46edo

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[[toc|flat]]
----

=<span style="color: #300094; font-family: 'Times New Roman',Times,serif; font-size: 113%;">46 tone equal temperament</span>= 
The 46 equal temperament, often abbreviated 46-tET, 46-EDO, or 46-ET, is the scale derived by dividing the [[octave]] into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 [[cent]]s, an interval close in size to [[66_65|66/65]], the interval from [[13_11|13/11]] to [[6_5|6/5]].

46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with varied consequences it would take a very long article to describe. [[Rank two temperaments]] it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The [[11-limit]] [[Target tunings|minimax]] tuning for [[Starling family|valentine temperament]], (11/7)^(1/10), is only 0.01 cents flat of 3/46 octaves. In the opinion of some 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]]. In fact, while 41 is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]] but not a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.

The fifth of 46 equal is 2.39 cents sharp, which some people (eg, Margo Schulter) prefer, sometimes strongly, over both the [[just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.

46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.

=46edo srutis= 
[[Magic22 as srutis#shrutar22assrutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.

=Linear temperaments= 
|| Periods
per octave || Generator || Cents || Temperaments ||
|| 1 || 1\46 || 26.087 ||   ||
||   || 3\46 || 78.261 ||   ||
||   || 5\46 || 130.435 ||   ||
||   || 7 || 182.609 ||   ||
||   || 9 || 234.783 ||   ||
||   || 11 || 286.957 ||   ||
||   || 13 || 339.13 ||   ||
||   || 15 || 391.304 ||   ||
||   || 17 || 443.478 ||   ||
||   || 19 || 495.652 ||   ||
||   || 21 || 547.826 ||   ||
|| 2 || 1 || 26.087 ||   ||
||   || 2 || 52.174 ||   ||
||   || 3 || 78.261 ||   ||
||   || 4 || 104.348 ||   ||
||   || 5 || 130.435 ||   ||
||   || 6 || 156.522 ||   ||
||   || 7 || 182.609 ||   ||
||   || 8 || 208.696 ||   ||
||   || 9 || 234.783 ||   ||
||   || 10 || 260.87 ||   ||
||   || 11 || 286.957 ||   ||
|| 23 || 1 || 26.087 ||   ||

=Intervals= 
|| degrees of 46edo || cents value ||
|| 0 || 0.00 ||
|| 1 || 26.087 ||
|| 2 || 52.174 ||
|| 3 || 78.261 ||
|| 4 || 104.348 ||
|| 5 || 130.435 ||
|| 6 || 156.522 ||
|| 7 || 182.609 ||
|| 8 || 208.696 ||
|| 9 || 234.783 ||
|| 10 || 260.87 ||
|| 11 || 286.957 ||
|| 12 || 313.043 ||
|| 13 || 339.13 ||
|| 14 || 365.217 ||
|| 15 || 391.304 ||
|| 16 || 417.391 ||
|| 17 || 443.478 ||
|| 18 || 469.565 ||
|| 19 || 495.652 ||
|| 20 || 521.739 ||
|| 21 || 547.826 ||
|| 22 || 573.913 ||
|| 23 || 600 ||
|| 24 || 626.087 ||
|| 25 || 652.174 ||
|| 26 || 628.261 ||
|| 27 || 704.348 ||
|| 28 || 730.435 ||
|| 29 || 756.522 ||
|| 30 || 782.609 ||
|| 31 || 808.696 ||
|| 32 || 834.783 ||
|| 33 || 860.87 ||
|| 34 || 886.957 ||
|| 35 || 913.043 ||
|| 36 || 939.13 ||
|| 37 || 965.217 ||
|| 38 || 991.304 ||
|| 39 || 1017.391 ||
|| 40 || 1043.478 ||
|| 41 || 1069.565 ||
|| 42 || 1095.652 ||
|| 43 || 1121.739 ||
|| 44 || 1147.826 ||
|| 45 || 1173.913 ||

=Approximation to Mode 8 of the Harmonic Series= 

46edo represents [[overtone]]s 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.

