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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | The ''59 equal division'' divides the octave into 59 equal steps of 20.339 cents each. Its best fifth is very (9.9 cents) sharp, and yet its [[major_third|major third]] is nearly pure. It is a good [[Porcupine_family|porcupine]] tuning, giving in fact the [[Optimal_patent_val|optimal patent val]] for [[11-limit|11-limit]] porcupine. This patent val tempers out 250/243 in the [[5-limit|5-limit]], 64/63 and 16875/16807 in the [[7-limit|7-limit]], and 55/54, 100/99 and 176/175 in the [[11-limit|11-limit]]. 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N_subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas as 118et. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&59 temperament with a subminor third generator provides an interesting temperament. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-01 03:40:06 UTC</tt>.<br>
| |
| : The original revision id was <tt>341632678</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">The //59 equal division// divides the octave into 59 equal steps of 20.339 cents each. Its best fifth is very (9.9 cents) sharp, and yet its [[major third]] is nearly pure. It is a good [[Porcupine family|porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out 250/243 in the [[5-limit]], 64/63 and 16875/16807 in the [[7-limit]], and 55/54, 100/99 and 176/175 in the [[11-limit]]. 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas as 118et. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&59 temperament with a subminor third generator provides an interesting temperament.
| |
|
| |
|
| Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to [[Schismatic family|garibaldi temperament]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. | | Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to [[Schismatic_family|garibaldi temperament]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. |
|
| |
|
| 59edo is the 17th [[prime numbers|prime]] edo. | | 59edo is the 17th [[prime_numbers|prime]] edo. |
|
| |
|
| || Degrees || Cents Value || | | {| class="wikitable" |
| || 1 || 20.339 || | | |- |
| || 2 || 40.678 || | | | | Degrees |
| || 3 || 61.017 || | | | | Cents Value |
| || 4 || 81.356 || | | |- |
| || 5 || 101.695 || | | | | 1 |
| || 6 || 122.034 || | | | | 20.339 |
| || 7 || 142.373 || | | |- |
| || 8 || 162.712 || | | | | 2 |
| || 9 || 183.051 || | | | | 40.678 |
| || 10 || 203.390 || | | |- |
| || 11 || 223.729 || | | | | 3 |
| || 12 || 244.068 || | | | | 61.017 |
| || 13 || 264.407 || | | |- |
| || 14 || 284.746 || | | | | 4 |
| || 15 || 305.085 || | | | | 81.356 |
| || 16 || 325.424 || | | |- |
| || 17 || 345.763 || | | | | 5 |
| || 18 || 366.102 || | | | | 101.695 |
| || 19 || 386.441 || | | |- |
| || 20 || 406.780 || | | | | 6 |
| || 21 || 427.119 || | | | | 122.034 |
| || 22 || 447.458 || | | |- |
| || 23 || 467.797 || | | | | 7 |
| || 24 || 488.136 || | | | | 142.373 |
| || 25 || 508.475 || | | |- |
| || 26 || 528.814 || | | | | 8 |
| || 27 || 549.153 || | | | | 162.712 |
| || 28 || 569.492 || | | |- |
| || 29 || 589.831 || | | | | 9 |
| || 30 || 610.169 || | | | | 183.051 |
| || 31 || 630.508 || | | |- |
| || 32 || 650.847 || | | | | 10 |
| || 33 || 671.186 || | | | | 203.390 |
| || 34 || 691.525 || | | |- |
| || 35 || 711.864 || | | | | 11 |
| || 36 || 732.203 || | | | | 223.729 |
| || 37 || 752.542 || | | |- |
| || 38 || 772.881 || | | | | 12 |
