42edo

IMPORTED REVISION FROM WIKISPACES

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This revision was by author TallKite and made on 2017-11-11 03:00:01 UTC.

The original revision id was 621434943.

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Original Wikitext content:

The //42 equal division// divides the octave into 42 equal parts of 28.571 [[cent]]s each. It has a 3 (the size of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7n) and a 5 both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12, which means it [[tempering out|tempers out]] 128/125. In the [[7-limit]], it tempers out 64/63 and 126/125, making it a tuning supporting [[Augmented family|augene temperament]].

While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 [[k*N subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo]]. On this subgroup 42 has the same [[comma]]s as 84.

42edo is a [[xenharmonic/Ups and Downs Notation#Other%20EDOs|regular]] edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult regular edos to notate, because no other regular edo's 5th is as sharp (see 47edo for the opposite extreme). The major 2nd is 8 edosteps and the minor is only one. The naturals create a roughly 5edo-ish scale, with two of the notes inflected by a comma-sized edostep:

D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D

D# is next to E. The notation requires quadruple ups and downs, even more if chortds are to be spelled properly. For example, a 4:5:6:7 chord with a root on the key or fret midway between G and A would be written either

==Intervals of 42edo== 

||~ Degree ||~ Size (Cents) ||||~ relative notation ||~ absolute notation ||
||= 0 ||= 0 ||= perfect unison ||= P1 ||= D ||
||= 1 ||= 29 ||= up 1sn, minor 2nd ||= ^1, m2 ||= D# ||
||= 2 ||= 57 ||= double-up 1sn, upminor 2nd ||= ^^1, ^m2 ||= Dx ||
||= 3 ||= 86 ||= triple-up 1sn, double-up minor 2nd ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>1, ^^m2 ||= D#<span style="font-size: 90%; vertical-align: super;">3</span>, Eb<span style="font-size: 90%; vertical-align: super;">4</span> ||
||= 4 ||= 114 ||= downmid 2nd ||= v~2 ||= Eb<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 5 ||= 143 ||= upmid 2nd ||= ^~2 ||= Ebb ||
||= 6 ||= 171 ||= double-down major 2nd ||= vvM2 ||= Eb ||
||= 7 ||= 200 ||= downmajor 2nd ||= vM2 ||= E ||
||= 8 ||= 229 ||= major 2nd ||= M2 ||= E# ||
||= 9 ||= 257 ||= minor 3rd ||= m3 ||= Ex ||
||= 10 ||= 286 ||= upminor 3rd ||= ^m3 ||= E#<span style="font-size: 90%; vertical-align: super;">3</span>, Fb<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 11 ||= 314 ||= double-up minor 3rd ||= ^^m3 ||= Fbb ||
||= 12 ||= 343 ||= downmid 3rd ||= v~3 ||= Fb ||
||= 13 ||= 371 ||= upmid 3rd ||= ^~3 ||= F ||
||= 14 ||= 400 ||= double-down major 3rd ||= vvM3 ||= F# ||
||= 15 ||= 429 ||= downmajor 3rd ||= vM3 ||= Fx ||
||= 16 ||= 457 ||= major 3rd, down 4th ||= M3, v4 ||= F#<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 17 ||= 486 ||= perfect 4th ||= P4 ||= F#<span style="font-size: 90%; vertical-align: super;">4</span>, Gb<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 18 ||= 514 ||= up 4th ||= ^4 ||= Gbb ||
||= 19 ||= 543 ||= double-up 4th ||= ^^4 ||= Gb ||
||= 20 ||= 571 ||= triple-up 4th ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>4 ||= G ||
||= 21 ||= 600 ||= triple-down aug 4th, triple-up dim 5th ||= v<span style="font-size: 90%; vertical-align: super;">3</span>A4, ^<span style="font-size: 90%; vertical-align: super;">3</span>d5 ||= G# ||
||= 22 ||= 629 ||= triple-down 5th ||= v<span style="font-size: 90%; vertical-align: super;">3</span>5 ||= Gx ||
||= 23 ||= 657 ||= double-down 5th ||= vv5 ||= G#<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 24 ||= 686 ||= down 5th ||= v5 ||= Ab<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 25 ||= 714 ||= perfect 5th ||= P5 ||= Abb ||
||= 26 ||= 743 ||= up 5th, minor 6th ||= ^5, m6 ||= Ab ||
||= 27 ||= 771 ||= upminor 6th ||= ^m6 ||= A ||
||= 28 ||= 800 ||= double-up minor 5th ||= ^^m6 ||= A# ||
||= 29 ||= 829 ||= downmid 5th ||= v~6 ||= Ax ||
||= 30 ||= 857 ||= upmid 6th ||= ^~6 ||= A#<span style="font-size: 90%; vertical-align: super;">3</span>, Bb<span style="font-size: 90%; vertical-align: super;">4</span> ||
||= 31 ||= 886 ||= double-down major 6th ||= vvM6 ||= Bb<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 32 ||= 914 ||= downmajor 6th ||= vM6 ||= Bbb ||
||= 33 ||= 943 ||= major 6th ||= M6 ||= Bb ||
||= 34 ||= 971 ||= minor 7th ||= m7 ||= B ||
||= 35 ||= 1000 ||= upminor 7th ||= ^m7 ||= B# ||
||= 36 ||= 1029 ||= double-up minor 7th ||= ^^m7 ||= Bx ||
||= 37 ||= 1057 ||= downmid 7th ||= v~7 ||= B#<span style="font-size: 90%; vertical-align: super;">3</span>, Cb<span style="font-size: 90%; vertical-align: super;">3</span> ||
||= 38 ||= 1086 ||= upmid 7th ||= ^~7 ||= Cbb ||
||= 39 ||= 1114 ||= double-down major 7th ||= vvM7 ||= Cb ||
||= 40 ||= 1143 ||= downmajor 7th ||= vM7 ||= C ||
||= 41 ||= 1171 ||= major 7th ||= M7 ||= C# ||
||= 42 ||= 1200 ||= perfect 8ve ||= P8 ||= Cx ||

