2L 8s: Difference between revisions
Wikispaces>JosephRuhf **Imported revision 551973612 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 565456799 - 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:JosephRuhf|JosephRuhf]] and made on <tt>2015- | : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-11-06 11:08:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>565456799</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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|| || || || || || || || || || 600/(4+e) || || | || || || || || || || || || || 600/(4+e) || || | ||
|| || || || 2\26 || || || || || || 92.31 ||= || | || || || || 2\26 || || || || || || 92.31 ||= || | ||
|| || || || || || 5\64 || || || || 93.75 || || | |||
|| || || || || || || || 13\166 || || 93.98 || || | |||
|| || || || || || || || || 21\268 || 94.03 ||= <span style="display: block; text-align: center;">Golden decatonic (Injera [bad tuning])</span> || | |||
|| || || || || || || 8\102 || || || 94.12 || || | |||
|| || || || || 3\38 || || || || || 94.74 ||= || | || || || || || 3\38 || || || || || 94.74 ||= || | ||
|| || || || || || 4\50 || || || || 96 ||= Injera is around here || | || || || || || || 4\50 || || || || 96 ||= Injera is around here || | ||
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larger than this are proper) || | larger than this are proper) || | ||
|| || || || || 4\46 || || || || || 104.35 ||= || | || || || || || 4\46 || || || || || 104.35 ||= || | ||
|| || || || || || || || || || 600/(4+sqrt(3)) || || | |||
|| || || || 3\34 || || || || || || 105.88 ||= || | || || || || 3\34 || || || || || || 105.88 ||= || | ||
|| || || || || || 8\90 || || || || 106.67 ||= | || || || || || || 8\90 || || || || 106.67 ||= around here 8g=18/11 || | ||
|| || || || || || || || 21\236 || || 106.78 ||= || | || || || || || || || || 21\236 || || 106.78 ||= || | ||
|| || || || || || || || || 34\382 || 106.81 ||= Golden decatonic (Srutal/pajara) || | || || || || || || || || || 34\382 || 106.81 ||= Golden decatonic (Srutal/pajara) || | ||
|| || || || || || || 13\146 || || || 106.85 ||= || | || || || || || || || 13\146 || || || 106.85 ||= || | ||
|| || || || || 5\56 || || || || || 107.14 ||= Srutal/pajara decatonic is around here || | || || || || || 5\56 || || || || || 107.14 ||= Srutal/pajara decatonic is around here || | ||
|| || || || || || || || || || 600/(4+pi/2) || || | |||
|| || || 2\22 || || || || || || || 109.09 ||= Optimum rank range (L/s=3/2) decatonic || | || || || 2\22 || || || || || || || 109.09 ||= Optimum rank range (L/s=3/2) decatonic || | ||
|| || || || 3\32 || || || || || || 112.5 ||= || | || || || || 3\32 || || || || || || 112.5 ||= || | ||
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<br /> | <br /> | ||
The only significant harmonic entropy minimum that is <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> is the decatonic scale itself (<a class="wiki_link" href="/Diaschismic%20family">pajara</a>[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes <a class="wiki_link" href="/4_3">4/3</a>, and another generator down makes <a class="wiki_link" href="/5_4">5/4</a>. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:<br /> | The only significant harmonic entropy minimum that is <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> is the decatonic scale itself (<a class="wiki_link" href="/Diaschismic%20family">pajara</a>[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes <a class="wiki_link" href="/4_3">4/3</a>, and another generator down makes <a class="wiki_link" href="/5_4">5/4</a>. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:1188:http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf --><a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf</a><!-- ws:end:WikiTextUrlRule:1188 --><br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:1189:http://www.lumma.org/tuning/erlich/ --><a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow">http://www.lumma.org/tuning/erlich/</a><!-- ws:end:WikiTextUrlRule:1189 --><br /> | ||
<br /> | <br /> | ||
Improper harmonic entropy minima include <a class="wiki_link" href="/Meantone%20family#Injera">injera</a> (which is similar to pajara except that 5/4 is now four generators <strong>up</strong> and no periods) and <a class="wiki_link" href="/Diaschismic%20family#Shrutar">shrutar</a> (which is basically pajara with the generator divided in two).<br /> | Improper harmonic entropy minima include <a class="wiki_link" href="/Meantone%20family#Injera">injera</a> (which is similar to pajara except that 5/4 is now four generators <strong>up</strong> and no periods) and <a class="wiki_link" href="/Diaschismic%20family#Shrutar">shrutar</a> (which is basically pajara with the generator divided in two).<br /> | ||
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</td> | </td> | ||
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<td>5\64<br /> | |||
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<td>93.75<br /> | |||
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<td>13\166<br /> | |||
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<td>93.98<br /> | |||
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<td>21\268<br /> | |||
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<td>94.03<br /> | |||
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<td style="text-align: center;"><span style="display: block; text-align: center;">Golden decatonic (Injera [bad tuning])</span><br /> | |||
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<td>8\102<br /> | |||
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<td>94.12<br /> | |||
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<td>600/(4+sqrt(3))<br /> | |||
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<td>106.67<br /> | <td>106.67<br /> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;">around here 8g=18/11<br /> | ||
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<td style="text-align: center;">Srutal/pajara decatonic is around here<br /> | <td style="text-align: center;">Srutal/pajara decatonic is around here<br /> | ||
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<td>600/(4+pi/2)<br /> | |||
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