| || 39 || 793.220 || | | | | 244.068 |
| || 40 || 813.559 || | | |- |
| || 41 || 833.898 || | | | | 13 |
| || 42 || 854.237 || | | | | 264.407 |
| || 43 || 874.576 || | | |- |
| || 44 || 894.915 || | | | | 14 |
| || 45 || 915.254 || | | | | 284.746 |
| || 46 || 935.593 || | | |- |
| || 47 || 955.932 || | | | | 15 |
| || 48 || 976.271 || | | | | 305.085 |
| || 49 || 996.610 || | | |- |
| || 50 || 1016.949 || | | | | 16 |
| || 51 || 1037.288 || | | | | 325.424 |
| || 52 || 1057.627 || | | |- |
| || 53 || 1077.966 || | | | | 17 |
| || 54 || 1098.305 || | | | | 345.763 |
| || 55 || 1118.644 || | | |- |
| || 56 || 1138.983 || | | | | 18 |
| || 57 || 1159.322 || | | | | 366.102 |
| || 58 || 1179.661 ||</pre></div> | | |- |
| <h4>Original HTML content:</h4>
| | | | 19 |
| <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>59edo</title></head><body>The <em>59 equal division</em> divides the octave into 59 equal steps of 20.339 cents each. Its best fifth is very (9.9 cents) sharp, and yet its <a class="wiki_link" href="/major%20third">major third</a> is nearly pure. It is a good <a class="wiki_link" href="/Porcupine%20family">porcupine</a> tuning, giving in fact the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for <a class="wiki_link" href="/11-limit">11-limit</a> porcupine. This patent val tempers out 250/243 in the <a class="wiki_link" href="/5-limit">5-limit</a>, 64/63 and 16875/16807 in the <a class="wiki_link" href="/7-limit">7-limit</a>, and 55/54, 100/99 and 176/175 in the <a class="wiki_link" href="/11-limit">11-limit</a>. 59edo is an excellent tuning for the 2.9.5.21.11 11-limit <a class="wiki_link" href="/k%2AN%20subgroups">2*59 subgroup</a>, on which it takes the same tuning and tempers out the same commas as 118et. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&amp;59 temperament with a subminor third generator provides an interesting temperament.<br />
| | | | 386.441 |
| <br />
| | |- |
| Using the flat fifth instead of the sharp one allows for the 12&amp;35 temperament, which is a kind of bizarre cousin to <a class="wiki_link" href="/Schismatic%20family">garibaldi temperament</a> with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth.<br />
| | | | 20 |
| <br />
| | | | 406.780 |
| 59edo is the 17th <a class="wiki_link" href="/prime%20numbers">prime</a> edo.<br />
| | |- |
| <br />
| | | | 21 |
| | | | | 427.119 |
| | | |- |
| <table class="wiki_table">
| | | | 22 |
| <tr>
| | | | 447.458 |
| <td>Degrees<br />
| | |- |
| </td>
| | | | 23 |
| <td>Cents Value<br />
| | | | 467.797 |
| </td>
| | |- |
| </tr>
| | | | 24 |
| <tr>
| | | | 488.136 |
| <td>1<br />
| | |- |
| </td>
| | | | 25 |
| <td>20.339<br />
| | | | 508.475 |
| </td>
| | |- |
| </tr>
| | | | 26 |
| <tr>
| | | | 528.814 |
| <td>2<br />
| | |- |
| </td>
| | | | 27 |
| <td>40.678<br />
| | | | 549.153 |
| </td>
| | |- |
| </tr>
| | | | 28 |
| <tr>
| | | | 569.492 |
| <td>3<br />
| | |- |
| </td>
| | | | 29 |
| <td>61.017<br />
| | | | 589.831 |
| </td>
| | |- |
| </tr>
| | | | 30 |
| <tr>
| | | | 610.169 |
| <td>4<br />
| | |- |
| </td>
| | | | 31 |
| <td>81.356<br />
| | | | 630.508 |
| </td>
| | |- |
| </tr>
| | | | 32 |
| <tr>
| | | | 650.847 |
| <td>5<br />
| | |- |
| </td>
| | | | 33 |
| <td>101.695<br />
| | | | 671.186 |
| </td>
| | |- |
| </tr>
| | | | 34 |
| <tr>
| | | | 691.525 |
| <td>6<br />
| | |- |
| </td>
| | | | 35 |
| <td>122.034<br />
| | | | 711.864 |
| </td>
| | |- |
| </tr>
| | | | 36 |
| <tr>
| | | | 732.203 |
| <td>7<br />
