100edo

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author MasonGreen1 and made on 2016-06-17 21:45:48 UTC.

The original revision id was 585770669.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note scales= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd<span style="vertical-align: sub;">10</span> || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd<span style="vertical-align: sub;">10</span> || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd<span style="vertical-align: sub;">10</span> || Augmented unison || 168¢ [14] || 156<span style="line-height: 1.5;">¢ [8]</span> ||
|| 4 || Minor 3rd<span style="vertical-align: sub;">10</span> || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd<span style="vertical-align: sub;">10</span> || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th<span style="vertical-align: sub;">10</span> || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th<span style="vertical-align: sub;">10</span> || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th<span style="vertical-align: sub;">10</span>
<span style="vertical-align: sub;">Diminished </span>5th<span style="vertical-align: sub;">10</span> || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th<span style="vertical-align: sub;">10</span> || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th<span style="vertical-align: sub;">10</span>
Diminished 6th<span style="vertical-align: sub;">10</span> || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th<span style="vertical-align: sub;">10</span> || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th<span style="vertical-align: sub;">10</span>
Diminished 7th<span style="vertical-align: sub;">10</span> || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th<span style="vertical-align: sub;">10</span> || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th<span style="vertical-align: sub;">10</span>
Diminished 8th<span style="vertical-align: sub;">10</span> || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th<span style="vertical-align: sub;">10</span> || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th<span style="vertical-align: sub;">10</span> || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th<span style="vertical-align: sub;">10</span> || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th<span style="vertical-align: sub;">10</span> || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th<span style="vertical-align: sub;">10</span> || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th<span style="vertical-align: sub;">10</span> || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th<span style="vertical-align: sub;">10</span>
Diminished 11th<span style="vertical-align: sub;">10</span> || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th<span style="vertical-align: sub;">10</span> || Octave || 1200¢ [22] || N/A ||

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th<span style="vertical-align: sub;">10 </span>is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[https://en.wikipedia.org/wiki/Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales.

Original HTML content:

