | 168/121<br />(55/54) |
| | | [[Octowerck]] / octowerckis |
| | |- |
| | | 26 |
| | | 43\104<br />(1\104) |
| | | 496.15<br />(11.54) |
| | | 4/3<br />(225/224) |
| | | [[Bosonic]] |
| | |} |
| | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct |
| | |
| | == Intervals == |
| | {| class="wikitable center-1 right-2" |
| | |- |
| | ! rowspan="2" | # |
| | ! rowspan="2" | Cents |
| | ! colspan="3" | Approximate ratios |
| | |- |
| | ! Of 2.3.25.7.11.13.17.19<br>subgroup |
| | ! Additional ratios of 5<br>tending sharp (104c val) |
| | ! Additional ratios of 5<br>tending flat (patent val) |
| | |- |
| | | 0 |
| | | 0.0 |
| | | [[1/1]] |
| | | |
| | | |
| | |- |
| | | 1 |
| | | 11.5 |
| | | [[144/143]], [[169/168]] |
| | | ''[[91/90]]'', [[121/120]] |
| | | [[105/104]], [[196/195]] |
| | |- |
| | | 2 |
| | | 23.1 |
| | | [[64/63]], [[99/98]] |
| | | [[81/80]], [[100/99]], ''[[105/104]]'' |
| | | ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]'' |
| | |- |
| | | 3 |
| | | 34.6 |
| | | [[49/48]], [[50/49]] |
| | | [[55/54]] |
| | | ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]'' |
| | |- |
| | | 4 |
| | | 46.2 |
| | | |
| | | [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]'' |
| | | |
| | |- |
| | | 5 |
| | | 57.7 |
| | | [[28/27]], [[33/32]] |
| | | ''[[26/25]]'' |
| | | ''[[25/24]]'', ''[[36/35]]'' |
| | |- |
| | | 6 |
| | | 69.2 |
| | | [[25/24]], [[26/25]], [[27/26]] |
| | | |
| | | |
| | |- |
| | | 7 |
| | | 80.8 |
| | | [[22/21]] |
| | | [[21/20]], ''[[25/24]]'' |
| | | ''[[20/19]]'', ''[[26/25]]'' |
| | |- |
| | | 8 |
| | | 92.3 |
| | | [[19/18]] |
| | | [[20/19]] |
| | | ''[[21/20]]'' |
| | |- |
| | | 9 |
| | | 103.8 |
| | | [[17/16]], [[18/17]] |
| | | ''[[16/15]]'' |
| | | |
| | |- |
| | | 10 |
| | | 115.4 |
| | | |
| | | |
| | | [[16/15]], [[15/14]] |
| | |- |
| | | 11 |
| | | 126.9 |
| | | [[14/13]] |
| | | ''[[15/14]]'' |
| | | |
| | |- |
| | | 12 |
| | | 138.5 |
| | | [[13/12]] |
| | | |
| | | |
| | |- |
| | | 13 |
| | | 150.0 |
| | | [[12/11]] |
| | | |
| | | |
| | |- |
| | | 14 |
| | | 161.5 |
| | | |
| | | [[11/10]] |
| | | |
| | |- |
| | | 15 |
| | | 173.1 |
| | | [[21/19]] |
| | | |
| | | ''[[10/9]]'', ''[[11/10]]'' |
| | |- |
| | | 16 |
| | | 184.6 |
| | | |
| | | [[10/9]] |
| | | |
| | |- |
| | | 17 |
| | | 196.2 |
| | | [[19/17]], [[28/25]] |
| | | |
| | | |
| | |- |
| | | 18 |
| | | 207.7 |
| | | [[9/8]] |
| | | ''[[17/15]]'' |
| | | |
| | |- |
| | | 19 |
| | | 219.2 |
| | | [[25/22]] |
| | | |
| | | [[17/15]] |
| | |- |
| | | 20 |
| | | 230.8 |
| | | [[8/7]] |
| | | |
| | | |
| | |- |
| | | 21 |
