41edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~guest~~|~~guest~~]] and made on <tt>2011-02-~~12 05~~:45:35 UTC</tt>.

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-02-22 15:50:28 UTC</tt>.

: The original revision id was <tt>~~201114834~~</tt>.

: The original revision id was <tt>204104438</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<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 //41 equal temperament//, often abbreviated 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.27 cents, an interval close in size to 64/63, the [[http://en.wikipedia.org/wiki/Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the [[http://en.wikipedia.org/wiki/Schismatic_temperament|Garibaldi temperament]] <ref>[http://x31eq.com/schismic.htm "Schismic Temperaments "], ''Intonation Information''.</ref> , the [[http://en.wikipedia.org/wiki/Schismatic_temperament|miracle temperament]], <ref>[http://x31eq.com/decimal_lattice.htm "Lattices with Decimal Notation"], ''Intonation Information''.</ref> the [[http://en.wikipedia.org/wiki/Magic_temperament|magic temperament]] and the valentine (41&26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of 12-ET, and is the seventh [[http://www.research.att.com/%7Enjas/sequences/A117538|Zeta integral tuning]] after 31. The latter has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp.

==Intervals==

|| degrees of 41edo || cents value || generator for ||

|| 0 || 0.00 || ||

|| 1 || 29.27 || ||

|| 2 || 58.54 || ||

|| 3 || 87.80 || 88cET (approx) ||

|| 4 || 117.07 || Miracle ||

|| 5 || 146.34 || Bohlen-Pierce (approx) ||

|| 6 || 175.61 || ||

|| 7 || 204.88 || ||

|| 8 || 234.15 || ||

|| 9 || 263.41 || ||

|| 10 || 292.68 || ||

|| 11 || 321.95 || ||

|| 12 || 351.22 || ||

|| 13 || 380.49 || ||

|| 14 || 409.76 || ||

|| 15 || 439.02 || ||

|| 16 || 468.29 || ||

|| 17 || 497.56 || Pythagorean ||

|| 18 || 526.83 || ||

|| 19 || 556.10 || ||

|| 20 || 585.37 || ||

|| 21 || 614.63 || ||

|| 22 || 643.90 || ||

|| 23 || 673.17 || ||

|| 24 || 702.44 || Pythagorean ||

|| 25 || 731.71 || ||

|| 26 || 760.98 || ||

|| 27 || 790.24 || ||

|| 28 || 819.51 || ||

|| 29 || 848.78 || ||

|| 30 || 878.05 || ||

|| 31 || 907.32 || ||

|| 32 || 936.59 || ||

|| 33 || 965.85 || ||

|| 34 || 995.12 || ||

|| 35 || 1024.39 || ||

|| 36 || 1053.66 || ||

|| 37 || 1082.93 || ||

|| 38 || 1112.20 || ||

|| 39 || 1141.46 || ||

|| 40 || 1170.73 || ||

==Harmonic Scale==

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<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>41edo</title></head><body>The 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.27 cents, an interval close in size to 64/63, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a>. 41-ET can be seen as a tuning of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">Garibaldi temperament</a> <a href="#cite_note-1">[1]</a> , the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">miracle temperament</a>, <a href="#cite_note-2">[2]</a> the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Magic_temperament" rel="nofollow">magic temperament</a> and the valentine (41&amp;26) temperament. It is the second smallest equal temperament (after <a class="wiki_link" href="/29edo">29edo</a>) whose perfect fifth is closer to just intonation than that of 12-ET, and is the seventh <a class="wiki_link_ext" href="http://www.research.att.com/%7Enjas/sequences/A117538" rel="nofollow">Zeta integral tuning</a> after 31. The latter has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp.

<h2 id="toc0"><a name="x-Harmonic Scale"></a>Harmonic Scale</h2>

<h2 id="toc0"><a name="x-Intervals"></a>Intervals</h2>

<tr>

<td>degrees of 41edo

</td>

<td>cents value

</td>

<td>generator for

</td>

</tr>

<tr>

<td>0

</td>

<td>0.00

</td>

<td>

</td>

</tr>

<tr>

<td>1

</td>

<td>29.27

</td>

<td>

</td>

</tr>

<tr>

<td>2

</td>

<td>58.54

</td>

<td>

</td>

</tr>

<tr>

<td>3

</td>

<td>87.80

</td>

<td>88cET (approx)

</td>

</tr>

<tr>

<td>4

</td>

<td>117.07

</td>

<td>Miracle

</td>

</tr>

<tr>

<td>5

</td>

<td>146.34

</td>

<td>Bohlen-Pierce (approx)

