5L 7s
IMPORTED REVISION FROM WIKISPACES
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Original Wikitext content:
This is the MOS pattern of the Pythagorean/[[Schismatic family|schismatic/Helmholtz/Garibaldi]] chromatic scale, and also the [[Archytas clan|superpyth]] chromatic scale. In contrast to the [[7L 5s|meantone chromatic scale]], in which "diatonic" semitones are larger than "chromatic" semitones, here the reverse is true: diatonic semitones are smaller than chromatic semitones, so the [[5L 2s|diatonic scale]] subset is actually [[Rothenberg propriety|improper]]. The two distinct harmonic entropy minima with this MOS pattern are, on the one hand, scales very close to Pythagorean such that 64/63 is not tempered out, such as the schismatic temperaments known as "Helmholtz" and "Garibaldi", and on the other hand, the much simpler and less accurate scale known as "superpyth" in which 64/63 is tempered out. The Pythagorean/schismatic version is proper, but the superpyth version is improper (it doesn't become proper until you add 5 more notes to form the superpyth "enharmonic" scale, superpyth[17]). ||||||||||||~ Generator ||~ in cents ||~ Comments || ||= 5\12 ||= ||= ||= ||= || ||= 500.000 ||= || || || || || || ||= 27\65 ||= 498.462 ||= Photia || ||= ||= ||= ||= ||= ||= 49\118 ||= 498.305 ||= Helmholtz/Pontiac/Nestoria || ||= ||= ||= ||= ||= ||= 71\171 ||= 498.246 ||= Helmholtz/Pontiac/Nestoria || ||= ||= ||= ||= ||= 22\53 || ||= 498.113 ||= Helenus || || || || || || || 39\94 ||= 497.872 ||= Garibaldi || ||= ||= ||= ||= 17\41 ||= || ||= 497.591 ||= Cassandra || || || || || || || 46\111 || 497.297 || || ||= ||= ||= ||= ||= 29\70 || ||= 497.143 ||= Undecental || || || || || || || 41\99 || 496.97 ||= Undecental || ||= ||= ||= 12\29 ||= ||= || ||= 496.552 ||= Optimum rank range (L/s=3/2) Edson || || || || || || || 43\104 || 496.154 || || ||= ||= ||= ||= ||= 31\75 || ||= 496.000 ||= || || || || || || || 50\121 || 495.868 ||= Leapday/[[Peppermint-24|Peppermint]]/Pepperoni || ||= ||= ||= ||= 19\46 ||= || ||= 495.652 ||= Leapday || || || || || || || 45\109 || 495.413 || || ||= ||= ||= ||= ||= 26\63 || ||= 495.238 ||= || || || || || || || 33\80 || 495 || || ||= ||= 7\17 ||= ||= ||= || ||= 494.118 ||= Boundary of propriety (generators larger than this are proper) Supraphon || || || || || || || 30\73 || 493.151 || || ||= ||= ||= ||= ||= 23\56 || ||= 492.857 ||= || || || || || || || 39\95 || 492.632 || || ||= ||= ||= ||= 16\39 ||= || ||= 492.308 ||= || || || || || || || 41\100 || 492 || || ||= ||= ||= ||= ||= 25\61 || ||= 491.803 ||= || || || || || || || 34\83 || 491.566 || || ||= ||= ||= 9\22 ||= ||= || ||= 490.909 ||= Suprapyth/Supra || || || || || || || 29\71 || 490.141 || || ||= ||= ||= ||= ||= 20\49 || ||= 489.796 ||= Superpyth || || || || || || || 31\76 || 489.474 || || ||= ||= ||= ||= 11\27 ||= || ||= 488.889 ||= Archy || || || || || || || 24\59 || 488.136 || || ||= ||= ||= ||= ||= 13\32 || ||= 487.500 ||= || || || || || || || 15\37 || 486.486 || || ||= 2\5 ||= ||= ||= ||= || ||= 480.000 ||= ||
Original HTML content:
<html><head><title>5L 7s</title></head><body>This is the MOS pattern of the Pythagorean/<a class="wiki_link" href="/Schismatic%20family">schismatic/Helmholtz/Garibaldi</a> chromatic scale, and also the <a class="wiki_link" href="/Archytas%20clan">superpyth</a> chromatic scale. In contrast to the <a class="wiki_link" href="/7L%205s">meantone chromatic scale</a>, in which "diatonic" semitones are larger than "chromatic" semitones, here the reverse is true: diatonic semitones are smaller than chromatic semitones, so the <a class="wiki_link" href="/5L%202s">diatonic scale</a> subset is actually <a class="wiki_link" href="/Rothenberg%20propriety">improper</a>.<br />
<br />
The two distinct harmonic entropy minima with this MOS pattern are, on the one hand, scales very close to Pythagorean such that 64/63 is not tempered out, such as the schismatic temperaments known as "Helmholtz" and "Garibaldi", and on the other hand, the much simpler and less accurate scale known as "superpyth" in which 64/63 is tempered out.<br />
<br />
The Pythagorean/schismatic version is proper, but the superpyth version is improper (it doesn't become proper until you add 5 more notes to form the superpyth "enharmonic" scale, superpyth[17]).<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="6">Generator<br />
</th>
<th>in cents<br />
</th>
<th>Comments<br />
</th>
</tr>
<tr>
<td style="text-align: center;">5\12<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 style="text-align: center;">500.000<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 style="text-align: center;">27\65<br />
</td>
<td style="text-align: center;">498.462<br />
</td>
<td style="text-align: center;">Photia<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;"><br />
</td>
<td style="text-align: center;">49\118<br />
</td>
<td style="text-align: center;">498.305<br />