<html><head><title>100edo</title></head><body><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;w=560&quot; class=&quot;WikiMedia WikiMediaYoutube&quot; id=&quot;wikitext@@media@@type=&amp;quot;youtube&amp;quot; key=&amp;quot;shcrw2vtmJU&amp;quot; width=&amp;quot;560&amp;quot; height=&amp;quot;315&amp;quot;&quot; title=&quot;YouTube Video&quot;height=&quot;315&quot; width=&quot;560&quot;/&gt; --><iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen></iframe><!-- ws:end:WikiTextMediaRule:0 --><br />
<br />
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.<br />
<br />
Like <a class="wiki_link" href="/88edo">88edo</a>, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to <a class="wiki_link" href="/12edo">12edo</a>.<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:1 -->Scales</h1>
 <a class="wiki_link" href="/greeley8">greeley8</a><br />
<a class="wiki_link" href="/greeley15">greeley15</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc1"><!-- ws:end:WikiTextHeadingRule:3 --> </h2>
 <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="x100bddd and the 22-note scales"></a><!-- ws:end:WikiTextHeadingRule:5 -->100bddd and the 22-note scales</h1>
 <br />
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to <a class="wiki_link" href="/22edo">22edo</a> for <a class="wiki_link" href="/pajara">pajara</a> temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range <a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow">favored by George Secor</a> for neomedieval compositions.<br />
<br />
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &quot;wolf&quot; fifths. Much like meantone, this tuning has &quot;wolf&quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the &quot;wolves&quot; are still usable (albeit xenharmonic), it might be better to use the term <em>dog</em> rather than wolf for these intervals. Dog intervals frequently provide <em>closer</em> matches to intervals involving the 7th and 11th harmonics. Even <em>if</em> the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.<br />
<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Steps of 22-note MODMOS<br />
</td>
        <td>Interval name (decatonic)<br />
</td>
        <td>Interval name (superpyth diatonic)<br />
</td>
        <td>Pure interval size [multiplicity]<br />
</td>
        <td>Dog interval size [multiplicity]<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>Diminished 2nd<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Minor second<br />
</td>
        <td>60¢ [12]<br />
</td>
        <td>48¢ [10]<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>Minor 2nd<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Augmented seventh<br />
</td>
        <td>108¢ [20]<br />
</td>
        <td>120¢ [2]<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>Major 2nd<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Augmented unison<br />
</td>
        <td>168¢ [14]<br />
</td>
        <td>156<span style="line-height: 1.5;">¢ [8]</span><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>Minor 3rd<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Major second<br />
</td>
        <td>216¢ [18]<br />
</td>
        <td>228¢ [4]<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>Major 3rd<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Minor third<br />
</td>
        <td>276¢ [16]<br />
</td>
        <td>264¢ [6]<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>Minor 4th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Augmented second<br />
</td>
        <td>324¢ [16]<br />
</td>
        <td>336¢ [6]<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>Major 4th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Diminished fourth<br />
</td>
        <td>384¢ [18]<br />
</td>
        <td>372¢ [4]<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>Augmented 4th<span style="vertical-align: sub;">10</span><br />
<span style="vertical-align: sub;">Diminished </span>5th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Major third<br />
</td>
        <td>432¢ [14]<br />
</td>
        <td>444¢ [8]<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>Perfect 5th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Perfect fourth<br />
</td>
        <td>492¢ [20]<br />
</td>
        <td>480¢ [2]<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>Augmented 5th<span style="vertical-align: sub;">10</span><br />
Diminished 6th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Diminished fifth<br />
</td>
        <td>540¢ [12]<br />
</td>
        <td>552¢ [10]<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>Perfect 6th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Augmented third<br />
Diminished sixth<br />
</td>
        <td>600¢ [20]<br />
</td>
        <td>588¢ [1]<br />
612¢ [1]<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>Augmented 6th<span style="vertical-align: sub;">10</span><br />
Diminished 7th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Augmented fourth<br />
</td>
        <td>660¢ [12]<br />
</td>
        <td>648¢ [10]<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>Perfect 7th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Perfect fifth<br />
</td>
        <td>708¢ [20]<br />
</td>
        <td>720¢ [2]<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>Augmented 7th<span style="vertical-align: sub;">10</span><br />
Diminished 8th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Minor sixth<br />
</td>
        <td>768¢ [14]<br />
</td>
        <td>756¢ [8]<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>Minor 8th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Diminished seventh<br />
</td>
        <td>816¢ [18]<br />
</td>
        <td>828¢ [4]<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>Major 8th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Augmented sixth<br />
</td>
        <td>876¢ [16]<br />
</td>
        <td>864¢ [6]<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>Minor 9th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Major sixth<br />
</td>
        <td>924¢ [16]<br />
</td>
        <td>936¢ [6]<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>Major 9th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Minor seventh<br />
</td>
        <td>984¢ [18]<br />
</td>
        <td>972¢ [4]<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>Minor 10th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Diminished octave<br />
</td>
        <td>1032¢ [14]<br />
</td>
        <td>1044¢ [8]<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>Major 10th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Diminished second<br />
</td>
        <td>1092¢ [20]<br />
</td>
        <td>1080¢ [2]<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>Augmented 10th<span style="vertical-align: sub;">10</span><br />
Diminished 11th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Major seventh<br />
</td>
        <td>1140¢ [12]<br />
</td>
        <td>1152¢ [10]<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>11th<span style="vertical-align: sub;">10</span><br />
</td>
        <td>Octave<br />
</td>
        <td>1200¢ [22]<br />
</td>
        <td>N/A<br />
</td>
    </tr>
</table>

<br />
Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th<span style="vertical-align: sub;">10 </span>is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Augmented-fourths_tuning" rel="nofollow">are tuned in tritones</a>. This makes guitar construction much easier compared to other non-equally-tempered scales.</body></html>