47edo

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This revision was by author TallKite and made on 2017-09-05 03:57:05 UTC.
The original revision id was 617184721.
The revision comment was:

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Original Wikitext content:

=<span style="color: #616161; font-size: 99%;">47 tone Equal Temperament</span>= 

**//47-EDO//** divides the octave into 47 equal parts of 25.5319 cents each. It has a fifth which is 12.5933 cents flat, unless you use the alternative fifth which is 12.9386 cents sharp, similar to 35edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third of a cent sharp. It does a good job of approximating the 2.9.5.7.33.13.17.57.69 23-limit [[k*N subgroups|2*47 subgroup]] of the [[23-limit]], on which it tempers out the same commas as [[94edo]]. It provides a good tuning for [[Chromatic pairs#Baldy|baldy]] and [[Chromatic pairs#Silver|silver]] temperaments and relatives.

47edo is the 15th [[prime numbers|prime]] edo, following [[43edo]] and preceding [[53edo]].

47edo is a regular edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. (Its alternate 5th does as well.) 47edo is the most difficult regular edo to notate, because no other regular edo's 5th is as far off from a just 3/2.

A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a very 7edo-ish scale: 
D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D
This notation requires triple, quadruple and even quintuple sharps and flats.

Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a very 5edo-ish scale, with 2 of the notes inflected by a comma: 
D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D
This notation requires quintuple, sextuple, and even septuple ups and downs.


==Intervals of 47edo== 

||~ Degree ||~ Size (Cents) ||~ relative notation ||
||= 0 ||> 0.0000 ||= P1 ||
||= 1 ||> 25.5319 || A1 ||
||= 2 ||> 51.0638 || AA1 ||
||= 3 ||> 76.5957 || A31, d32 ||
||= 4 ||> 102.1277 || dd2 ||
||= 5 ||> 127.6596 || d2 ||
||= 6 ||> 153.1915 || m2 ||
||= 7 ||> 178.7234 || M2 ||
||= 8 ||> 204.2553 ||   ||
||= 9 ||> 229.7872 ||   ||
||= 10 ||> 255.3191 ||   ||
||= 11 ||> 280.8511 ||   ||
||= 12 ||> 306.3830 ||   ||
||= 13 ||> 331.9149 ||   ||
||= 14 ||> 357.4468 || M3 ||
||= 15 ||> 382.9787 ||   ||
||= 16 ||> 408.5106 ||   ||
||= 17 ||> 434.0426 ||   ||
||= 18 ||> 459.5745 ||   ||
||= 19 ||> 485.1064 ||   ||
||= 20 ||> 510.6383 ||   ||
||= 21 ||> 536.1702 ||   ||
||= 22 ||> 561.7021 ||   ||
||= 23 ||> 587.2340 ||   ||
||= 24 ||> 612.7660 ||   ||
||= 25 ||> 638.2979 ||   ||
||= 26 ||> 663.8298 ||   ||
||= 27 ||> 689.3617 ||   ||
||= 28 ||> 714.8936 ||   ||
||= 29 ||> 740.4255 ||   ||
||= 30 ||> 765.9574 ||   ||
||= 31 ||> 791.4894 ||   ||
||= 32 ||> 817.0213 ||   ||
||= 33 ||> 842.5532 ||   ||
||= 34 ||> 868.0851 ||   ||
||= 35 ||> 893.6170 ||   ||
||= 36 ||> 919.1489 ||   ||
||= 37 ||> 944.6809 ||   ||
||= 38 ||> 970.2128 ||   ||
||= 39 ||> 995.7447 ||   ||
||= 40 ||> 1021.2766 ||   ||
||= 41 ||> 1046.8085 ||   ||
||= 42 ||> 1072.3404 ||   ||
||= 43 ||> 1097.8723 ||   ||
||= 44 ||> 1123.4043 ||   ||
||= 45 ||> 1148.9362 ||   ||
||= 46 ||> 1174.4681 ||   ||
||= 47 ||> 1200.0000 ||   ||

