5L 7s: Difference between revisions
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Wikispaces>keenanpepper **Imported revision 228069886 - Original comment: ** |
Wikispaces>guest **Imported revision 249051197 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:guest|guest]] and made on <tt>2011-08-29 04:36:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>249051197</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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]). | 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 || | ||||||||||||~ Generator ||~ in cents ||~ Comments || | ||
||= 5\12 ||= ||= ||= ||= ||= 500.000 ||= || | ||= 5\12 ||= ||= ||= ||= || ||= 500.000 ||= || | ||
||= ||= ||= ||= ||= 22\53 ||= 498.113 ||= Pythagorean/Helmholtz/Garibaldi | ||= ||= ||= ||= ||= 22\53 || ||= 498.113 ||= Pythagorean/Helmholtz/Garibaldi | ||
is around here || | is around here || | ||
||= ||= ||= ||= 17\41 ||= ||= 497.591 ||= || | ||= ||= ||= ||= 17\41 ||= || ||= 497.591 ||= || | ||
||= ||= ||= ||= ||= 29\70 ||= 497.143 ||= || | ||= ||= ||= ||= ||= 29\70 || ||= 497.143 ||= || | ||
||= ||= ||= 12\29 ||= ||= ||= 496.552 ||= | ||= ||= ||= 12\29 ||= ||= || ||= 496.552 ||= Optimum rank range (L/s=3/2) || | ||
||= ||= ||= ||= ||= 31\75 ||= 496.000 ||= || | ||= ||= ||= ||= ||= 31\75 || ||= 496.000 ||= || | ||
||= ||= ||= ||= 19\46 ||= ||= 495.652 ||= || | || || || || || || 50\121 || 495.868 ||= Golden leapday (a.k.a. "[[Peppermint-24|Peppermint]]") || | ||
||= ||= ||= ||= ||= 26\63 ||= 495.238 ||= || | ||= ||= ||= ||= 19\46 ||= || ||= 495.652 ||= || | ||
||= ||= 7\17 ||= ||= ||= ||= 494.118 ||= Boundary of propriety (generators | ||= ||= ||= ||= ||= 26\63 || ||= 495.238 ||= || | ||
||= ||= 7\17 ||= ||= ||= || ||= 494.118 ||= Boundary of propriety (generators | |||
larger than this are proper) || | larger than this are proper) || | ||
||= ||= ||= ||= ||= 23\56 ||= 492.857 ||= || | ||= ||= ||= ||= ||= 23\56 || ||= 492.857 ||= || | ||
||= ||= ||= ||= 16\39 ||= ||= 492.308 ||= || | ||= ||= ||= ||= 16\39 ||= || ||= 492.308 ||= || | ||
||= ||= ||= ||= ||= 25\61 ||= 491.803 ||= || | ||= ||= ||= ||= ||= 25\61 || ||= 491.803 ||= || | ||
||= ||= ||= 9\22 ||= ||= ||= 490.909 ||= Superpyth is in this region || | ||= ||= ||= 9\22 ||= ||= || ||= 490.909 ||= Superpyth is in this region || | ||
||= ||= ||= ||= ||= 20\49 ||= 489.796 ||= || | ||= ||= ||= ||= ||= 20\49 || ||= 489.796 ||= || | ||
||= ||= ||= ||= 11\27 ||= ||= 488.889 ||= || | ||= ||= ||= ||= 11\27 ||= || ||= 488.889 ||= || | ||
||= ||= ||= ||= ||= 13\32 ||= 487.500 ||= || | ||= ||= ||= ||= ||= 13\32 || ||= 487.500 ||= || | ||
||= 2\5 ||= ||= ||= ||= ||= 480.000 ||= ||</pre></div> | ||= 2\5 ||= ||= ||= ||= || ||= 480.000 ||= ||</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 /> | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th colspan=" | <th colspan="6">Generator<br /> | ||
</th> | </th> | ||
<th>in cents<br /> | <th>in cents<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">500.000<br /> | <td style="text-align: center;">500.000<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">22\53<br /> | <td style="text-align: center;">22\53<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">498.113<br /> | <td style="text-align: center;">498.113<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">497.591<br /> | <td style="text-align: center;">497.591<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">29\70<br /> | <td style="text-align: center;">29\70<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">497.143<br /> | <td style="text-align: center;">497.143<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">496.552<br /> | <td style="text-align: center;">496.552<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Optimum rank range (L/s=3/2)<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">31\75<br /> | <td style="text-align: center;">31\75<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">496.000<br /> | <td style="text-align: center;">496.000<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <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> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">495.652<br /> | <td style="text-align: center;">495.652<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">26\63<br /> | <td style="text-align: center;">26\63<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">495.238<br /> | <td style="text-align: center;">495.238<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">494.118<br /> | <td style="text-align: center;">494.118<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">23\56<br /> | <td style="text-align: center;">23\56<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">492.857<br /> | <td style="text-align: center;">492.857<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">492.308<br /> | <td style="text-align: center;">492.308<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">25\61<br /> | <td style="text-align: center;">25\61<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">491.803<br /> | <td style="text-align: center;">491.803<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">490.909<br /> | <td style="text-align: center;">490.909<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">20\49<br /> | <td style="text-align: center;">20\49<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">489.796<br /> | <td style="text-align: center;">489.796<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">488.889<br /> | <td style="text-align: center;">488.889<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">13\32<br /> | <td style="text-align: center;">13\32<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">487.500<br /> | <td style="text-align: center;">487.500<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">480.000<br /> | <td style="text-align: center;">480.000<br /> |
Revision as of 04:36, 29 August 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author guest and made on 2011-08-29 04:36:49 UTC.
- The original revision id was 249051197.
- 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:
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 ||= || ||= ||= ||= ||= ||= 22\53 || ||= 498.113 ||= Pythagorean/Helmholtz/Garibaldi is around here || ||= ||= ||= ||= 17\41 ||= || ||= 497.591 ||= || ||= ||= ||= ||= ||= 29\70 || ||= 497.143 ||= || ||= ||= ||= 12\29 ||= ||= || ||= 496.552 ||= Optimum rank range (L/s=3/2) || ||= ||= ||= ||= ||= 31\75 || ||= 496.000 ||= || || || || || || || 50\121 || 495.868 ||= Golden leapday (a.k.a. "[[Peppermint-24|Peppermint]]") || ||= ||= ||= ||= 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 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 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 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 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 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 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. "<a class="wiki_link" href="/Peppermint-24">Peppermint</a>")<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 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 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 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 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 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 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 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 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 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 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>