Original HTML content:

<html><head><title>42edo</title></head><body>The <em>42 equal division</em> divides the octave into 42 equal parts of 28.571 <a class="wiki_link" href="/cent">cent</a>s each. It has a 3 (the size of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7n) and a 5 both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12, which means it <a class="wiki_link" href="/tempering%20out">tempers out</a> 128/125. In the <a class="wiki_link" href="/7-limit">7-limit</a>, it tempers out 64/63 and 126/125, making it a tuning supporting <a class="wiki_link" href="/Augmented%20family">augene temperament</a>.<br />
<br />
While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 <a class="wiki_link" href="/k%2AN%20subgroups">2*42 subgroup</a>, having the same tuning on it as does <a class="wiki_link" href="/84edo">84edo</a>. On this subgroup 42 has the same <a class="wiki_link" href="/comma">comma</a>s as 84.<br />
<br />
42edo is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Other%20EDOs">regular</a> edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult regular edos to notate, because no other regular edo's 5th is as sharp (see 47edo for the opposite extreme). The major 2nd is 8 edosteps and the minor is only one. The naturals create a roughly 5edo-ish scale, with two of the notes inflected by a comma-sized edostep:<br />
<br />
D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D<br />
<br />
D# is next to E. The notation requires quadruple ups and downs, even more if chortds are to be spelled properly. For example, a 4:5:6:7 chord with a root on the key or fret midway between G and A would be written either<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Intervals of 42edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals of 42edo</h2>
 <br />


