46edo
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[[toc|flat]] ---- =<span style="color: #300094; font-size: 111%;">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]]. 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 26.09 cents. =46edo srutis= [[Magic22 as srutis#shrutar22assrutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music. =Intervals= || degrees of 46edo || cents value || || 0 || 0.00 || || 1 || 26.09 || || 2 || 52.17 || || 3 || 78.26 || || 4 || 104.35 || || 5 || 130.43 || || 6 || 156.52 || || 7 || 182.61 || || 8 || 208.70 || || 9 || 234.78 || || 10 || 260.87 || || 11 || 286.96 || || 12 || 313.04 || || 13 || 339.13 || || 14 || 365.22 || || 15 || 391.30 || || 16 || 417.39 || || 17 || 443.48 || || 18 || 469.57 || || 19 || 495.65 || || 20 || 521.74 || || 21 || 547.83 || || 22 || 573.91 || || 23 || 600.00 || || 24 || 626.09 || || 25 || 652.17 || || 26 || 628.26 || || 27 || 704.35 || || 28 || 730.43 || || 29 || 756.52 || || 30 || 782.61 || || 31 || 808.70 || || 32 || 834.78 || || 33 || 860.87 || || 34 || 886.96 || || 35 || 913.04 || || 36 || 939.13 || || 37 || 965.22 || || 38 || 991.30 || || 39 || 1017.39 || || 40 || 1043.48 || || 41 || 1069.56 || || 42 || 1095.65 || || 43 || 1121.74 || || 44 || 1147.83 || || 45 || 1173.91 || =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= [[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:12:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#x46 tone equal temperament">46 tone equal temperament</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#x46edo srutis">46edo srutis</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Approximation to Mode 8 of the Harmonic Series">Approximation to Mode 8 of the Harmonic Series</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: -->
<!-- ws:end:WikiTextTocRule:19 --><hr />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x46 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #300094; font-size: 111%;">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>.<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 26.09 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><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:<h1> --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->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>26.09<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>52.17<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>78.26<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>104.35<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>130.43<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>156.52<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>182.61<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>208.70<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>234.78<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>260.87<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>286.96<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>313.04<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>339.13<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>365.22<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>391.30<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>417.39<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>443.48<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>469.57<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>495.65<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>521.74<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>547.83<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>573.91<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>600.00<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>626.09<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>652.17<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>628.26<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>704.35<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>730.43<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>756.52<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>782.61<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>808.70<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>834.78<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>860.87<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>886.96<br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>913.04<br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>939.13<br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>965.22<br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>991.30<br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>1017.39<br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>1043.48<br />
</td>
</tr>
<tr>
<td>41<br />
</td>
<td>1069.56<br />
</td>
</tr>
<tr>
<td>42<br />
</td>
<td>1095.65<br />
</td>
</tr>
<tr>
<td>43<br />
</td>
<td>1121.74<br />
</td>
</tr>
<tr>
<td>44<br />
</td>
<td>1147.83<br />
</td>
</tr>
<tr>
<td>45<br />
</td>
<td>1173.91<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Approximation to Mode 8 of the Harmonic Series"></a><!-- ws:end:WikiTextHeadingRule:6 -->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:8:<h1> --><h1 id="toc4"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:8 -->Scales</h1>
<a class="wiki_link" href="/sensi5">sensi5</a><br />
<a class="wiki_link" href="/sensi8">sensi8</a><br />
<a class="wiki_link" href="/sensi11">sensi11</a><br />
<a class="wiki_link" href="/sensi19">sensi19</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:10 -->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>