</td>

</tr>

<tr>

<td>6

</td>

<td>175.61

</td>

<td>

</td>

</tr>

<tr>

<td>7

</td>

<td>204.88

</td>

<td>

</td>

</tr>

<tr>

<td>8

</td>

<td>234.15

</td>

<td>

</td>

</tr>

<tr>

<td>9

</td>

<td>263.41

</td>

<td>

</td>

</tr>

<tr>

<td>10

</td>

<td>292.68

</td>

<td>

</td>

</tr>

<tr>

<td>11

</td>

<td>321.95

</td>

<td>

</td>

</tr>

<tr>

<td>12

</td>

<td>351.22

</td>

<td>

</td>

</tr>

<tr>

<td>13

</td>

<td>380.49

</td>

<td>

</td>

</tr>

<tr>

<td>14

</td>

<td>409.76

</td>

<td>

</td>

</tr>

<tr>

<td>15

</td>

<td>439.02

</td>

<td>

</td>

</tr>

<tr>

<td>16

</td>

<td>468.29

</td>

<td>

</td>

</tr>

<tr>

<td>17

</td>

<td>497.56

</td>

<td>Pythagorean

</td>

</tr>

<tr>

<td>18

</td>

<td>526.83

</td>

<td>

</td>

</tr>

<tr>

<td>19

</td>

<td>556.10

</td>

<td>

</td>

</tr>

<tr>

<td>20

</td>

<td>585.37

</td>

<td>

</td>

</tr>

<tr>

<td>21

</td>

<td>614.63

</td>

<td>

</td>

</tr>

<tr>

<td>22

</td>

<td>643.90

</td>

<td>

</td>

</tr>

<tr>

<td>23

</td>

<td>673.17

</td>

<td>

</td>

</tr>

<tr>

<td>24

</td>

<td>702.44

</td>

<td>Pythagorean

</td>

</tr>

<tr>

<td>25

</td>

<td>731.71

</td>

<td>

</td>

</tr>

<tr>

<td>26

</td>

<td>760.98

</td>

<td>

</td>

</tr>

<tr>

<td>27

</td>

<td>790.24

</td>

<td>

</td>

</tr>

<tr>

<td>28

</td>

<td>819.51

</td>

<td>

</td>

</tr>

<tr>

<td>29

</td>

<td>848.78

</td>

<td>

</td>

</tr>

<tr>

<td>30

</td>

<td>878.05

</td>

<td>

</td>

</tr>

<tr>

<td>31

</td>

<td>907.32

</td>

<td>

</td>

</tr>

<tr>

<td>32

</td>

<td>936.59

</td>

<td>

</td>

</tr>

<tr>

<td>33

</td>

<td>965.85

</td>

<td>

</td>

</tr>

<tr>

<td>34

</td>

<td>995.12

</td>

<td>

</td>

</tr>

<tr>

<td>35

</td>

<td>1024.39

</td>

<td>

</td>

</tr>

<tr>

<td>36

</td>

<td>1053.66

</td>

<td>

</td>

</tr>

<tr>

<td>37

</td>

<td>1082.93

</td>

<td>

</td>

</tr>

<tr>

<td>38

</td>

<td>1112.20

</td>

<td>

</td>

</tr>

<tr>

<td>39

</td>

<td>1141.46

</td>

<td>

</td>

</tr>

<tr>

<td>40

</td>

<td>1170.73

</td>

<td>

</td>

</tr>

</table>

<h2 id="toc1"><a name="x-Harmonic Scale"></a>Harmonic Scale</h2>

41edo is the first edo to do some justice to Mode 8 of the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>, which Dante Rosati calls the &quot;<a class="wiki_link" href="/overtone%20scales">Diatonic Harmonic Series Scale</a>,&quot; consisting of overtones 8 through 16 (sometimes made to repeat at the octave).

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The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.

<h2 id="~~toc1~~"><a name="x-Nonoctave Temperaments"></a>Nonoctave Temperaments</h2>

<h2 id="toc2"><a name="x-Nonoctave Temperaments"></a>Nonoctave Temperaments</h2>

Taking every third degree of 41edo produces a scale extremely close to <a class="wiki_link" href="/88cET">88cET</a> or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <a class="wiki_link" href="/BP">Bohlen-Pierce</a><a class="wiki_link" href="/BP"> Scale</a> (or the 13th root of 3). See chart:

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<h2 id="~~toc2~~"><a name="x-Links"></a>Links</h2>

<h2 id="toc3"><a name="x-Links"></a>Links</h2>

<ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow">Wikipedia article on 41edo</a></li><li><a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis">Magic22 as srutis</a> describes a possible use of 41edo for <a class="wiki_link" href="/indian">indian</a> music.</li><li>see also <a class="wiki_link" href="/Magic%20family">Magic family</a></li><li>Sword, Ron.<a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank"> &quot;Tetracontamonophonic Scales for Guitar&quot;</a></li></ul><hr class="references" /><ol class="references">

<ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow">Wikipedia article on 41edo</a></li><li><a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis">Magic22 as srutis</a> describes a possible use of 41edo for <a class="wiki_link" href="/indian">indian</a> music.</li><li>see also <a class="wiki_link" href="/Magic%20family">Magic family</a></li><li>Sword, Ron.<a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank"> &quot;Tetracontamonophonic Scales for Guitar&quot;</a></li></ul><hr class="references" /><ol class="references">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> [<a class="wiki_link_ext" href="http://x31eq.com/schismic.htm" rel="nofollow">http://x31eq.com/schismic.htm</a> &quot;Schismic Temperaments &quot;], ''Intonation Information''.</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> [<a class="wiki_link_ext" href="http://x31eq.com/decimal_lattice.htm" rel="nofollow">http://x31eq.com/decimal_lattice.htm</a> &quot;Lattices with Decimal Notation&quot;], ''Intonation Information''.</li>

</ol></body></html></pre></div>

</ol></body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2011-02-12 05:45:35 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-02-22 15:50:28 UTC</tt>.<br>
-: The original revision id was <tt>201114834</tt>.<br>
+: The original revision id was <tt>204104438</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 //41 equal temperament//, often abbreviated 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.27 cents, an interval close in size to 64/63, the [[http://en.wikipedia.org/wiki/Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the [[http://en.wikipedia.org/wiki/Schismatic_temperament|Garibaldi temperament]] &lt;ref&gt;[http://x31eq.com/schismic.htm "Schismic Temperaments "], ''Intonation Information''.&lt;/ref&gt; , the [[http://en.wikipedia.org/wiki/Schismatic_temperament|miracle temperament]], &lt;ref&gt;[http://x31eq.com/decimal_lattice.htm "Lattices with Decimal Notation"], ''Intonation Information''.&lt;/ref&gt; the [[http://en.wikipedia.org/wiki/Magic_temperament|magic temperament]] and the valentine (41&amp;26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of 12-ET, and is the seventh [[http://www.research.att.com/%7Enjas/sequences/A117538|Zeta integral tuning]] after 31. The latter has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp.
+==Intervals==
+|| degrees of 41edo || cents value || generator for ||
+|| 0 || 0.00 ||   ||
+|| 1 || 29.27 ||   ||
+|| 2 || 58.54 ||   ||
+|| 3 || 87.80 || 88cET (approx) ||
+|| 4 || 117.07 || Miracle ||
+|| 5 || 146.34 || Bohlen-Pierce (approx) ||
+|| 6 || 175.61 ||   ||
+|| 7 || 204.88 ||   ||
+|| 8 || 234.15 ||   ||
+|| 9 || 263.41 ||   ||
+|| 10 || 292.68 ||   ||
+|| 11 || 321.95 ||   ||
+|| 12 || 351.22 ||   ||
+|| 13 || 380.49 ||   ||
+|| 14 || 409.76 ||   ||
+|| 15 || 439.02 ||   ||
+|| 16 || 468.29 ||   ||
+|| 17 || 497.56 || Pythagorean ||
+|| 18 || 526.83 ||   ||
+|| 19 || 556.10 ||   ||
+|| 20 || 585.37 ||   ||
+|| 21 || 614.63 ||   ||
+|| 22 || 643.90 ||   ||
+|| 23 || 673.17 ||   ||
+|| 24 || 702.44 || Pythagorean ||
+|| 25 || 731.71 ||   ||
+|| 26 || 760.98 ||   ||
+|| 27 || 790.24 ||   ||
+|| 28 || 819.51 ||   ||
+|| 29 || 848.78 ||   ||
+|| 30 || 878.05 ||   ||
+|| 31 || 907.32 ||   ||
+|| 32 || 936.59 ||   ||
+|| 33 || 965.85 ||   ||
+|| 34 || 995.12 ||   ||
+|| 35 || 1024.39 ||   ||
+|| 36 || 1053.66 ||   ||
+|| 37 || 1082.93 ||   ||
+|| 38 || 1112.20 ||   ||
+|| 39 || 1141.46 ||   ||
+|| 40 || 1170.73 ||   ||
 ==Harmonic Scale==
@@ Line 100: / Line 145: @@
 <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;41edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;41 equal temperament&lt;/em&gt;, often abbreviated 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.27 cents, an interval close in size to 64/63, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow"&gt;septimal comma&lt;/a&gt;. 41-ET can be seen as a tuning of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow"&gt;Garibaldi temperament&lt;/a&gt; &lt;!-- ws:start:WikiTextRefRule:1:&amp;amp;lt;ref&amp;amp;gt;[http://x31eq.com/schismic.htm &amp;amp;quot;Schismic Temperaments &amp;amp;quot;], ''Intonation Information''.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-1" class="reference"&gt;&lt;a href="#cite_note-1"&gt;[1]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:1 --&gt; , the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow"&gt;miracle temperament&lt;/a&gt;, &lt;!-- ws:start:WikiTextRefRule:3:&amp;amp;lt;ref&amp;amp;gt;[http://x31eq.com/decimal_lattice.htm &amp;amp;quot;Lattices with Decimal Notation&amp;amp;quot;], ''Intonation Information''.