41edo: Difference between revisions

Revision as of 16:09, 1 November 2010

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Original Wikitext content:

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.

==Harmonic Scale== 
41edo is the first edo to do some justice to Mode 8 of the [[OverToneSeries|harmonic series]], which Dante Rosati calls the "[[overtone scales|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).

|| Overtones in "Mode 8": || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 ||
|| ...as JI Ratio from 1/1: || 1/1 || 9/8 || 5/4 || 11/8 || 3/2 || 13/8 || 7/4 || 15/8 || 2/1 ||
|| ...in cents: || 0 || 203.9 || 386.3 || 551.3 || 702.0 || 840.5 || 968.8 || 1088.3 || 1200.0 ||
|| Nearest degree of 41edo: || 0 || 7 || 13 || 19 || 24 || 29 || 33 || 37 || 41 ||
|| ...in cents: || 0 || 204.9 || 380.5 || 556.1 || 702.4 || 848.8 || 965.9 || 1082.9 || 1200.0 ||

While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)

7\41 (7 degrees of 41edo) (203.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).

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

==Nonoctave Temperaments== 
Taking every third degree of 41edo produces a scale extremely close to [[88cET]] 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 <span class="wiki_link_new">[[BP|Bohlen-Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See chart:

||||||= 3 degrees of 41edo (near 88cET) ||= overlap ||||||= 5 degrees of 41edo (near BP) ||
||~ deg of 41edo ||~ deg of 88cET ||~ cents ||~ cents ||~ cents ||~ deg of BP ||~ deg of 41edo ||
||= 0 ||= 0 ||=   ||= 0 ||=   ||= 0 ||= 0 ||
||= 3 ||= 1 ||= 87.8 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 146.3 ||= 1 ||= 5 ||
||= 6 ||= 2 ||= 175.6 ||=   ||=   ||=   ||=   ||
||= 9 ||= 3 ||= 263.4 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 292.7 ||= 2 ||= 10 ||
||= 12 ||= 4 ||= 351.2 ||=   ||=   ||=   ||=   ||
||= 15 ||= 5 ||=   ||= 439.0 ||=   ||= 3 ||= 15 ||
||= 18 ||= 6 ||= 526.8 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 585.4 ||= 4 ||= 20 ||
||= 21 ||= 7 ||= 614.6 ||=   ||=   ||=   ||=   ||
||= 24 ||= 8 ||= 702.4 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 731.7 ||= 5 ||= 25 ||
||= 27 ||= 9 ||= 790.2 ||=   ||=   ||=   ||=   ||
||= 30 ||= 10 ||=   ||= 878.0 ||=   ||= 6 ||= 30 ||
||= 33 ||= 11 ||= 965.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1024.4 ||= 7 ||= 35 ||
||= 36 ||= 12 ||= 1053.7 ||=   ||=   ||=   ||=   ||
||= 39 ||= 13 ||= 1141.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1170.7 ||= 8 ||= 40 ||
||||||||||||||~ [ second octave ] ||
||= 1 ||= 14 ||= 29.2 ||=   ||=   ||=   ||=   ||
||= 4 ||= 15 ||=   ||= 117.1 ||=   ||= 9 ||= 4 ||
||= 7 ||= 16 ||= 204.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 263.4 ||= 10 ||= 9 ||
||= 10 ||= 17 ||= 292.7 ||=   ||=   ||=   ||=   ||
||= 13 ||= 18 ||= 380.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 409.8 ||= 11 ||= 14 ||
||= 16 ||= 19 ||= 468.3 ||=   ||=   ||=   ||=   ||
||= 19 ||= 20 ||=   ||= 556.1 ||=   ||= 12 ||= 19 ||
||= 22 ||= 21 ||= 643.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 702.4 ||= 13 ||= 24 ||
||= 25 ||= 22 ||= 731.7 ||=   ||=   ||=   ||=   ||
||= 28 ||= 23 ||= 819.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 848.8 ||= 14 ||= 29 ||
||= 31 ||= 24 ||= 907.3 ||=   ||=   ||=   ||=   ||
||= 34 ||= 25 ||=   ||= 995.1 ||=   ||= 15 ||= 34 ||
||= 37 ||= 26 ||= 1082.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1141.5 ||= 16 ||= 39 ||
||= 40 ||= 27 ||= 1170.7 ||=   ||=   ||=   ||=   ||
||||||||||||||~ [ third octave ] ||
||= 2 ||= 28 ||= 58.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 87.8 ||= 17 ||= 3 ||
||= 5 ||= 29 ||= 146.3 ||=   ||=   ||=   ||=   ||
||= 8 ||= 30 ||=   ||= 234.1 ||=   ||= 18 ||= 8 ||
||= 11 ||= 31 ||= 322.0 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 380.5 ||= 19 ||= 13 ||
||= 14 ||= 32 ||= 409.8 ||=   ||=   ||=   ||=   ||
||= 17 ||= 33 ||= 497.6 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 526.8 ||= 20 ||= 18 ||
||= 20 ||= 34 ||= 585.3 ||=   ||=   ||=   ||=   ||
||= 23 ||= 35 ||=   ||= 673.2 ||=   ||= 21 ||= 23 ||
||= 26 ||= 36 ||= 761.0 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 819.5 ||= 22 ||= 28 ||
||= 29 ||= 37 ||= 848.8 ||=   ||=   ||=   ||=   ||
||= 32 ||= 38 ||= 936.6 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 965.9 ||= 23 ||= 33 ||
||= 35 ||= 39 ||= 1024.4 ||=   ||=   ||=   ||=   ||
||= 38 ||= 40 ||=   ||= 1112.2 ||=   ||= 24 ||= 38 ||


