Semicomma family

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[[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, 243, 296, 645c

=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.

=Orwell=
[[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]]: ~7/6 = 271.509
Algebraic generators: Sabra3, the real root of 12x^3-7x-48.

Map: [<1 0 3 1|, <0 7 -3 8|]
Wedgie: <<7 -3 8 -21 -7 27||
EDOs: 22, 31, 53, 84, 137, 221d, 358d
Badness: 0.0207

==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|84e]]
Badness: 0.0152

==13-limit==
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: 22, 31, 53, 84e, 137e
Badness: 0.0197

==Blair==
Commas: 65/64, 78/77, 91/90, 99/98

POTE generator: ~7/6 = 271.301

Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]
EDOs: 9, 22, 31f
Badness: 0.0231

==Newspeak==
Commas: 225/224, 441/440, 1728/1715

POTE tuning: ~7/6 = 271.288

Map: [<1 0 3 1 -4|, <0 7 -3 8 33|]
EDOs: 31, 84, 115, 376b, 491bd, 606bde
Badness: 0.0314

==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: 22f, 31
Badness: 0.0199

=Doublethink=
Commas: 99/98, 121/120, 169/168, 176/175

POTE tuning: ~13/12 = 135.723

Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]
EDOs: 9, 44, 53, 115ef, 168ef
Badness: 0.0271

=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: 31, 106, 137, 442bd
Badness: 0.0384

=Triwell=
Commas: 1029/1024, 235298/234375

POTE generator: ~448/375 = 309.472

Map: [<1 7 0 1|, <0 -21 9 7]]
Wedgie: <<21 -9 -7 -63 -70 9||
EDOs: 31, 97, 128, 159, 190
Badness: 0.0806

==11-limit==
Commas: 385/384, 441/440, 456533/455625

POTE generator: ~448/375 = 309.471

Map: [<1 7 0 1 13|, <0 -21 9 7 -37]]
EDOs: 31, 97, 128, 159, 190
Badness: 0.0298

=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:26:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div>
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 2em;"><a href="#Seven limit children-Orwell">Orwell</a></div>
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 1em;"><a href="#Orwell">Orwell</a></div>
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Orwell-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#Orwell-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Orwell-Blair">Blair</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 2em;"><a href="#Orwell-Newspeak">Newspeak</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Orwell-Winston">Winston</a></div>
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 1em;"><a href="#Doublethink">Doublethink</a></div>
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Borwell">Borwell</a></div>
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#Triwell">Triwell</a></div>
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 2em;"><a href="#Triwell-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --></div>
<!-- ws:end:WikiTextTocRule:40 -->The 5-limit parent comma for the <strong>semicomma family</strong> 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. <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: [&lt;1 0 3|, &lt;0 7 -3|]<br />
EDOs: 22, 31, 53, 190, 243, 296, 645c<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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&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;h2&gt; --><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&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 <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:&lt;h1&gt; --><h1 id="toc2"><a name="Orwell"></a><!-- ws:end:WikiTextHeadingRule:4 -->Orwell</h1>
<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>: ~7/6 = 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 />
Wedgie: &lt;&lt;7 -3 8 -21 -7 27||<br />
EDOs: 22, 31, 53, 84, 137, 221d, 358d<br />
Badness: 0.0207<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Orwell-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">84e</a><br />
Badness: 0.0152<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Orwell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</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: 22, 31, 53, 84e, 137e<br />
Badness: 0.0197<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Orwell-Blair"></a><!-- ws:end:WikiTextHeadingRule:10 -->Blair</h2>
Commas: 65/64, 78/77, 91/90, 99/98<br />
<br />
POTE generator: ~7/6 = 271.301<br />
<br />
Map: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]<br />
EDOs: 9, 22, 31f<br />
Badness: 0.0231<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Orwell-Newspeak"></a><!-- ws:end:WikiTextHeadingRule:12 -->Newspeak</h2>
Commas: 225/224, 441/440, 1728/1715<br />
<br />
POTE tuning: ~7/6 = 271.288<br />
<br />
Map: [&lt;1 0 3 1 -4|, &lt;0 7 -3 8 33|]<br />
EDOs: 31, 84, 115, 376b, 491bd, 606bde<br />
Badness: 0.0314<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Orwell-Winston"></a><!-- ws:end:WikiTextHeadingRule:14 -->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: 22f, 31<br />
Badness: 0.0199<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Doublethink"></a><!-- ws:end:WikiTextHeadingRule:16 -->Doublethink</h1>
Commas: 99/98, 121/120, 169/168, 176/175<br />
<br />
POTE tuning: ~13/12 = 135.723<br />
<br />
Map: [&lt;1 0 3 1 3 2|, &lt;0 14 -6 16 4 15|]<br />
EDOs: 9, 44, 53, 115ef, 168ef<br />
Badness: 0.0271<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Borwell"></a><!-- ws:end:WikiTextHeadingRule:18 -->Borwell</h1>
 Commas: 225/224, 243/242, 1728/1715<br />
<br />
POTE generator: ~55/36 = 735.752<br />
<br />
Map: [&lt;1 7 0 9 17|, &lt;0 -14 6 -16 -35|]<br />
EDOs: 31, 106, 137, 442bd<br />
Badness: 0.0384<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Triwell"></a><!-- ws:end:WikiTextHeadingRule:20 -->Triwell</h1>
Commas: 1029/1024, 235298/234375<br />
<br />
POTE generator: ~448/375 = 309.472<br />
<br />
Map: [&lt;1 7 0 1|, &lt;0 -21 9 7]]<br />
Wedgie: &lt;&lt;21 -9 -7 -63 -70 9||<br />
EDOs: 31, 97, 128, 159, 190<br />
Badness: 0.0806<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Triwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2>
Commas: 385/384, 441/440, 456533/455625<br />
<br />
POTE generator: ~448/375 = 309.471<br />
<br />
Map: [&lt;1 7 0 1 13|, &lt;0 -21 9 7 -37]]<br />
EDOs: 31, 97, 128, 159, 190<br />
Badness: 0.0298<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:24 -->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>