5L 7s
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author keenanpepper and made on 2011-05-12 21:29:31 UTC.
- The original revision id was 228068766.
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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 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 ||= || ||= ||= ||= ||= ||= 22\53 ||= 498.113 ||= Pythagorean/Helmholtz/Garibaldi is around here || ||= ||= ||= ||= 17\41 ||= ||= 497.591 ||= || ||= ||= ||= ||= ||= 29\70 ||= 497.143 ||= || ||= ||= ||= 12\29 ||= ||= ||= 496.552 ||= || ||= ||= ||= ||= ||= 31\75 ||= 496.000 ||= || ||= ||= ||= ||= 19\46 ||= ||= 495.652 ||= || ||= ||= ||= ||= ||= 26\63 ||= 495.238 ||= || ||= ||= 7\17 ||= ||= ||= ||= 494.118 ||= Boundary of propriety (generators larger than this are proper) || ||= ||= ||= ||= ||= 23\56 ||= 492.857 ||= || ||= ||= ||= ||= 16\39 ||= ||= 492.308 ||= || ||= ||= ||= ||= ||= 25\61 ||= 491.803 ||= || ||= ||= ||= 9\22 ||= ||= ||= 490.909 ||= Superpyth is in this region || ||= ||= ||= ||= ||= 20\49 ||= 489.796 ||= || ||= ||= ||= ||= 11\27 ||= ||= 488.889 ||= || ||= ||= ||= ||= ||= 13\32 ||= 487.500 ||= || ||= 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 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="5">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 style="text-align: center;">500.000<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;">22\53<br />
</td>
<td style="text-align: center;">498.113<br />
</td>
<td style="text-align: center;">Pythagorean/Helmholtz/Garibaldi<br />
is around here<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 style="text-align: center;">497.591<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;">29\70<br />
</td>
<td style="text-align: center;">497.143<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;">12\29<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">496.552<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;">31\75<br />
</td>
<td style="text-align: center;">496.000<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;">19\46<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">495.652<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;">26\63<br />
</td>
<td style="text-align: center;">495.238<br />
</td>
<td style="text-align: center;"><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 style="text-align: center;">494.118<br />
</td>
<td style="text-align: center;">Boundary of propriety (generators<br />
larger than this are proper)<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 style="text-align: center;">492.857<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;">16\39<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">492.308<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;">25\61<br />
</td>
<td style="text-align: center;">491.803<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;">9\22<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">490.909<br />
</td>
<td style="text-align: center;">Superpyth is in this region<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 style="text-align: center;">489.796<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;">11\27<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">488.889<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;">13\32<br />
</td>
<td style="text-align: center;">487.500<br />
</td>
<td style="text-align: center;"><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 style="text-align: center;">480.000<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
</body></html>