Superpyth
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author keenanpepper and made on 2011-08-17 17:28:42 UTC.
- The original revision id was 246562811.
- 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:
Superpyth, a member of the [[Archytas clan]], has 4/3 as a generator, and the Archytas comma 64/63 is tempered out, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is "the opposite of" septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex. If intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98. This temperament is called "supra", or "suprapyth" if you include 5 as well. MOSes include 5, 7, 12, 17, and 22. ==Interval chains== ===Basic superpyth (2.3.7)=== || 1146.61 || 437.29 || 927.97 || 218.64 || 709.32 || 0 || 490.68 || 981.36 || 272.03 || 762.71 || 53.39 || || 27/14 || 9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9 || 28/27 || ===Full 7-limit superpyth=== || 613.20 || 1102.91 || 392.62 || 882.33 || 172.04 || 661.75 || 1151.46 || 441.16 || 930.87 || 220.58 || 710.29 || 0 || 489.71 || 979.42 || 269.13 || 758.84 || 48.54 || 538.25 || 1027.96 || 317.67 || 807.38 || 97.09 || 586.80 || || 10/7 || 15/8 || 5/4 || 5/3 || 10/9 || || 27/14 || 9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9 || 28/27 || || 9/5 || 6/5 || 8/5 || 16/15 || 7/5 || ===Supra (2.3.7.11)=== || 857.54 || 150.35 || 643.15 || 1135.96 || 428.77 || 921.58 || 214.38 || 707.19 || 0 || 492.81 || 985.62 || 278.42 || 771.23 || 64.04 || 556.85 || 1049.65 || 342.46 || || 18/11 || 12/11 || 16/11 || 27/14 || 14/11~9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9~11/7 || 33/32~28/27 || 11/8 || 11/6 || 11/9 || ===Full 11-limit suprapyth=== || 604.44 || 1094.94 || 385.45 || 875.96 || 166.46 || 656.97 || 1147.47 || 437.98 || 928.48 || 218.99 || 709.49 || 0 || 490.51 || 981.01 || 271.52 || 762.02 || 52.53 || 543.03 || 1033.54 || 324.04 || 814.55 || 105.06 || 595.56 || || 10/7 || 15/8 || 5/4 || 18/11~5/3 || 12/11~10/9 || 16/11 || 27/14 || 14/11~9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9~11/7 || 33/32~28/27 || 11/8 || 9/5~11/6 || 6/5~11/9 || 8/5 || 16/15 || 7/5 || ==MOSes== ===5-note (LsLss, proper)=== See [[2L 3s]]. ===7-note (LLLsLLs, improper)=== See [[5L 2s]]. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper. ===12-note (LsLsLssLsLss, borderline improper)=== See [[5L 7s]]. The boundary of propriety is [[17edo]].
Original HTML content:
<html><head><title>Superpyth</title></head><body>Superpyth, a member of the <a class="wiki_link" href="/Archytas%20clan">Archytas clan</a>, has 4/3 as a generator, and the Archytas comma 64/63 is tempered out, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for <a class="wiki_link" href="/meantone">meantone</a> and <a class="wiki_link" href="/12edo">12edo</a>, with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical).<br />
<br />
If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is "the opposite of" septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.<br />
<br />
If intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98. This temperament is called "supra", or "suprapyth" if you include 5 as well.<br />
<br />
MOSes include 5, 7, 12, 17, and 22.<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Interval chains"></a><!-- ws:end:WikiTextHeadingRule:0 -->Interval chains</h2>
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Interval chains-Basic superpyth (2.3.7)"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basic superpyth (2.3.7)</h3>
<table class="wiki_table">
<tr>
<td>1146.61<br />
</td>
<td>437.29<br />
</td>
<td>927.97<br />
</td>
<td>218.64<br />
</td>
<td>709.32<br />
</td>
<td>0<br />
</td>
<td>490.68<br />
</td>
<td>981.36<br />
</td>
<td>272.03<br />
</td>
<td>762.71<br />
</td>
<td>53.39<br />
</td>
</tr>
<tr>
<td>27/14<br />
</td>
<td>9/7<br />
</td>
<td>12/7<br />
</td>
<td>9/8~8/7<br />
</td>
<td>3/2<br />
</td>
<td>1/1<br />
</td>
<td>4/3<br />
</td>
<td>7/4~16/9<br />
</td>
<td>7/6<br />
</td>
<td>14/9<br />
</td>
<td>28/27<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Interval chains-Full 7-limit superpyth"></a><!-- ws:end:WikiTextHeadingRule:4 -->Full 7-limit superpyth</h3>
