Semicomma 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 2011-02-16 18:22:44 UTC.

The original revision id was 202539020.

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 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=== 
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-EDO]] and [[84edo]], 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 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

==Music== 
http://www.archive.org/details/TrioInOrwell 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 by [[Chris Vaisvil]]

Original HTML content:

<html><head><title>Semicomma family</title></head><body>The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = |-21 3 7&gt;. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. Orson, 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: [&lt;1 0 3|, &lt;0 7 -3|]<br />
EDOs: 22, 31, 53, 190, 253, 296<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><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. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;159 temperament with wedgie &lt;&lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;243 temperament with wedgie &lt;&lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;243 temperament with wedgie &lt;&lt;7 -3 61 -21 77 150||.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Seven limit children-Orwell"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orwell</h3>
 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-EDO</a> and <a class="wiki_link" href="/84edo">84edo</a>, and may be described as the 22&amp;31 temperament, or &lt;&lt;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 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Seven limit children-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:4 -->Vital statistics</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 1728/1715<br />
<br />
7-limit<br />
[|1 0 0 0&gt;, |14/11 0 -7/11 7/11&gt;, <br />
|27/11 0 3/11 -3/11&gt;, |27/11 0 -8/11 8/11&gt;]<br />
<a class="wiki_link" href="/Fractional%20monzos">Eigenmonzos</a>: 2, 7/5<br />
<br />
9-limit<br />
[|1 0 0 0&gt;, |21/17 14/17 -7/17 0&gt;, <br />
|42/17 -6/17 3/17 0&gt;, |41/17 16/17 -8/17 0&gt;]<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: [&lt;1 0 3 1|, &lt;0 7 -3 8|]<br />
EDOs: 22, 31, 53, 84, 137<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2>
<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&gt;, |14/11 0 -7/11 7/11 0&gt;, |27/11 0 3/11 -3/11 0&gt;,<br />
 |27/11 0 -8/11 8/11 0&gt;, |37/11 0 -2/11 2/11 0&gt;]<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: [&lt;1 0 3 1 3|, &lt;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:&lt;h2&gt; --><h2 id="toc4"><a name="x-Winston"></a><!-- ws:end:WikiTextHeadingRule:8 -->Winston</h2>
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: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]<br />
EDOs: 9, 22, 31<br />
Badness: 0.0199<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Julia"></a><!-- ws:end:WikiTextHeadingRule:10 -->Julia</h2>
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: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]<br />
EDOs: 9, 22, 31, 53, 137<br />
Badness: 0.0197<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->Music</h2>
 <!-- ws:start:WikiTextUrlRule:116:http://www.archive.org/details/TrioInOrwell --><a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">http://www.archive.org/details/TrioInOrwell</a><!-- ws:end:WikiTextUrlRule:116 --> 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 />
<!-- ws:start:WikiTextUrlRule:117:http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 --><a class="wiki_link_ext" href="http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3" rel="nofollow">http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3</a><!-- ws:end:WikiTextUrlRule:117 --> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></body></html>