Porcupine family
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-03-12 22:41:34 UTC.
- The original revision id was 209935778.
- 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:
The 5-limit parent comma for the porcupine family is 250/243, the maximal [[diesis]] or porcupine comma. Its [[monzo]] is |1 -5 3>, and flipping that yields <<3 5 1|| for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the 10/9 interval, and that three of these add up to a fourth, with two more giving the minor sixth. In fact, (10/9)^3 = 4/3 * 250/243, and (10/9)^5 = 8/5 * (250/243)^2. 3/22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities. [[POTE tuning|POTE generator]]: 163.950 Map: [<1 2 3|, <0 -3 -5|] EDOs: 22, 161, 183 ==Seven limit children== The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. That means 64/63, the Archytas comma, for porcupine, 36/35, the septimal quarter tone, for hystrix, 50/49, the jubilisma, for hedgehog, and 49/48, the slendro diesis, for nautilus. [[toc|flat]] =Porcupine= Porcupine, with wedgie <<3 5 -6 1 -18 -28||, uses six of its minor tone generator steps to get to 7/4. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3/22 is a good tuning choice, though we might pick in preference 8/59, 11/81, or 19/140 for our generator. Commas: 250/243, 64/63 [[POTE tuning|POTE generator]]: ~10/9 = 162.880 Map: [<1 2 3 2|, <0 -3 -5 6|] EDOs: 22, 59, 81, 140 ==11-limit== Commas: 55/54, 64/63, 100/99 POTE generator: ~10/9 = 162.747 Map: [<1 2 3 2 4|, <0 -3 -5 6 -4|] EDOs: 7, 15, 22, 37, 59 Badness: 0.0217 =Hystrix= Hystrix, with wedgie <<3 5 1 1 -7 -12||, provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2/15 or 9/68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits. Commas: 36/35, 160/147 [[POTE tuning|POTE generator]]: 158.868 Map: [<1 2 3 3|, <0 -3 -5 -1|] EDOs: 15, 68 =Hedgehog= Hedgehog, with wedgie <<6 10 10 2 -1 -5||, has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out 245/243, the sensamagic comma. 22et provides the obvious tuning, but if you are looking for an alternative, you could try the <146 232 338 411| val with generator 10/73, or you could try 164 cents if you are fond of round numbers. The 14 note MOS gives scope for harmony while stopping well short of 22. Commas: 50/49, 245/243 [[POTE tuning|POTE generator]]: 164.352 Map: [<2 1 1 2|, <0 3 5 5|] EDOs: 22, 146
Original HTML content:
<html><head><title>Porcupine family</title></head><body>The 5-limit parent comma for the porcupine family is 250/243, the maximal <a class="wiki_link" href="/diesis">diesis</a> or porcupine comma. Its <a class="wiki_link" href="/monzo">monzo</a> is |1 -5 3>, and flipping that yields <<3 5 1|| for the <a class="wiki_link" href="/wedgie">wedgie</a>. This tells us the <a class="wiki_link" href="/generator">generator</a> is a minor whole tone, the 10/9 interval, and that three of these add up to a fourth, with two more giving the minor sixth. In fact, (10/9)^3 = 4/3 * 250/243, and (10/9)^5 = 8/5 * (250/243)^2. 3/22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 163.950<br /> <br /> Map: [<1 2 3|, <0 -3 -5|]<br /> <br /> EDOs: 22, 161, 183<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. That means 64/63, the Archytas comma, for porcupine, 36/35, the septimal quarter tone, for hystrix, 50/49, the jubilisma, for hedgehog, and 49/48, the slendro diesis, for nautilus.<br /> <br /> <!-- ws:start:WikiTextTocRule:10:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Porcupine">Porcupine</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Hystrix">Hystrix</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Hedgehog">Hedgehog</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> <!-- ws:end:WikiTextTocRule:16 --><br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Porcupine"></a><!-- ws:end:WikiTextHeadingRule:2 -->Porcupine</h1> Porcupine, with wedgie <<3 5 -6 1 -18 -28||, uses six of its minor tone generator steps to get to 7/4. For this to work you need a small minor tone such as <a class="wiki_link" href="/22edo">22edo</a> provides, and once again 3/22 is a good tuning choice, though we might pick in preference 8/59, 11/81, or 19/140 for our generator.<br /> <br /> Commas: 250/243, 64/63<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~10/9 = 162.880<br /> <br /> Map: [<1 2 3 2|, <0 -3 -5 6|]<br /> EDOs: 22, 59, 81, 140<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Porcupine-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> Commas: 55/54, 64/63, 100/99<br /> <br /> POTE generator: ~10/9 = 162.747<br /> <br /> Map: [<1 2 3 2 4|, <0 -3 -5 6 -4|]<br /> EDOs: 7, 15, 22, 37, 59<br /> Badness: 0.0217<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Hystrix"></a><!-- ws:end:WikiTextHeadingRule:6 -->Hystrix</h1> Hystrix, with wedgie <<3 5 1 1 -7 -12||, provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2/15 or 9/68 can be used, is a temperament for the adventurous souls who have probably already tried <a class="wiki_link" href="/15edo">15edo</a>. They can try the even sharper fifth of hystrix in <a class="wiki_link" href="/68edo">68edo</a> and see how that suits.<br /> <br /> Commas: 36/35, 160/147<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 158.868<br /> <br /> Map: [<1 2 3 3|, <0 -3 -5 -1|]<br /> <br /> EDOs: 15, 68<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Hedgehog"></a><!-- ws:end:WikiTextHeadingRule:8 -->Hedgehog</h1> Hedgehog, with wedgie <<6 10 10 2 -1 -5||, has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out 245/243, the sensamagic comma. 22et provides the obvious tuning, but if you are looking for an alternative, you could try the <146 232 338 411| val with generator 10/73, or you could try 164 cents if you are fond of round numbers. The 14 note MOS gives scope for harmony while stopping well short of 22.<br /> <br /> Commas: 50/49, 245/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 164.352<br /> <br /> Map: [<2 1 1 2|, <0 3 5 5|]<br /> <br /> EDOs: 22, 146</body></html>