Semicomma family: Difference between revisions

Wikispaces>Andrew_Heathwaite
**Imported revision 151402009 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 164175691 - 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-07-02 17:04:47 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-21 04:20:41 UTC</tt>.<br>
: The original revision id was <tt>151402009</tt>.<br>
: The original revision id was <tt>164175691</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
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==Seven limit children==  
==Seven limit children==  
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 64625/65536 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||.
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&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||.


===Orwell===  
===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 . 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. [[53edo]] may 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. [[53edo]] may 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.
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.
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===
Commas: 225/224, 1728/1715
7-limit
Eigenmonzos: 2, 7/5
9-limit
Eigenmonzos: 2, 10/9
11-limit
Commas: 99/98, 121/120, 176/175
Minimax tuning
Eigenmonzos: 2, 7/5
Map: [&lt;1 0 3 1 3|, &lt;0 7 -3 8 2|]
Edos: 31, 53, 84


==Music==  
==Music==  
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
  The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding 64625/65536 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;amp;159 temperament with wedgie &amp;lt;&amp;lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;7 -3 61 -21 77 150||.&lt;br /&gt;
  The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; 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;amp;159 temperament with wedgie &amp;lt;&amp;lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;7 -3 61 -21 77 150||.&lt;br /&gt;
&lt;br /&gt;
&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;
&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 . 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. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; may 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. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; may 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;
&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;
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;
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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.&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Music&lt;/h2&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;!-- ws:start:WikiTextUrlRule:35:http://www.archive.org/details/TrioInOrwell --&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow"&gt;http://www.archive.org/details/TrioInOrwell&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:35 --&gt;&lt;br /&gt;
Commas: 225/224, 1728/1715&lt;br /&gt;
&lt;br /&gt;
7-limit&lt;br /&gt;
Eigenmonzos: 2, 7/5&lt;br /&gt;
&lt;br /&gt;
9-limit&lt;br /&gt;
Eigenmonzos: 2, 10/9&lt;br /&gt;
&lt;br /&gt;
11-limit&lt;br /&gt;
Commas: 99/98, 121/120, 176/175&lt;br /&gt;
&lt;br /&gt;
Minimax tuning&lt;br /&gt;
Eigenmonzos: 2, 7/5&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 3 1 3|, &amp;lt;0 7 -3 8 2|]&lt;br /&gt;
Edos: 31, 53, 84&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Music&lt;/h2&gt;
  &lt;!-- ws:start:WikiTextUrlRule:54:http://www.archive.org/details/TrioInOrwell --&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow"&gt;http://www.archive.org/details/TrioInOrwell&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:54 --&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow"&gt;one drop of rain&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow"&gt;i've come with a bucket of roses&lt;/a&gt;, and &lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow"&gt;my own house&lt;/a&gt; by Andrew Heathwaite&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow"&gt;one drop of rain&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow"&gt;i've come with a bucket of roses&lt;/a&gt;, and &lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow"&gt;my own house&lt;/a&gt; by Andrew Heathwaite&lt;/body&gt;&lt;/html&gt;</pre></div>