8\46edo (208.70¢) stands in for frequency ratio [[9_8|9:8]] (203.91¢).
7\46edo (182.61¢) stands in for [[10_9|10:9]] (182.40¢).
6\46edo (156.52¢) stands in for [[11_10|11:10]] (165.00¢) and [[12_11|12:11]] (150.64¢).
5\46edo (130.43¢) stands in for [[13_12|13:12]] (138.57¢), [[14_13|14:13]] (128.30¢) and [[15_14|15:14]] (119.44¢).
4\46edo (104.35¢) stands in for [[16_15|16:15]] (111.73¢).

=Scales= 
* [[plum]]
* [[sensi5]]
* [[sensi8]]
* [[sensi11]]
* [[sensi19]]

=Music= 
by [[Gene Ward Smith]]
[[http://www.archive.org/details/Chromosounds|Chromosounds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3|play]]
[[http://www.archive.org/details/MusicForYourEars|Music For Your Ears]] [[http://www.archive.org/download/MusicForYourEars/musicfor.mp3|play]] The central portion is in [[27edo]], the rest in 46edo.

Original HTML content:

<html><head><title>46edo</title></head><body><!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#x46 tone equal temperament">46 tone equal temperament</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#x46edo srutis">46edo srutis</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Approximation to Mode 8 of the Harmonic Series">Approximation to Mode 8 of the Harmonic Series</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
<!-- ws:end:WikiTextTocRule:22 --><hr />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x46 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #300094; font-family: 'Times New Roman',Times,serif; font-size: 113%;">46 tone equal temperament</span></h1>
 The 46 equal temperament, often abbreviated 46-tET, 46-EDO, or 46-ET, is the scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 <a class="wiki_link" href="/cent">cent</a>s, an interval close in size to <a class="wiki_link" href="/66_65">66/65</a>, the interval from <a class="wiki_link" href="/13_11">13/11</a> to <a class="wiki_link" href="/6_5">6/5</a>.<br />
<br />
46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with varied consequences it would take a very long article to describe. <a class="wiki_link" href="/Rank%20two%20temperaments">Rank two temperaments</a> it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The <a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/Target%20tunings">minimax</a> tuning for <a class="wiki_link" href="/Starling%20family">valentine temperament</a>, (11/7)^(1/10), is only 0.01 cents flat of 3/46 octaves. In the opinion of some 46et is the first equal division to deal adequately with the <a class="wiki_link" href="/13-limit">13-limit</a>, though others award that distinction to <a class="wiki_link" href="/41edo">41edo</a>. In fact, while 41 is a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta integral edo</a> but not a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta gap edo</a>, 46 is zeta gap but not zeta integral.<br />
<br />
The fifth of 46 equal is 2.39 cents sharp, which some people (eg, Margo Schulter) prefer, sometimes strongly, over both the <a class="wiki_link" href="/just%20fifth">just fifth</a> and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.<br />
<br />
46edo can be treated as two <a class="wiki_link" href="/23edo">23edo</a>'s separated by an interval of 26.087 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x46edo srutis"></a><!-- ws:end:WikiTextHeadingRule:2 -->46edo srutis</h1>
 <a class="wiki_link" href="/Magic22%20as%20srutis#shrutar22assrutis">Shrutar22 as srutis</a> describes a possible use of 46edo for <a class="wiki_link" href="/Indian">Indian</a> music.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Linear temperaments</h1>
 

