5L 1s
IMPORTED REVISION FROM WIKISPACES
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Original Wikitext content:
5L 1s refers to [[MOSScales]] with 5 large steps and 1 small step. When L=s we have [[6edo]], the equal-tempered "whole tone scale" of impressionistic fame. At the other end of the spectrum, we approach [[5edo]], with five equal whole tones of 240 cents. In between, we find relatively even hexatonic scales with one irregularity: a "whole tone" which is smaller than all the others — perhaps not a "whole tone" at all. The only notable low-harmonic-entropy scale with this MOS pattern is [[Gamelismic clan|slendric]], in which the large step is 8/7 and three of them make a 3/2. Scales with this pattern are always [[Rothenberg propriety|proper]], because there is only one small step. ||||||||||||||~ generator ||~ scale ||~ large step (L) ||~ small step (s) ||~ comments || ||= 1\[[5edo|5]] ||= ||= ||= ||= || || ||= 1 1 1 1 1 0 ||= 240 ||= 0 ||= || || || || || || || || 7\36 ||= 7 7 7 7 7 1 ||= 233.3 ||= 33.3 ||= Slendric is around here || || || || || || || 6\31 || ||= 6 6 6 6 6 1 ||= 232.3 ||= 38.7 ||= || ||= ||= ||= ||= ||= 5\[[26edo|26]] || || ||= 5 5 5 5 5 1 ||= 230.8 ||= 46.2 ||= || ||= ||= ||= ||= 4\[[21edo|21]] ||= || || ||= 4 4 4 4 4 1 ||= 228.6 ||= 57.1 ||= L/s = 4 || ||= ||= ||= ||= ||= 7\[[37edo|37]] || || ||= 7 7 7 7 7 2 ||= 227.0 ||= 64.9 ||= || || || || || || || || ||= pi pi pi pi pi 1 ||= 225.6 ||= 71.8 ||= <span style="display: block; text-align: center;">L/s = pi</span> || ||= ||= ||= 3\[[16edo|16]] ||= ||= || || ||= 3 3 3 3 3 1 ||= 225 ||= 75 ||= Gorgo is around here L/s = 3 || || || || || || || || ||= e e e e e 1 ||= 223.55 ||= 82.2 ||= <span style="display: block; text-align: center;">L/s = e</span> || ||= ||= ||= ||= ||= 8\[[43edo|43]] || || ||= 8 8 8 8 8 3 ||= 223.3 ||= 83.7 ||= || || || || || || || || ||= <span style="display: block; text-align: center;">phi+1 phi+1 phi+1 phi+1 phi+1 1</span> ||= 223 ||= 85.2 || || ||= ||= ||= ||= 5\[[27edo|27]] ||= || || ||= 5 5 5 5 5 2 ||= 222.2 ||= 88.9 ||= || ||= ||= ||= ||= ||= 7\[[38edo|38]] || || ||= 7 7 7 7 7 3 ||= 221.1 ||= 94.7 ||= || ||= ||= 2\[[11edo|11]] ||= ||= ||= || || ||= 2 2 2 2 2 1 ||= 218.2 ||= 109.1 ||= Optimum rank range (L/s=2/1) machine || ||= ||= ||= ||= ||= 7\[[39edo|39]] || || ||= 7 7 7 7 7 4 ||= 215.4 ||= 123.1 ||= || || || || || || || || ||= <span style="background-color: #ffffff;">√3 √3 √3 √3 √3 1</span> ||= 215.2 ||= 124.2 || || ||= ||= ||= ||= 5\[[28edo|28]] ||= || || ||= 5 5 5 5 5 3 ||= 214.3 ||= 128.6 ||= || || || || || || || 13\73 || ||= 13 13 13 13 8 ||= 213.7 ||= 131.5 ||= || || || || || || || || ||= phi phi phi phi phi 1 ||= 213.6 ||= 1200/(1+5phi) ||= Golden machine || ||= ||= ||= ||= ||= 8\[[45edo|45]] || || ||= 8 8 8 8 8 5 ||= 213.3 ||= 133.3 ||= || || || || || || || || ||= <span style="display: block; text-align: center;">pi pi pi pi pi 2</span> ||= 212.9 ||= 135.5 || || ||= ||= ||= 3\[[17edo|17]] ||= ||= || || ||= 3 3 3 3 3 2 ||= 211.8 ||= 141.2 ||= || ||= ||= ||= ||= ||= 7\[[40edo|40]] || || ||= 7 7 7 7 7 5 ||= 210 ||= 150 ||= || ||= ||= ||= ||= 4\[[23edo|23]] ||= || || ||= 4 4 4 4 4 3 ||= 208.7 ||= 156.5 ||= || ||= ||= ||= ||= ||= 5\[[29edo|29]] || || ||= 5 5 5 5 5 4 ||= 206.9 ||= 165.5 ||= || ||= 1\[[6edo|6]] ||= ||= ||= ||= || || ||= 1 1 1 1 1 1 ||||= 200 ||= ||
Original HTML content:
<html><head><title>5L 1s</title></head><body>5L 1s refers to <a class="wiki_link" href="/MOSScales">MOSScales</a> with 5 large steps and 1 small step. When L=s we have <a class="wiki_link" href="/6edo">6edo</a>, the equal-tempered "whole tone scale" of impressionistic fame. At the other end of the spectrum, we approach <a class="wiki_link" href="/5edo">5edo</a>, with five equal whole tones of 240 cents. In between, we find relatively even hexatonic scales with one irregularity: a "whole tone" which is smaller than all the others — perhaps not a "whole tone" at all.<br />
<br />
The only notable low-harmonic-entropy scale with this MOS pattern is <a class="wiki_link" href="/Gamelismic%20clan">slendric</a>, in which the large step is 8/7 and three of them make a 3/2.<br />
<br />
Scales with this pattern are always <a class="wiki_link" href="/Rothenberg%20propriety">proper</a>, because there is only one small step.<br />
<table class="wiki_table">
<tr>
<th colspan="7">generator<br />
</th>
<th>scale<br />
</th>
<th>large step (L)<br />
</th>
<th>small step (s)<br />
</th>
<th>comments<br />
</th>
</tr>
<tr>
<td style="text-align: center;">1\<a class="wiki_link" href="/5edo">5</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">1 1 1 1 1 0<br />
</td>
<td style="text-align: center;">240<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\36<br />
</td>
<td style="text-align: center;">7 7 7 7 7 1<br />
</td>
<td style="text-align: center;">233.3<br />
</td>
<td style="text-align: center;">33.3<br />
</td>
<td style="text-align: center;">Slendric is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>6\31<br />
</td>
<td><br />
</td>
<td style="text-align: center;">6 6 6 6 6 1<br />
</td>
<td style="text-align: center;">232.3<br />
</td>
<td style="text-align: center;">38.7<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">5\<a class="wiki_link" href="/26edo">26</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">5 5 5 5 5 1<br />
</td>
<td style="text-align: center;">230.8<br />
</td>
<td style="text-align: center;">46.2<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">4\<a class="wiki_link" href="/21edo">21</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">4 4 4 4 4 1<br />