<table class="wiki_table">
    <tr>
        <th>Degree<br />
</th>
        <th>Size (Cents)<br />
</th>
        <th colspan="2">relative notation<br />
</th>
        <th>absolute notation<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;">perfect unison<br />
</td>
        <td style="text-align: center;">P1<br />
</td>
        <td style="text-align: center;">D<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">up 1sn, minor 2nd<br />
</td>
        <td style="text-align: center;">^1, m2<br />
</td>
        <td style="text-align: center;">D#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">57<br />
</td>
        <td style="text-align: center;">double-up 1sn, upminor 2nd<br />
</td>
        <td style="text-align: center;">^^1, ^m2<br />
</td>
        <td style="text-align: center;">Dx<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">86<br />
</td>
        <td style="text-align: center;">triple-up 1sn, double-up minor 2nd<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>1, ^^m2<br />
</td>
        <td style="text-align: center;">D#<span style="font-size: 90%; vertical-align: super;">3</span>, Eb<span style="font-size: 90%; vertical-align: super;">4</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">114<br />
</td>
        <td style="text-align: center;">downmid 2nd<br />
</td>
        <td style="text-align: center;">v~2<br />
</td>
        <td style="text-align: center;">Eb<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">143<br />
</td>
        <td style="text-align: center;">upmid 2nd<br />
</td>
        <td style="text-align: center;">^~2<br />
</td>
        <td style="text-align: center;">Ebb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">171<br />
</td>
        <td style="text-align: center;">double-down major 2nd<br />
</td>
        <td style="text-align: center;">vvM2<br />
</td>
        <td style="text-align: center;">Eb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">200<br />
</td>
        <td style="text-align: center;">downmajor 2nd<br />
</td>
        <td style="text-align: center;">vM2<br />
</td>
        <td style="text-align: center;">E<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">229<br />
</td>
        <td style="text-align: center;">major 2nd<br />
</td>
        <td style="text-align: center;">M2<br />
</td>
        <td style="text-align: center;">E#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">257<br />
</td>
        <td style="text-align: center;">minor 3rd<br />
</td>
        <td style="text-align: center;">m3<br />
</td>
        <td style="text-align: center;">Ex<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">286<br />
</td>
        <td style="text-align: center;">upminor 3rd<br />
</td>
        <td style="text-align: center;">^m3<br />
</td>
        <td style="text-align: center;">E#<span style="font-size: 90%; vertical-align: super;">3</span>, Fb<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">314<br />
</td>
        <td style="text-align: center;">double-up minor 3rd<br />
</td>
        <td style="text-align: center;">^^m3<br />
</td>
        <td style="text-align: center;">Fbb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">343<br />
</td>
        <td style="text-align: center;">downmid 3rd<br />
</td>
        <td style="text-align: center;">v~3<br />
</td>
        <td style="text-align: center;">Fb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">371<br />
</td>
        <td style="text-align: center;">upmid 3rd<br />
</td>
        <td style="text-align: center;">^~3<br />
</td>
        <td style="text-align: center;">F<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">400<br />
</td>
        <td style="text-align: center;">double-down major 3rd<br />
</td>
        <td style="text-align: center;">vvM3<br />
</td>
        <td style="text-align: center;">F#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">429<br />
</td>
        <td style="text-align: center;">downmajor 3rd<br />
</td>
        <td style="text-align: center;">vM3<br />
</td>
        <td style="text-align: center;">Fx<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">457<br />
</td>
        <td style="text-align: center;">major 3rd, down 4th<br />
</td>
        <td style="text-align: center;">M3, v4<br />
</td>
        <td style="text-align: center;">F#<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">486<br />
</td>
        <td style="text-align: center;">perfect 4th<br />
</td>
        <td style="text-align: center;">P4<br />
</td>
        <td style="text-align: center;">F#<span style="font-size: 90%; vertical-align: super;">4</span>, Gb<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">514<br />
</td>
        <td style="text-align: center;">up 4th<br />
</td>
        <td style="text-align: center;">^4<br />
</td>
        <td style="text-align: center;">Gbb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">543<br />
</td>
        <td style="text-align: center;">double-up 4th<br />
</td>
        <td style="text-align: center;">^^4<br />
</td>
        <td style="text-align: center;">Gb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">571<br />
</td>
        <td style="text-align: center;">triple-up 4th<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>4<br />
</td>
        <td style="text-align: center;">G<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">600<br />
</td>
        <td style="text-align: center;">triple-down aug 4th, triple-up dim 5th<br />
</td>