<table class="wiki_table">
    <tr>
        <td>Periods<br />
per octave<br />
</td>
        <td>Generator<br />
</td>
        <td>Cents<br />
</td>
        <td>Temperaments<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\46<br />
</td>
        <td>26.087<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\46<br />
</td>
        <td>78.261<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5\46<br />
</td>
        <td>130.435<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7<br />
</td>
        <td>182.609<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>9<br />
</td>
        <td>234.783<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>11<br />
</td>
        <td>286.957<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>13<br />
</td>
        <td>339.13<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>15<br />
</td>
        <td>391.304<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>17<br />
</td>
        <td>443.478<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>19<br />
</td>
        <td>495.652<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>21<br />
</td>
        <td>547.826<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>1<br />
</td>
        <td>26.087<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>2<br />
</td>
        <td>52.174<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3<br />
</td>
        <td>78.261<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>4<br />
</td>
        <td>104.348<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5<br />
</td>
        <td>130.435<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>6<br />
</td>
        <td>156.522<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7<br />
</td>
        <td>182.609<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>8<br />
</td>
        <td>208.696<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>9<br />
</td>
        <td>234.783<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>10<br />
</td>
        <td>260.87<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>11<br />
</td>
        <td>286.957<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>1<br />
</td>
        <td>26.087<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:6 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>degrees of 46edo<br />
</td>
        <td>cents value<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>26.087<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>52.174<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>78.261<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>104.348<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>130.435<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>156.522<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>182.609<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>208.696<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>234.783<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>260.87<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>286.957<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>313.043<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>339.13<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>365.217<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>391.304<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>417.391<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>443.478<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>469.565<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>495.652<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>521.739<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>547.826<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>573.913<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>600<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>626.087<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>652.174<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>628.261<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>704.348<br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>730.435<br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>756.522<br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>782.609<br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>808.696<br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>834.783<br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>860.87<br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>886.957<br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>913.043<br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>939.13<br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>965.217<br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>991.304<br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>1017.391<br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>1043.478<br />
</td>
    </tr>
    <tr>
        <td>41<br />
</td>
        <td>1069.565<br />
</td>
    </tr>
    <tr>
        <td>42<br />
</td>
        <td>1095.652<br />
</td>
    </tr>
    <tr>
        <td>43<br />
</td>
        <td>1121.739<br />
</td>
    </tr>
    <tr>
        <td>44<br />
</td>
        <td>1147.826<br />
</td>
    </tr>
    <tr>
        <td>45<br />
</td>
        <td>1173.913<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Approximation to Mode 8 of the Harmonic Series"></a><!-- ws:end:WikiTextHeadingRule:8 -->Approximation to Mode 8 of the Harmonic Series</h1>
 <br />
46edo represents <a class="wiki_link" href="/overtone">overtone</a>s 8 through 16 (written as <a class="wiki_link" href="/JI">JI</a> ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.<br />
<br />
8\46edo (208.70¢) stands in for frequency ratio <a class="wiki_link" href="/9_8">9:8</a> (203.91¢).<br />
7\46edo (182.61¢) stands in for <a class="wiki_link" href="/10_9">10:9</a> (182.40¢).<br />
6\46edo (156.52¢) stands in for <a class="wiki_link" href="/11_10">11:10</a> (165.00¢) and <a class="wiki_link" href="/12_11">12:11</a> (150.64¢).<br />
5\46edo (130.43¢) stands in for <a class="wiki_link" href="/13_12">13:12</a> (138.57¢), <a class="wiki_link" href="/14_13">14:13</a> (128.30¢) and <a class="wiki_link" href="/15_14">15:14</a> (119.44¢).<br />
4\46edo (104.35¢) stands in for <a class="wiki_link" href="/16_15">16:15</a> (111.73¢).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->Scales</h1>
 <ul><li><a class="wiki_link" href="/plum">plum</a></li><li><a class="wiki_link" href="/sensi5">sensi5</a></li><li><a class="wiki_link" href="/sensi8">sensi8</a></li><li><a class="wiki_link" href="/sensi11">sensi11</a></li><li><a class="wiki_link" href="/sensi19">sensi19</a></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->Music</h1>
 by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a><br />
<a class="wiki_link_ext" href="http://www.archive.org/details/Chromosounds" rel="nofollow">Chromosounds</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3" rel="nofollow">play</a><br />
<a class="wiki_link_ext" href="http://www.archive.org/details/MusicForYourEars" rel="nofollow">Music For Your Ears</a> <a class="wiki_link_ext" href="http://www.archive.org/download/MusicForYourEars/musicfor.mp3" rel="nofollow">play</a> The central portion is in <a class="wiki_link" href="/27edo">27edo</a>, the rest in 46edo.</body></html>