Original HTML content:

<html><head><title>47edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x47 tone Equal Temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #616161; font-size: 99%;">47 tone Equal Temperament</span></h1>
 <br />
<strong><em>47-EDO</em></strong> divides the octave into 47 equal parts of 25.5319 cents each. It has a fifth which is 12.5933 cents flat, unless you use the alternative fifth which is 12.9386 cents sharp, similar to 35edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third of a cent sharp. It does a good job of approximating the 2.9.5.7.33.13.17.57.69 23-limit <a class="wiki_link" href="/k%2AN%20subgroups">2*47 subgroup</a> of the <a class="wiki_link" href="/23-limit">23-limit</a>, on which it tempers out the same commas as <a class="wiki_link" href="/94edo">94edo</a>. It provides a good tuning for <a class="wiki_link" href="/Chromatic%20pairs#Baldy">baldy</a> and <a class="wiki_link" href="/Chromatic%20pairs#Silver">silver</a> temperaments and relatives.<br />
<br />
47edo is the 15th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/43edo">43edo</a> and preceding <a class="wiki_link" href="/53edo">53edo</a>.<br />
<br />
47edo is a regular edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. (Its alternate 5th does as well.) 47edo is the most difficult regular edo to notate, because no other regular edo's 5th is as far off from a just 3/2.<br />
<br />
A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a very 7edo-ish scale: <br />
D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D<br />
This notation requires triple, quadruple and even quintuple sharps and flats.<br />
<br />
Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a very 5edo-ish scale, with 2 of the notes inflected by a comma: <br />
D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D<br />
This notation requires quintuple, sextuple, and even septuple ups and downs.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x47 tone Equal Temperament-Intervals of 47edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals of 47edo</h2>
 <br />


<table class="wiki_table">
    <tr>
        <th>Degree<br />
</th>
        <th>Size (Cents)<br />
</th>
        <th>relative notation<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: right;">0.0000<br />
</td>
        <td style="text-align: center;">P1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: right;">25.5319<br />
</td>
        <td>A1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: right;">51.0638<br />
</td>
        <td>AA1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: right;">76.5957<br />
</td>
        <td>A31, d32<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: right;">102.1277<br />
</td>
        <td>dd2<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: right;">127.6596<br />
</td>
        <td>d2<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: right;">153.1915<br />
</td>
        <td>m2<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: right;">178.7234<br />
</td>
        <td>M2<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: right;">204.2553<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: right;">229.7872<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: right;">255.3191<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: right;">280.8511<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: right;">306.3830<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: right;">331.9149<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: right;">357.4468<br />
</td>
        <td>M3<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: right;">382.9787<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: right;">408.5106<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: right;">434.0426<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: right;">459.5745<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: right;">485.1064<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: right;">510.6383<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: right;">536.1702<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: right;">561.7021<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: right;">587.2340<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: right;">612.7660<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: right;">638.2979<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: right;">663.8298<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: right;">689.3617<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: right;">714.8936<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: right;">740.4255<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: right;">765.9574<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: right;">791.4894<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: right;">817.0213<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: right;">842.5532<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: right;">868.0851<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: right;">893.6170<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: right;">919.1489<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: right;">944.6809<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: right;">970.2128<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: right;">995.7447<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: right;">1021.2766<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">41<br />
</td>
        <td style="text-align: right;">1046.8085<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">42<br />
</td>
        <td style="text-align: right;">1072.3404<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">43<br />
</td>
        <td style="text-align: right;">1097.8723<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">44<br />
</td>
        <td style="text-align: right;">1123.4043<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">45<br />
</td>
        <td style="text-align: right;">1148.9362<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">46<br />
</td>
        <td style="text-align: right;">1174.4681<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">47<br />
</td>
        <td style="text-align: right;">1200.0000<br />
</td>
        <td><br />
</td>
    </tr>
</table>

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