        <td style="text-align: center;">v<span style="font-size: 90%; vertical-align: super;">3</span>A4, ^<span style="font-size: 90%; vertical-align: super;">3</span>d5<br />
</td>
        <td style="text-align: center;">G#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">629<br />
</td>
        <td style="text-align: center;">triple-down 5th<br />
</td>
        <td style="text-align: center;">v<span style="font-size: 90%; vertical-align: super;">3</span>5<br />
</td>
        <td style="text-align: center;">Gx<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">657<br />
</td>
        <td style="text-align: center;">double-down 5th<br />
</td>
        <td style="text-align: center;">vv5<br />
</td>
        <td style="text-align: center;">G#<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">686<br />
</td>
        <td style="text-align: center;">down 5th<br />
</td>
        <td style="text-align: center;">v5<br />
</td>
        <td style="text-align: center;">Ab<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;">714<br />
</td>
        <td style="text-align: center;">perfect 5th<br />
</td>
        <td style="text-align: center;">P5<br />
</td>
        <td style="text-align: center;">Abb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">743<br />
</td>
        <td style="text-align: center;">up 5th, minor 6th<br />
</td>
        <td style="text-align: center;">^5, m6<br />
</td>
        <td style="text-align: center;">Ab<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">771<br />
</td>
        <td style="text-align: center;">upminor 6th<br />
</td>
        <td style="text-align: center;">^m6<br />
</td>
        <td style="text-align: center;">A<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">800<br />
</td>
        <td style="text-align: center;">double-up minor 5th<br />
</td>
        <td style="text-align: center;">^^m6<br />
</td>
        <td style="text-align: center;">A#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">829<br />
</td>
        <td style="text-align: center;">downmid 5th<br />
</td>
        <td style="text-align: center;">v~6<br />
</td>
        <td style="text-align: center;">Ax<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;">857<br />
</td>
        <td style="text-align: center;">upmid 6th<br />
</td>
        <td style="text-align: center;">^~6<br />
</td>
        <td style="text-align: center;">A#<span style="font-size: 90%; vertical-align: super;">3</span>, Bb<span style="font-size: 90%; vertical-align: super;">4</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">886<br />
</td>
        <td style="text-align: center;">double-down major 6th<br />
</td>
        <td style="text-align: center;">vvM6<br />
</td>
        <td style="text-align: center;">Bb<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">914<br />
</td>
        <td style="text-align: center;">downmajor 6th<br />
</td>
        <td style="text-align: center;">vM6<br />
</td>
        <td style="text-align: center;">Bbb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">943<br />
</td>
        <td style="text-align: center;">major 6th<br />
</td>
        <td style="text-align: center;">M6<br />
</td>
        <td style="text-align: center;">Bb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">971<br />
</td>
        <td style="text-align: center;">minor 7th<br />
</td>
        <td style="text-align: center;">m7<br />
</td>
        <td style="text-align: center;">B<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;">1000<br />
</td>
        <td style="text-align: center;">upminor 7th<br />
</td>
        <td style="text-align: center;">^m7<br />
</td>
        <td style="text-align: center;">B#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">1029<br />
</td>
        <td style="text-align: center;">double-up minor 7th<br />
</td>
        <td style="text-align: center;">^^m7<br />
</td>
        <td style="text-align: center;">Bx<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">1057<br />
</td>
        <td style="text-align: center;">downmid 7th<br />
</td>
        <td style="text-align: center;">v~7<br />
</td>
        <td style="text-align: center;">B#<span style="font-size: 90%; vertical-align: super;">3</span>, Cb<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">1086<br />
</td>
        <td style="text-align: center;">upmid 7th<br />
</td>
        <td style="text-align: center;">^~7<br />
</td>
        <td style="text-align: center;">Cbb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">1114<br />
</td>
        <td style="text-align: center;">double-down major 7th<br />
</td>
        <td style="text-align: center;">vvM7<br />
</td>
        <td style="text-align: center;">Cb<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;">1143<br />
</td>
        <td style="text-align: center;">downmajor 7th<br />
</td>
        <td style="text-align: center;">vM7<br />
</td>
        <td style="text-align: center;">C<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">41<br />
</td>
        <td style="text-align: center;">1171<br />
</td>
        <td style="text-align: center;">major 7th<br />
</td>
        <td style="text-align: center;">M7<br />
</td>
        <td style="text-align: center;">C#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">42<br />
</td>
        <td style="text-align: center;">1200<br />
</td>
        <td style="text-align: center;">perfect 8ve<br />
</td>
        <td style="text-align: center;">P8<br />
</td>
        <td style="text-align: center;">Cx<br />
</td>
    </tr>
</table>

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