<table class="wiki_table">
<tr>
<td>613.20<br />
</td>
<td>1102.91<br />
</td>
<td>392.62<br />
</td>
<td>882.33<br />
</td>
<td>172.04<br />
</td>
<td>661.75<br />
</td>
<td>1151.46<br />
</td>
<td>441.16<br />
</td>
<td>930.87<br />
</td>
<td>220.58<br />
</td>
<td>710.29<br />
</td>
<td>0<br />
</td>
<td>489.71<br />
</td>
<td>979.42<br />
</td>
<td>269.13<br />
</td>
<td>758.84<br />
</td>
<td>48.54<br />
</td>
<td>538.25<br />
</td>
<td>1027.96<br />
</td>
<td>317.67<br />
</td>
<td>807.38<br />
</td>
<td>97.09<br />
</td>
<td>586.80<br />
</td>
</tr>
<tr>
<td>10/7<br />
</td>
<td>15/8<br />
</td>
<td>5/4<br />
</td>
<td>5/3<br />
</td>
<td>10/9<br />
</td>
<td><br />
</td>
<td>27/14<br />
</td>
<td>9/7<br />
</td>
<td>12/7<br />
</td>
<td>9/8~8/7<br />
</td>
<td>3/2<br />
</td>
<td>1/1<br />
</td>
<td>4/3<br />
</td>
<td>7/4~16/9<br />
</td>
<td>7/6<br />
</td>
<td>14/9<br />
</td>
<td>28/27<br />
</td>
<td><br />
</td>
<td>9/5<br />
</td>
<td>6/5<br />
</td>
<td>8/5<br />
</td>
<td>16/15<br />
</td>
<td>7/5<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Interval chains-Supra (2.3.7.11)"></a><!-- ws:end:WikiTextHeadingRule:6 -->Supra (2.3.7.11)</h3>
<table class="wiki_table">
<tr>
<td>857.54<br />
</td>
<td>150.35<br />
</td>
<td>643.15<br />
</td>
<td>1135.96<br />
</td>
<td>428.77<br />
</td>
<td>921.58<br />
</td>
<td>214.38<br />
</td>
<td>707.19<br />
</td>
<td>0<br />
</td>
<td>492.81<br />
</td>
<td>985.62<br />
</td>
<td>278.42<br />
</td>
<td>771.23<br />
</td>
<td>64.04<br />
</td>
<td>556.85<br />
</td>
<td>1049.65<br />
</td>
<td>342.46<br />
</td>
</tr>
<tr>
<td>18/11<br />
</td>
<td>12/11<br />
</td>
<td>16/11<br />
</td>
<td>27/14<br />
</td>
<td>14/11~9/7<br />
</td>
<td>12/7<br />
</td>
<td>9/8~8/7<br />
</td>
<td>3/2<br />
</td>
<td>1/1<br />
</td>
<td>4/3<br />
</td>
<td>7/4~16/9<br />
</td>
<td>7/6<br />
</td>
<td>14/9~11/7<br />
</td>
<td>33/32~28/27<br />
</td>
<td>11/8<br />
</td>
<td>11/6<br />
</td>
<td>11/9<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Interval chains-Full 11-limit suprapyth"></a><!-- ws:end:WikiTextHeadingRule:8 -->Full 11-limit suprapyth</h3>
<table class="wiki_table">
<tr>
<td>604.44<br />
</td>
<td>1094.94<br />
</td>
<td>385.45<br />
</td>
<td>875.96<br />
</td>
<td>166.46<br />
</td>
<td>656.97<br />
</td>
<td>1147.47<br />
</td>
<td>437.98<br />
</td>
<td>928.48<br />
</td>
<td>218.99<br />
</td>
<td>709.49<br />
</td>
<td>0<br />
</td>
<td>490.51<br />
</td>
<td>981.01<br />
</td>
<td>271.52<br />
</td>
<td>762.02<br />
</td>
<td>52.53<br />
</td>
<td>543.03<br />
</td>
<td>1033.54<br />
</td>
<td>324.04<br />
</td>
<td>814.55<br />
</td>
<td>105.06<br />
</td>
<td>595.56<br />
</td>
</tr>
<tr>
<td>10/7<br />
</td>
<td>15/8<br />
</td>
<td>5/4<br />
</td>
<td>18/11~5/3<br />
</td>
<td>12/11~10/9<br />
</td>
<td>16/11<br />
</td>
<td>27/14<br />
</td>
<td>14/11~9/7<br />
</td>
<td>12/7<br />
</td>
<td>9/8~8/7<br />
</td>
<td>3/2<br />
</td>
<td>1/1<br />
</td>
<td>4/3<br />
</td>
<td>7/4~16/9<br />
</td>
<td>7/6<br />
</td>
<td>14/9~11/7<br />
</td>
<td>33/32~28/27<br />
</td>
<td>11/8<br />
</td>
<td>9/5~11/6<br />
</td>
<td>6/5~11/9<br />
</td>
<td>8/5<br />
</td>
<td>16/15<br />
</td>
<td>7/5<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-MOSes"></a><!-- ws:end:WikiTextHeadingRule:10 -->MOSes</h2>
<!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x-MOSes-5-note (LsLss, proper)"></a><!-- ws:end:WikiTextHeadingRule:12 -->5-note (LsLss, proper)</h3>
See <a class="wiki_link" href="/2L%203s">2L 3s</a>.<br />
<!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x-MOSes-7-note (LLLsLLs, improper)"></a><!-- ws:end:WikiTextHeadingRule:14 -->7-note (LLLsLLs, improper)</h3>
See <a class="wiki_link" href="/5L%202s">5L 2s</a>. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.<br />
<!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x-MOSes-12-note (LsLsLssLsLss, borderline improper)"></a><!-- ws:end:WikiTextHeadingRule:16 -->12-note (LsLsLssLsLss, borderline improper)</h3>
See <a class="wiki_link" href="/5L%207s">5L 7s</a>. The boundary of propriety is <a class="wiki_link" href="/17edo">17edo</a>.</body></html>