==Links== 
* [[http://en.wikipedia.org/wiki/41_equal_temperament|Wikipedia article on 41edo]]
* [[Magic22 as srutis#magic22assrutis]] describes a possible use of 41edo for [[indian]] music.
* see also [[Magic family]]

Original HTML content:

<html><head><title>41edo</title></head><body>The <em>41 equal temperament</em>, 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> <!-- ws:start:WikiTextRefRule:1:&amp;lt;ref&amp;gt;[http://x31eq.com/schismic.htm &amp;quot;Schismic Temperaments &amp;quot;], ''Intonation Information''.&amp;lt;/ref&amp;gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:1 --> , the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">miracle temperament</a>, <!-- ws:start:WikiTextRefRule:3:&amp;lt;ref&amp;gt;[http://x31eq.com/decimal_lattice.htm &amp;quot;Lattices with Decimal Notation&amp;quot;], ''Intonation Information''.&amp;lt;/ref&amp;gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:3 --> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc0"><a name="x-Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->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).<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Overtones in &quot;Mode 8&quot;:<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
        <td>13<br />
</td>
        <td>14<br />
</td>
        <td>15<br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>...as JI Ratio from 1/1:<br />
</td>
        <td>1/1<br />
</td>
        <td>9/8<br />
</td>
        <td>5/4<br />
</td>
        <td>11/8<br />
</td>
        <td>3/2<br />
</td>
        <td>13/8<br />
</td>
        <td>7/4<br />
</td>
        <td>15/8<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>203.9<br />
</td>
        <td>386.3<br />
</td>
        <td>551.3<br />
</td>
        <td>702.0<br />
</td>
        <td>840.5<br />
</td>
        <td>968.8<br />
</td>
        <td>1088.3<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 41edo:<br />
</td>
        <td>0<br />
</td>
        <td>7<br />
</td>
        <td>13<br />
</td>
        <td>19<br />
</td>
        <td>24<br />
</td>
        <td>29<br />
</td>
        <td>33<br />
</td>
        <td>37<br />
</td>
        <td>41<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>204.9<br />
</td>
        <td>380.5<br />
</td>
        <td>556.1<br />
</td>
        <td>702.4<br />
</td>
        <td>848.8<br />
</td>
        <td>965.9<br />
</td>
        <td>1082.9<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
</table>

<br />
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)<br />
<br />
7\41 (7 degrees of 41edo) (203.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.<br />
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).<br />
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).<br />
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).<br />
<br />
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc1"><a name="x-Nonoctave Temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->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 <span class="wiki_link_new"><a class="wiki_link" href="/BP">Bohlen-Pierce</a></span><a class="wiki_link" href="/BP"> Scale</a> (or the 13th root of 3). See chart:<br />
<br />


<table class="wiki_table">
    <tr>
        <td colspan="3" style="text-align: center;">3 degrees of 41edo (near 88cET)<br />
</td>
        <td style="text-align: center;">overlap<br />
</td>
        <td colspan="3" style="text-align: center;">5 degrees of 41edo (near BP)<br />
</td>
    </tr>
    <tr>
        <th>deg of 41edo<br />
</th>
        <th>deg of 88cET<br />
</th>
        <th>cents<br />
</th>
        <th>cents<br />
</th>
        <th>cents<br />
</th>
        <th>deg of BP<br />
</th>
        <th>deg of 41edo<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">87.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">146.3<br />
</td>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">175.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">263.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">292.7<br />
</td>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">10<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">351.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">439.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">15<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">526.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">585.4<br />
</td>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">20<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">614.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">702.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">731.7<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">25<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">790.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">878.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">30<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">965.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1024.4<br />
</td>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">35<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">1053.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">1141.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1170.7<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">40<br />
</td>
    </tr>
    <tr>
        <th colspan="7">[ second octave ]<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">29.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">117.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">4<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">204.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">263.4<br />
</td>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">9<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">292.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">380.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">409.8<br />
</td>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">14<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">468.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">556.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">19<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">643.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">702.4<br />
</td>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">24<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">731.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">819.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">848.8<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">29<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">907.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">995.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">34<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">1082.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1141.5<br />
</td>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">39<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">1170.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <th colspan="7">[ third octave ]<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">58.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">87.8<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">3<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">146.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">234.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">8<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">322.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">380.5<br />
</td>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">13<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">409.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">497.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">526.8<br />
</td>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">18<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">585.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">673.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">23<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">761.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">819.5<br />
</td>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">28<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">848.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">936.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">965.9<br />
</td>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">33<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">1024.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1112.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">38<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc2"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:8 -->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></ul><!-- ws:start:WikiTextReferencesRule:1574: --><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><!-- ws:end:WikiTextReferencesRule:1574 --></body></html>