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-2" class="reference"&gt;&lt;a href="#cite_note-2"&gt;[2]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:3 --&gt; the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Magic_temperament" rel="nofollow"&gt;magic temperament&lt;/a&gt; and the valentine (41&amp;amp;26) temperament. It is the second smallest equal temperament (after &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;) whose perfect fifth is closer to just intonation than that of 12-ET, and is the seventh &lt;a class="wiki_link_ext" href="http://www.research.att.com/%7Enjas/sequences/A117538" rel="nofollow"&gt;Zeta integral tuning&lt;/a&gt; after 31. The latter has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Harmonic Scale&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h2&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;degrees of 41edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;cents value&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;generator for&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0.00&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;29.27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;58.54&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;87.80&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;88cET (approx)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;117.07&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Miracle&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;146.34&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Bohlen-Pierce (approx)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;175.61&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Harmonic Scale&lt;/h2&gt;
 edo is the first edo to do some justice to Mode 8 of the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;, which Dante Rosati calls the &amp;quot;&lt;a class="wiki_link" href="/overtone%20scales"&gt;Diatonic Harmonic Series Scale&lt;/a&gt;,&amp;quot; consisting of overtones 8 through 16 (sometimes made to repeat at the octave).&lt;br /&gt;
 &lt;br /&gt;
@@ Line 228: / Line 617: @@
 The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Nonoctave Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Nonoctave Temperaments&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Nonoctave Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Nonoctave Temperaments&lt;/h2&gt;
   Taking every third degree of 41edo produces a scale extremely close to &lt;a class="wiki_link" href="/88cET"&gt;88cET&lt;/a&gt; or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered &lt;span class="wiki_link_new"&gt;&lt;a class="wiki_link" href="/BP"&gt;Bohlen-Pierce&lt;/a&gt;&lt;/span&gt;&lt;a class="wiki_link" href="/BP"&gt; Scale&lt;/a&gt; (or the 13th root of 3). See chart:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,182: / Line 1,571: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Links&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Links&lt;/h2&gt;
-  &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow"&gt;Wikipedia article on 41edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis"&gt;Magic22 as srutis&lt;/a&gt; describes a possible use of 41edo for &lt;a class="wiki_link" href="/indian"&gt;indian&lt;/a&gt; music.&lt;/li&gt;&lt;li&gt;see also &lt;a class="wiki_link" href="/Magic%20family"&gt;Magic family&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Sword, Ron.&lt;a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank"&gt; &amp;quot;Tetracontamonophonic Scales for Guitar&amp;quot;&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextReferencesRule:1577: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;
+  &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow"&gt;Wikipedia article on 41edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis"&gt;Magic22 as srutis&lt;/a&gt; describes a possible use of 41edo for &lt;a class="wiki_link" href="/indian"&gt;indian&lt;/a&gt; music.&lt;/li&gt;&lt;li&gt;see also &lt;a class="wiki_link" href="/Magic%20family"&gt;Magic family&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Sword, Ron.&lt;a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank"&gt; &amp;quot;Tetracontamonophonic Scales for Guitar&amp;quot;&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextReferencesRule:2045: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;
 &lt;li id="cite_note-1"&gt;&lt;a href="#cite_ref-1"&gt;^&lt;/a&gt; [&lt;a class="wiki_link_ext" href="http://x31eq.com/schismic.htm" rel="nofollow"&gt;http://x31eq.com/schismic.htm&lt;/a&gt; &amp;quot;Schismic Temperaments &amp;quot;], ''Intonation Information''.&lt;/li&gt;
 &lt;li id="cite_note-2"&gt;&lt;a href="#cite_ref-2"&gt;^&lt;/a&gt; [&lt;a class="wiki_link_ext" href="http://x31eq.com/decimal_lattice.htm" rel="nofollow"&gt;http://x31eq.com/decimal_lattice.htm&lt;/a&gt; &amp;quot;Lattices with Decimal Notation&amp;quot;], ''Intonation Information''.&lt;/li&gt;
-&lt;/ol&gt;&lt;!-- ws:end:WikiTextReferencesRule:1577 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;/ol&gt;&lt;!-- ws:end:WikiTextReferencesRule:2045 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>