Semicomma family: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 264935378 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 264935564 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-10-14 19: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-14 19:40:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>264935564</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 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. | The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. | ||
Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9 | Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything. | ||
==Vital statistics== | ==Vital statistics== | ||
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The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> | The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> | ||
<br /> | <br /> | ||
Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has <a class="wiki_link" href="/Retuning%2012edo%20to%20Orwell9"> | Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has <a class="wiki_link" href="/Retuning%2012edo%20to%20Orwell9">considerable harmonic resources</a> despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Seven limit children-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:4 -->Vital statistics</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Seven limit children-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:4 -->Vital statistics</h2> | ||
Revision as of 19:40, 14 October 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 genewardsmith and made on 2011-10-14 19:40:13 UTC.
- The original revision id was 264935564.
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Original Wikitext content:
[[toc]] The 5-limit parent comma for the **semicomma family** is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. **Orson**, the [[5-limit]] temperament tempering it out, has a [[generator]] of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell. [[POTE tuning|POTE generator]]: 271.627 Map: [<1 0 3|, <0 7 -3|] EDOs: 22, 31, 53, 190, 253, 296 =Seven limit children= The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||. ==Orwell== Main article: [[Orwell]] So called because 19\84 (as a [[fraction of the octave]]) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma. The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything. ==Vital statistics== [[Comma|Commas]]: 225/224, 1728/1715 7-limit [|1 0 0 0>, |14/11 0 -7/11 7/11>, |27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>] [[Fractional monzos|Eigenmonzos]]: 2, 7/5 9-limit [|1 0 0 0>, |21/17 14/17 -7/17 0>, |42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 10/9 [[POTE tuning|POTE generator]]: 271.509 Algebraic generators: Sabra3, the real root of 12x^3-7x-48. Map: [<1 0 3 1|, <0 7 -3 8|] EDOs: 22, 31, 53, 84, 137 =11-limit= [[Comma|Commas]]: 99/98, 121/120, 176/175 [[Minimax tuning]] [|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>, |27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 7/5 [[POTE tuning|POTE generator]]: ~7/6 = 271.426 Map: [<1 0 3 1 3|, <0 7 -3 8 2|] [[edo|Edos]]: [[22edo|22]], [[31edo|31]], [[53edo|53]], [[84edo|84]] Badness: 99/98, 121/120, 176/175 =Winston= Commas: 66/65, 99/98, 105/104, 121/120 [[POTE tuning|POTE generator]]: ~7/6 = 271.088 Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|] EDOs: 9, 22, 31 Badness: 0.0199 =Julia= Commas: 99/98, 121/120, 176/175, 275/273 [[POTE tuning|POTE generator]]: ~7/6 = 271.546 Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|] EDOs: 9, 22, 31, 53, 137 Badness: 0.0197 =Borwell= Commas: 225/224, 243/242, 1728/1715 POTE generator: ~55/36 = 735.752 Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|] EDOs: 5, 8, 13, 18, 31, 106, 137 Badness: 0.0384 =Music= [[http://www.archive.org/details/TrioInOrwell|Trio in Orwell]] [[http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3|play]] by [[Gene Ward Smith]] [[http://soundclick.com/share?songid=9101705|one drop of rain]], [[http://soundclick.com/share?songid=9101704|i've come with a bucket of roses]], and [[http://soundclick.com/share?songid=8839071|my own house]] by [[Andrew Heathwaite]] [[http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3|Orwellian Cameras]] by [[Chris Vaisvil]]
Original HTML content:
<html><head><title>Semicomma family</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Seven limit children-Orwell">Orwell</a></div> <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 2em;"><a href="#Seven limit children-Vital statistics">Vital statistics</a></div> <!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 1em;"><a href="#x11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 1em;"><a href="#Winston">Winston</a></div> <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Julia">Julia</a></div> <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#Borwell">Borwell</a></div> <!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div> <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --></div> <!-- ws:end:WikiTextTocRule:25 -->The 5-limit parent comma for the <strong>semicomma family</strong> is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. <strong>Orson</strong>, the <a class="wiki_link" href="/5-limit">5-limit</a> temperament tempering it out, has a <a class="wiki_link" href="/generator">generator</a> of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example <a class="wiki_link" href="/53edo">53edo</a> or <a class="wiki_link" href="/84edo">84edo</a>. These give tunings to the generator which are sharp of 7/6 by less than five <a class="wiki_link" href="/cent">cent</a>s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.627<br /> <br /> Map: [<1 0 3|, <0 7 -3|]<br /> EDOs: 22, 31, 53, 190, 253, 296<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h1> 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. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Seven limit children-Orwell"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orwell</h2> Main article: <a class="wiki_link" href="/Orwell">Orwell</a><br /> So called because 19\84 (as a <a class="wiki_link" href="/fraction%20of%20the%20octave">fraction of the octave</a>) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53</a> and <a class="wiki_link" href="/84edo">84</a> equal, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the <a class="wiki_link" href="/7-limit">7-limit</a> and naturally extends into the <a class="wiki_link" href="/11-limit">11-limit</a>. <a class="wiki_link" href="/84edo">84edo</a>, with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. <a class="wiki_link" href="/53edo">53edo</a> might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.<br /> <br /> The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> <br /> Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has <a class="wiki_link" href="/Retuning%2012edo%20to%20Orwell9">considerable harmonic resources</a> despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Seven limit children-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:4 -->Vital statistics</h2> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 1728/1715<br /> <br /> 7-limit<br /> [|1 0 0 0>, |14/11 0 -7/11 7/11>,<br /> |27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>]<br /> <a class="wiki_link" href="/Fractional%20monzos">Eigenmonzos</a>: 2, 7/5<br /> <br /> 9-limit<br /> [|1 0 0 0>, |21/17 14/17 -7/17 0>,<br /> |42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.509<br /> Algebraic generators: Sabra3, the real root of 12x^3-7x-48.<br /> <br /> Map: [<1 0 3 1|, <0 7 -3 8|]<br /> EDOs: 22, 31, 53, 84, 137<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="x11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h1> <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 176/175<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> [|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>,<br /> |27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.426<br /> <br /> Map: [<1 0 3 1 3|, <0 7 -3 8 2|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/84edo">84</a><br /> Badness: 99/98, 121/120, 176/175<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Winston"></a><!-- ws:end:WikiTextHeadingRule:8 -->Winston</h1> Commas: 66/65, 99/98, 105/104, 121/120<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.088<br /> <br /> Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]<br /> EDOs: 9, 22, 31<br /> Badness: 0.0199<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Julia"></a><!-- ws:end:WikiTextHeadingRule:10 -->Julia</h1> Commas: 99/98, 121/120, 176/175, 275/273<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.546<br /> <br /> Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]<br /> EDOs: 9, 22, 31, 53, 137<br /> Badness: 0.0197<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Borwell"></a><!-- ws:end:WikiTextHeadingRule:12 -->Borwell</h1> Commas: 225/224, 243/242, 1728/1715<br /> <br /> POTE generator: ~55/36 = 735.752<br /> <br /> Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|]<br /> EDOs: 5, 8, 13, 18, 31, 106, 137<br /> Badness: 0.0384<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:14 -->Music</h1> <a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">Trio in Orwell</a> <a class="wiki_link_ext" href="http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a><br /> <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow">one drop of rain</a>, <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow">i've come with a bucket of roses</a>, and <a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow">my own house</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3" rel="nofollow">Orwellian Cameras</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></body></html>