5L 7s: Difference between revisions

Revision as of 13:21, 25 April 2012

IMPORTED REVISION FROM WIKISPACES

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This revision was by author guest and made on 2012-04-25 13:21:48 UTC.

The original revision id was 325139392.

The revision comment was:

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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 ||   ||
||=   ||=   ||=   ||=   ||= 22\53 ||   ||= 498.113 ||= Pythagorean/Helmholtz/Garibaldi
is around here ||
||   ||   ||   ||   ||   || 39/94 || 497.872 ||   ||
||=   ||=   ||=   ||= 17\41 ||=   ||   ||= 497.591 ||=   ||
||   ||   ||   ||   ||   || 46/111 || 497.297 ||   ||
||=   ||=   ||=   ||=   ||= 29\70 ||   ||= 497.143 ||=   ||
||   ||   ||   ||   ||   || 41/99 || 496.97 ||   ||
||=   ||=   ||= 12\29 ||=   ||=   ||   ||= 496.552 ||= Optimum rank range (L/s=3/2) ||
||   ||   ||   ||   ||   || 43/104 || 496.154 ||   ||
||=   ||=   ||=   ||=   ||= 31\75 ||   ||= 496.000 ||=   ||
||   ||   ||   ||   ||   || 50\121 || 495.868 ||= Golden leapday (a.k.a. "[[Peppermint-24|Peppermint]]") ||
||=   ||=   ||=   ||= 19\46 ||=   ||   ||= 495.652 ||=   ||
||   ||   ||   ||   ||   || 45/109 || 495.413 ||   ||
||=   ||=   ||=   ||=   ||= 26\63 ||   ||= 495.238 ||=   ||
||   ||   ||   ||   ||   || 33/80 || 495 ||   ||
||=   ||= 7\17 ||=   ||=   ||=   ||   ||= 494.118 ||= Boundary of propriety (generators
larger than this are proper) ||
||   ||   ||   ||   ||   || 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 ||= Superpyth is in this region ||
||   ||   ||   ||   ||   || 29/71 || 490.141 ||   ||
||=   ||=   ||=   ||=   ||= 20\49 ||   ||= 489.796 ||=   ||
||   ||   ||   ||   ||   || 31/76 || 489.474 ||   ||
||=   ||=   ||=   ||= 11\27 ||=   ||   ||= 488.889 ||=   ||
||   ||   ||   ||   ||   || 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 &quot;diatonic&quot; semitones are larger than &quot;chromatic&quot; 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 &quot;Helmholtz&quot; and &quot;Garibaldi&quot;, and on the other hand, the much simpler and less accurate scale known as &quot;superpyth&quot; 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 &quot;enharmonic&quot; 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>27/65<br />
</td>
        <td>498.462<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;">22\53<br />
</td>
        <td><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>39/94<br />
</td>
        <td>497.872<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;">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;"><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;"><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><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)<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;">Golden leapday (a.k.a. &quot;<a class="wiki_link" href="/Peppermint-24">Peppermint</a>&quot;)<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;"><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)<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;">Superpyth is in this region<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;"><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;"><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>

5L 7s: Difference between revisions

Revision as of 13:21, 25 April 2012

IMPORTED REVISION FROM WIKISPACES

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Original HTML content:

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