| | | 242.3 |
| | | [[38/33]] |
| | | |
| | | [[15/13]] |
| | |- |
| | | 22 |
| | | 253.8 |
| | | [[22/19]] |
| | | ''[[15/13]]'' |
| | | |
| | |- |
| | | 23 |
| | | 265.4 |
| | | [[7/6]] |
| | | |
| | | |
| | |- |
| | | 24 |
| | | 276.9 |
| | | [[75/64]] |
| | | |
| | | [[20/17]] |
| | |- |
| | | 25 |
| | | 288.5 |
| | | [[13/11]], [[32/27]] |
| | | ''[[20/17]]'' |
| | | |
| | |- |
| | | 26 |
| | | 300.0 |
| | | [[19/16]], [[25/21]] |
| | | |
| | | |
| | |- |
| | | 27 |
| | | 311.5 |
| | | |
| | | [[6/5]] |
| | | |
| | |- |
| | | 28 |
| | | 323.1 |
| | | |
| | | |
| | | ''[[6/5]]'', ''[[40/33]]'' |
| | |- |
| | | 29 |
| | | 334.6 |
| | | [[17/14]] |
| | | [[40/33]] |
| | | |
| | |- |
| | | 30 |
| | | 346.2 |
| | | [[11/9]], [[39/32]] |
| | | |
| | | |
| | |- |
| | | 31 |
| | | 357.7 |
| | | [[16/13]], [[27/22]] |
| | | |
| | | |
| | |- |
| | | 32 |
| | | 369.2 |
| | | [[21/17]], [[26/21]] |
| | | |
| | | |
| | |- |
| | | 33 |
| | | 380.8 |
| | | |
| | | |
| | | [[5/4]] |
| | |- |
| | | 34 |
| | | 392.3 |
| | | |
| | | ''[[5/4]]'' |
| | | |
| | |- |
| | | 35 |
| | | 403.8 |
| | | [[24/19]], [[63/50]] |
| | | [[19/15]] |
| | | |
| | |- |
| | | 36 |
| | | 415.4 |
| | | [[14/11]] |
| | | |
| | | ''[[19/15]]'' |
| | |- |
| | | 37 |
| | | 426.9 |
| | | [[32/25]] |
| | | |
| | | |
| | |- |
| | | 38 |
| | | 438.5 |
| | | [[9/7]] |
| | | |
| | | |
| | |- |
| | | 39 |
| | | 450.0 |
| | | [[22/17]] |
| | | [[13/10]] |
| | | |
| | |- |
| | | 40 |
| | | 461.5 |
| | | [[17/13]] |
| | | |
| | | ''[[13/10]]'' |
| | |- |
| | | 41 |
| | | 473.1 |
| | | [[21/16]] |
| | | |
| | | |
| | |- |
| | | 42 |
| | | 484.6 |
| | | |
| | | |
| | | |
| | |- |
| | | 43 |
| | | 496.2 |
| | | [[4/3]] |
| | | |
| | | |
| | |- |
| | | 44 |
| | | 507.7 |
| | | |
| | | |
| | | |
| | |- |
| | | 45 |
| | | 519.2 |
| | | |
| | | [[27/20]] |
| | | |
| | |- |
| | | 46 |
| | | 530.8 |
| | | [[19/14]] |
| | | |
| | | ''[[27/20]]'', ''[[15/11]]'' |
| | |- |
| | | 47 |
| | | 542.3 |
| | | [[26/19]] |
| | | [[15/11]] |
| | | |
| | |- |
| | | 48 |
| | | 553.8 |
| | | [[11/8]] |
| | | |
| | | |
| | |- |
| | | 49 |
| | | 565.4 |
| | | [[18/13]] |
| | | |
| | | |
| | |- |
| | | 50 |
| | | 576.9 |
| | | |
| | | [[7/5]] |
| | | |
| | |- |
| | | 51 |
| | | 588.5 |
| | | |
| | | |
| | | ''[[7/5]]'', [[45/32]] |
| | |- |
| | | 52 |
| | | 600.0 |
| | | [[17/12]], [[24/17]] |
| | | ''[[45/32]]'', ''[[64/45]]'' |
| | | |
| | |- |
| | | … |
| | | … |
| | | … |
| | | … |
| | | … |
| | |} |
| | |
| | [[Category:Apollo]] |
| | [[Category:Diaschismic]] |
| | [[Category:Magic]] |
| | [[Category:Necromancy]] |