Semicomma family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2011-02-~~16 18~~:22:44 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-05-22 00:32:48 UTC</tt>.<br>

: The original revision id was <tt>~~202539020~~</tt>.<br>

: The original revision id was <tt>230636974</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>

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===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.

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.

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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

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Badness: 99/98, 121/120, 176/175

==Winston==

Commas: 66/65, 99/98, 105/104, 121/120

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Badness: 0.0199

==Julia==

Commas: 99/98, 121/120, 176/175, 275/273

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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]]

http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 by [[Chris Vaisvil]]</pre></div>

</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 />

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<br />

<h3 id="toc1"><a name="x-Seven limit children-Orwell"></a>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 />

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 />

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<br />

<h3 id="toc2"><a name="x-Seven limit children-Vital statistics"></a>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 />

[|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 />

[|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 />

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 />

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<br />

<h2 id="toc3"><a name="x-11-limit"></a>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 />

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<br />

<h2 id="toc4"><a name="x-Winston"></a>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 />

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<br />

<h2 id="toc5"><a name="x-Julia"></a>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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-02-16 18:22:44 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-05-22 00:32:48 UTC</tt>.<br>
-: The original revision id was <tt>202539020</tt>.<br>
+: The original revision id was <tt>230636974</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>
@@ Line 17: / Line 17: @@
 ===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&amp;31 temperament, or &lt;&lt;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.
+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&amp;31 temperament, or &lt;&lt;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.
@@ Line 23: / Line 23: @@
 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
 -limit
 [|1 0 0 0&gt;, |14/11 0 -7/11 7/11&gt;,
 |27/11 0 3/11 -3/11&gt;, |27/11 0 -8/11 8/11&gt;]
 [[Fractional monzos|Eigenmonzos]]: 2, 7/5
 -limit
 [|1 0 0 0&gt;, |21/17 14/17 -7/17 0&gt;,
 |42/17 -6/17 3/17 0&gt;, |41/17 16/17 -8/17 0&gt;]
 [[Eigenmonzo|Eigenmonzos]]: 2, 10/9
 [[POTE tuning|POTE generator]]: 271.509
 Algebraic generators: Sabra3, the real root of 12x^3-7x-48.
 Map: [&lt;1 0 3 1|, &lt;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&gt;, |14/11 0 -7/11 7/11 0&gt;, |27/11 0 3/11 -3/11 0&gt;,
- |27/11 0 -8/11 8/11 0&gt;, |37/11 0 -2/11 2/11 0&gt;]
+|27/11 0 -8/11 8/11 0&gt;, |37/11 0 -2/11 2/11 0&gt;]
 [[Eigenmonzo|Eigenmonzos]]: 2, 7/5
@@ Line 56: / Line 56: @@
 Badness: 99/98, 121/120, 176/175
 ==Winston==
 Commas: 66/65, 99/98, 105/104, 121/120
@@ Line 65: / Line 65: @@
 Badness: 0.0199
 ==Julia==
 Commas: 99/98, 121/120, 176/175, 275/273
@@ Line 77: / Line 77: @@
 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]]
+http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 by [[Chris Vaisvil]]</pre></div>
-</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Semicomma family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = |-21 3 7&amp;gt;. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. Orson, the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; temperament tempering it out, has a &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt; 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 &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; or &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;. These give tunings to the generator which are sharp of 7/6 by less than five &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.&lt;br /&gt;
@@ Line 91: / Line 90: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Seven limit children-Orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Orwell&lt;/h3&gt;
-  So called because 19/84 (as a &lt;a class="wiki_link" href="/fraction%20of%20the%20octave"&gt;fraction of the octave&lt;/a&gt;) 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 &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53-EDO&lt;/a&gt; and &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;, and may be described as the 22&amp;amp;31 temperament, or &amp;lt;&amp;lt;7 -3 8 -21 -7 27||. It's a good system in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and naturally extends into the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;. &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt;, 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. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; 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.&lt;br /&gt;
+  So called because 19\84 (as a &lt;a class="wiki_link" href="/fraction%20of%20the%20octave"&gt;fraction of the octave&lt;/a&gt;) 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 &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53-EDO&lt;/a&gt; and &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;, and may be described as the 22&amp;amp;31 temperament, or &amp;lt;&amp;lt;7 -3 8 -21 -7 27||. It's a good system in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and naturally extends into the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;. &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt;, 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. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;
@@ Line 98: / Line 97: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Seven limit children-Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Vital statistics&lt;/h3&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 1728/1715&lt;br /&gt;
+ &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 1728/1715&lt;br /&gt;
 &lt;br /&gt;
 -limit&lt;br /&gt;
-[|1 0 0 0&amp;gt;, |14/11 0 -7/11 7/11&amp;gt;, &lt;br /&gt;
+[|1 0 0 0&amp;gt;, |14/11 0 -7/11 7/11&amp;gt;,&lt;br /&gt;
 |27/11 0 3/11 -3/11&amp;gt;, |27/11 0 -8/11 8/11&amp;gt;]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/5&lt;br /&gt;
 &lt;br /&gt;
 -limit&lt;br /&gt;
-[|1 0 0 0&amp;gt;, |21/17 14/17 -7/17 0&amp;gt;, &lt;br /&gt;
+[|1 0 0 0&amp;gt;, |21/17 14/17 -7/17 0&amp;gt;,&lt;br /&gt;
 |42/17 -6/17 3/17 0&amp;gt;, |41/17 16/17 -8/17 0&amp;gt;]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 271.509&lt;br /&gt;
-Algebraic generators: Sabra3, the real root of 12x^3-7x-48. &lt;br /&gt;
+Algebraic generators: Sabra3, the real root of 12x^3-7x-48.&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;1 0 3 1|, &amp;lt;0 7 -3 8|]&lt;br /&gt;
@@ Line 117: / Line 116: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 99/98, 121/120, 176/175&lt;br /&gt;
+ &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 99/98, 121/120, 176/175&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
 [|1 0 0 0 0&amp;gt;, |14/11 0 -7/11 7/11 0&amp;gt;, |27/11 0 3/11 -3/11 0&amp;gt;,&lt;br /&gt;
- |27/11 0 -8/11 8/11 0&amp;gt;, |37/11 0 -2/11 2/11 0&amp;gt;]&lt;br /&gt;
+|27/11 0 -8/11 8/11 0&amp;gt;, |37/11 0 -2/11 2/11 0&amp;gt;]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/5&lt;br /&gt;
 &lt;br /&gt;
@@ Line 131: / Line 130: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Winston"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Winston&lt;/h2&gt;
-Commas: 66/65, 99/98, 105/104, 121/120&lt;br /&gt;
+ Commas: 66/65, 99/98, 105/104, 121/120&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.088&lt;br /&gt;
@@ Line 140: / Line 139: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Julia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Julia&lt;/h2&gt;
-Commas: 99/98, 121/120, 176/175, 275/273&lt;br /&gt;
+ Commas: 99/98, 121/120, 176/175, 275/273&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.546&lt;br /&gt;