</td>
<td style="text-align: center;">228.6<br />
</td>
<td style="text-align: center;">57.1<br />
</td>
<td style="text-align: center;">L/s = 4<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">7\<a class="wiki_link" href="/37edo">37</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">7 7 7 7 7 2<br />
</td>
<td style="text-align: center;">227.0<br />
</td>
<td style="text-align: center;">64.9<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">pi pi pi pi pi 1<br />
</td>
<td style="text-align: center;">225.6<br />
</td>
<td style="text-align: center;">71.8<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">3\<a class="wiki_link" href="/16edo">16</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">3 3 3 3 3 1<br />
</td>
<td style="text-align: center;">225<br />
</td>
<td style="text-align: center;">75<br />
</td>
<td style="text-align: center;">Gorgo is around here<br />
L/s = 3<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">e e e e e 1<br />
</td>
<td style="text-align: center;">223.55<br />
</td>
<td style="text-align: center;">82.2<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = e</span><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">8\<a class="wiki_link" href="/43edo">43</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">8 8 8 8 8 3<br />
</td>
<td style="text-align: center;">223.3<br />
</td>
<td style="text-align: center;">83.7<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">phi+1 phi+1 phi+1 phi+1 phi+1 1</span><br />
</td>
<td style="text-align: center;">223<br />
</td>
<td style="text-align: center;">85.2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">5\<a class="wiki_link" href="/27edo">27</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">5 5 5 5 5 2<br />
</td>
<td style="text-align: center;">222.2<br />
</td>
<td style="text-align: center;">88.9<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">7\<a class="wiki_link" href="/38edo">38</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">7 7 7 7 7 3<br />
</td>
<td style="text-align: center;">221.1<br />
</td>
<td style="text-align: center;">94.7<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">2\<a class="wiki_link" href="/11edo">11</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">2 2 2 2 2 1<br />
</td>
<td style="text-align: center;">218.2<br />
</td>
<td style="text-align: center;">109.1<br />
</td>
<td style="text-align: center;">Optimum rank range (L/s=2/1) machine<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">7\<a class="wiki_link" href="/39edo">39</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">7 7 7 7 7 4<br />
</td>
<td style="text-align: center;">215.4<br />
</td>
<td style="text-align: center;">123.1<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="background-color: #ffffff;">√3 √3 √3 √3 √3 1</span><br />
</td>
<td style="text-align: center;">215.2<br />
</td>
<td style="text-align: center;">124.2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">5\<a class="wiki_link" href="/28edo">28</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">5 5 5 5 5 3<br />
</td>
<td style="text-align: center;">214.3<br />
</td>
<td style="text-align: center;">128.6<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\73<br />
</td>
<td><br />
</td>
<td style="text-align: center;">13 13 13 13 8<br />
</td>
<td style="text-align: center;">213.7<br />
</td>
<td style="text-align: center;">131.5<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">phi phi phi phi phi 1<br />
</td>
<td style="text-align: center;">213.6<br />
</td>
<td style="text-align: center;">1200/(1+5phi)<br />
</td>
<td style="text-align: center;">Golden machine<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">8\<a class="wiki_link" href="/45edo">45</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">8 8 8 8 8 5<br />
</td>
<td style="text-align: center;">213.3<br />
</td>
<td style="text-align: center;">133.3<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">pi pi pi pi pi 2</span><br />
</td>
<td style="text-align: center;">212.9<br />
</td>
<td style="text-align: center;">135.5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">3\<a class="wiki_link" href="/17edo">17</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">3 3 3 3 3 2<br />
</td>
<td style="text-align: center;">211.8<br />
</td>
<td style="text-align: center;">141.2<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">7\<a class="wiki_link" href="/40edo">40</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">7 7 7 7 7 5<br />
</td>
<td style="text-align: center;">210<br />
</td>
<td style="text-align: center;">150<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">4\<a class="wiki_link" href="/23edo">23</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">4 4 4 4 4 3<br />
</td>
<td style="text-align: center;">208.7<br />
</td>
<td style="text-align: center;">156.5<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">5\<a class="wiki_link" href="/29edo">29</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">5 5 5 5 5 4<br />
</td>
<td style="text-align: center;">206.9<br />
</td>
<td style="text-align: center;">165.5<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1\<a class="wiki_link" href="/6edo">6</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">1 1 1 1 1 1<br />
</td>
<td colspan="2" style="text-align: center;">200<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
</body></html>