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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&amp;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&amp;59 temperament with a subminor third generator provides an interesting temperament.


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