@@ 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-11-01 16:08:28 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-11-01 16:09:24 UTC</tt>.<br>
-: The original revision id was <tt>175449441</tt>.<br>
+: The original revision id was <tt>175449785</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>
@@ Line 29: / Line 29: @@
 Taking every third degree of 41edo produces a scale extremely close to [[88cET]] 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;[[BP|Bohlen-Pierce]]&lt;/span&gt;[[BP| Scale]] (or the 13th root of 3). See chart:
-|||||| 3 degrees of 41edo (near 88cET) || overlap |||||| 5 degrees of 41edo (near BP) ||
+||||||= 3 degrees of 41edo (near 88cET) ||= overlap ||||||= 5 degrees of 41edo (near BP) ||
 ||~ deg of 41edo ||~ deg of 88cET ||~ cents ||~ cents ||~ cents ||~ deg of BP ||~ deg of 41edo ||
 ||= 0 ||= 0 ||=   ||= 0 ||=   ||= 0 ||= 0 ||
@@ Line 79: / Line 79: @@
 ||=   ||=   ||=   ||=   ||= 380.5 ||= 19 ||= 13 ||
 ||= 14 ||= 32 ||= 409.8 ||=   ||=   ||=   ||=   ||
-|| 17 || 33 || 497.6 ||   ||   ||   ||   ||
+||= 17 ||= 33 ||= 497.6 ||=   ||=   ||=   ||=   ||
-||   ||   ||   ||   || 526.8 || 20 || 18 ||
+||=   ||=   ||=   ||=   ||= 526.8 ||= 20 ||= 18 ||
-|| 20 || 34 || 585.3 ||   ||   ||   ||   ||
+||= 20 ||= 34 ||= 585.3 ||=   ||=   ||=   ||=   ||
-|| 23 || 35 ||   || 673.2 ||   || 21 || 23 ||
+||= 23 ||= 35 ||=   ||= 673.2 ||=   ||= 21 ||= 23 ||
-|| 26 || 36 || 761.0 ||   ||   ||   ||   ||
+||= 26 ||= 36 ||= 761.0 ||=   ||=   ||=   ||=   ||
-||   ||   ||   ||   || 819.5 || 22 || 28 ||
+||=   ||=   ||=   ||=   ||= 819.5 ||= 22 ||= 28 ||
-|| 29 || 37 || 848.8 ||   ||   ||   ||   ||
+||= 29 ||= 37 ||= 848.8 ||=   ||=   ||=   ||=   ||
-|| 32 || 38 || 936.6 ||   ||   ||   ||   ||
+||= 32 ||= 38 ||= 936.6 ||=   ||=   ||=   ||=   ||
-||   ||   ||   ||   || 965.9 || 23 || 33 ||
+||=   ||=   ||=   ||=   ||= 965.9 ||= 23 ||= 33 ||
-|| 35 || 39 || 1024.4 ||   ||   ||   ||   ||
+||= 35 ||= 39 ||= 1024.4 ||=   ||=   ||=   ||=   ||
-|| 38 || 40 ||   || 1112.2 ||   || 24 || 38 ||
+||= 38 ||= 40 ||=   ||= 1112.2 ||=   ||= 24 ||= 38 ||
@@ Line 234: / Line 234: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td colspan="3"&gt;3 degrees of 41edo (near 88cET)&lt;br /&gt;
+         &lt;td colspan="3" style="text-align: center;"&gt;3 degrees of 41edo (near 88cET)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;overlap&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;overlap&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="3"&gt;5 degrees of 41edo (near BP)&lt;br /&gt;
+         &lt;td colspan="3" style="text-align: center;"&gt;5 degrees of 41edo (near BP)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,002: / Line 1,002: @@
      &lt;/tr&gt;
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-         &lt;td&gt;965.9&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;965.9&lt;br /&gt;
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-         &lt;td&gt;23&lt;br /&gt;
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-         &lt;td&gt;35&lt;br /&gt;
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-         &lt;td&gt;39&lt;br /&gt;
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-         &lt;td&gt;1024.4&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1024.4&lt;br /&gt;
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-         &lt;td&gt;&lt;br /&gt;
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+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
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      &lt;/tr&gt;
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-         &lt;td&gt;38&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;38&lt;br /&gt;
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-         &lt;td&gt;40&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;40&lt;br /&gt;
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-         &lt;td&gt;&lt;br /&gt;
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-         &lt;td&gt;1112.2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1112.2&lt;br /&gt;
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-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
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-         &lt;td&gt;24&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;24&lt;br /&gt;
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-         &lt;td&gt;38&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;38&lt;br /&gt;
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41edo: Difference between revisions

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