</td>
<td style="text-align: center;">Helmholtz/Pontiac/Nestoria<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;"><br />
</td>
<td style="text-align: center;">71\171<br />
</td>
<td style="text-align: center;">498.246<br />
</td>
<td style="text-align: center;">Helmholtz/Pontiac/Nestoria<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;">22\53<br />
</td>
<td><br />
</td>
<td style="text-align: center;">498.113<br />
</td>
<td style="text-align: center;">Helenus<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>39\94<br />
</td>
<td style="text-align: center;">497.872<br />
</td>
<td style="text-align: center;">Garibaldi<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;">17\41<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">497.591<br />
</td>
<td style="text-align: center;">Cassandra<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>46\111<br />
</td>
<td>497.297<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;"><br />
</td>
<td style="text-align: center;">29\70<br />
</td>
<td><br />
</td>
<td style="text-align: center;">497.143<br />
</td>
<td style="text-align: center;">Undecental<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>41\99<br />
</td>
<td>496.97<br />
</td>
<td style="text-align: center;">Undecental<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">12\29<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">496.552<br />
</td>
<td style="text-align: center;">Optimum rank range (L/s=3/2) Edson<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>43\104<br />
</td>
<td>496.154<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;"><br />
</td>
<td style="text-align: center;">31\75<br />
</td>
<td><br />
</td>
<td style="text-align: center;">496.000<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>50\121<br />
</td>
<td>495.868<br />
</td>
<td style="text-align: center;">Leapday/<a class="wiki_link" href="/Peppermint-24">Peppermint</a>/Pepperoni<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;">19\46<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">495.652<br />
</td>
<td style="text-align: center;">Leapday<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>45\109<br />
</td>
<td>495.413<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;"><br />
</td>
<td style="text-align: center;">26\63<br />
</td>
<td><br />
</td>
<td style="text-align: center;">495.238<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>33\80<br />
</td>
<td>495<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">7\17<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 style="text-align: center;">494.118<br />
</td>
<td style="text-align: center;">Boundary of propriety (generators<br />
larger than this are proper) Supraphon<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>30\73<br />
</td>
<td>493.151<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;"><br />
</td>
<td style="text-align: center;">23\56<br />
</td>
<td><br />
</td>
<td style="text-align: center;">492.857<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>39\95<br />
</td>
<td>492.632<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;">16\39<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">492.308<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>41\100<br />
</td>
<td>492<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;"><br />
</td>
<td style="text-align: center;">25\61<br />
</td>
<td><br />
</td>
<td style="text-align: center;">491.803<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>34\83<br />
</td>
<td>491.566<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;">9\22<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">490.909<br />
</td>
<td style="text-align: center;">Suprapyth/Supra<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>29\71<br />
</td>
<td>490.141<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;"><br />
</td>
<td style="text-align: center;">20\49<br />
</td>
<td><br />
</td>
<td style="text-align: center;">489.796<br />
</td>
<td style="text-align: center;">Superpyth<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>31\76<br />
</td>
<td>489.474<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;">11\27<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td style="text-align: center;">488.889<br />
</td>
<td style="text-align: center;">Archy<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>24\59<br />
</td>
<td>488.136<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;"><br />
</td>
<td style="text-align: center;">13\32<br />
</td>
<td><br />
</td>
<td style="text-align: center;">487.500<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>15\37<br />
</td>
<td>486.486<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2\5<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 style="text-align: center;">480.000<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
</body></html>