| | |- |
| </td>
| | | | 37 |
| <td>142.373<br />
| | | | 752.542 |
| </td>
| | |- |
| </tr>
| | | | 38 |
| <tr>
| | | | 772.881 |
| <td>8<br />
| | |- |
| </td>
| | | | 39 |
| <td>162.712<br />
| | | | 793.220 |
| </td>
| | |- |
| </tr>
| | | | 40 |
| <tr>
| | | | 813.559 |
| <td>9<br />
| | |- |
| </td>
| | | | 41 |
| <td>183.051<br />
| | | | 833.898 |
| </td>
| | |- |
| </tr>
| | | | 42 |
| <tr>
| | | | 854.237 |
| <td>10<br />
| | |- |
| </td>
| | | | 43 |
| <td>203.390<br />
| | | | 874.576 |
| </td>
| | |- |
| </tr>
| | | | 44 |
| <tr>
| | | | 894.915 |
| <td>11<br />
| | |- |
| </td>
| | | | 45 |
| <td>223.729<br />
| | | | 915.254 |
| </td>
| | |- |
| </tr>
| | | | 46 |
| <tr>
| | | | 935.593 |
| <td>12<br />
| | |- |
| </td>
| | | | 47 |
| <td>244.068<br />
| | | | 955.932 |
| </td>
| | |- |
| </tr>
| | | | 48 |
| <tr>
| | | | 976.271 |
| <td>13<br />
| | |- |
| </td>
| | | | 49 |
| <td>264.407<br />
| | | | 996.610 |
| </td>
| | |- |
| </tr>
| | | | 50 |
| <tr>
| | | | 1016.949 |
| <td>14<br />
| | |- |
| </td>
| | | | 51 |
| <td>284.746<br />
| | | | 1037.288 |
| </td>
| | |- |
| </tr>
| | | | 52 |
| <tr>
| | | | 1057.627 |
| <td>15<br />
| | |- |
| </td>
| | | | 53 |
| <td>305.085<br />
| | | | 1077.966 |
| </td>
| | |- |
| </tr>
| | | | 54 |
| <tr>
| | | | 1098.305 |
| <td>16<br />
| | |- |
| </td>
| | | | 55 |
| <td>325.424<br />
| | | | 1118.644 |
| </td>
| | |- |
| </tr>
| | | | 56 |
| <tr>
| | | | 1138.983 |
| <td>17<br />
| | |- |
| </td>
| | | | 57 |
| <td>345.763<br />
| | | | 1159.322 |
| </td>
| | |- |
| </tr>
| | | | 58 |
| <tr>
| | | | 1179.661 |
| <td>18<br />
| | |} |
| </td>
| | [[Category:edo]] |
| <td>366.102<br />
| | [[Category:porcupine]] |
| </td>
| | [[Category:prime_edo]] |
| </tr>
| | [[Category:subgroup]] |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>386.441<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>406.780<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>427.119<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>447.458<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>467.797<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>488.136<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>508.475<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>528.814<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>549.153<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>569.492<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>589.831<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>610.169<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
| |
| <td>630.508<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32<br />
| |
| </td>
| |
| <td>650.847<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>33<br />
| |
| </td>
| |
| <td>671.186<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>34<br />
| |
| </td>
| |
| <td>691.525<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>35<br />
| |
| </td>
| |
| <td>711.864<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>36<br />
| |
| </td>
| |
| <td>732.203<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>37<br />
| |
| </td>
| |
| <td>752.542<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>38<br />
| |
| </td>
| |
| <td>772.881<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>39<br />
| |
| </td>
| |
| <td>793.220<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>40<br />
| |
| </td>
| |
| <td>813.559<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>41<br />
| |
| </td>
| |
| <td>833.898<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>42<br />
| |
| </td>
| |
| <td>854.237<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>43<br />
| |
| </td>
| |
| <td>874.576<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>44<br />
| |
| </td>
| |
| <td>894.915<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>45<br />
| |
| </td>
| |
| <td>915.254<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>46<br />
| |
| </td>
| |
| <td>935.593<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>47<br />
| |
| </td>
| |
| <td>955.932<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>48<br />
| |
| </td>
| |
| <td>976.271<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>49<br />
| |
| </td>
| |
| <td>996.610<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>50<br />
| |
| </td>
| |
| <td>1016.949<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>51<br />
| |
| </td>
| |
| <td>1037.288<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>52<br />
| |
| </td>
| |
| <td>1057.627<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>53<br />
| |
| </td>
| |
| <td>1077.966<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>54<br />
| |
| </td>
| |
| <td>1098.305<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>55<br />
| |
| </td>
| |
| <td>1118.644<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>56<br />
| |
| </td>
| |
| <td>1138.983<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>57<br />
| |
| </td>
| |
| <td>1159.322<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>58<br />
| |
| </td>
| |
| <td>1179.661<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
| | |
| </body></html></pre></div>
| |
The 59 equal division divides the octave into 59 equal steps of 20.339 cents each. Its best fifth is very (9.9 cents) sharp, and yet its major third is nearly pure. It is a good porcupine tuning, giving in fact the optimal patent val for 11-limit porcupine. This patent val tempers out 250/243 in the 5-limit, 64/63 and 16875/16807 in the 7-limit, and 55/54, 100/99 and 176/175 in the 11-limit. 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas as 118et. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&59 temperament with a subminor third generator provides an interesting temperament.
Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to garibaldi temperament with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth.
59edo is the 17th prime edo.
| Degrees
|
Cents Value
|
| 1
|
20.339
|
| 2
|
40.678
|
| 3
|
61.017
|
| 4
|
81.356
|
| 5
|
101.695
|
| 6
|
122.034
|
| 7
|
142.373
|
| 8
|
162.712
|
| 9
|
183.051
|
| 10
|
203.390
|
| 11
|
223.729
|
| 12
|
244.068
|
| 13
|
264.407
|
| 14
|
284.746
|
| 15
|
305.085
|
| 16
|
325.424
|
| 17
|
345.763
|
| 18
|
366.102
|
| 19
|
386.441
|
| 20
|
406.780
|
| 21
|
427.119
|
| 22
|
447.458
|
| 23
|
467.797
|
| 24
|
488.136
|
| 25
|
508.475
|
| 26
|
528.814
|
| 27
|
549.153
|
| 28
|
569.492
|
| 29
|
589.831
|
| 30
|
610.169
|
| 31
|
630.508
|
| 32
|
650.847
|
| 33
|
671.186
|
| 34
|
691.525
|
| 35
|
711.864
|
| 36
|
732.203
|
| 37
|
752.542
|
| 38
|
772.881
|
| 39
|
793.220
|
| 40
|
813.559
|
| 41
|
833.898
|
| 42
|
854.237
|
| 43
|
874.576
|
| 44
|
894.915
|
| 45
|
915.254
|
| 46
|
935.593
|
| 47
|
955.932
|
| 48
|
976.271
|
| 49
|
996.610
|
| 50
|
1016.949
|
| 51
|
1037.288
|
| 52
|
1057.627
|
| 53
|
1077.966
|
| 54
|
1098.305
|
| 55
|
1118.644
|
| 56
|
1138.983
|
| 57
|
1159.322
|
| 58
|
1179.661
|