Starling temperaments: Difference between revisions

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<h4>Original Wikitext content:</h4>

In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&amp;31 temperament, or in terms of its wedgie &lt;&lt;10 9 7 -9 -17 -9||. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

7 and 9 limit minimax

[|1 0 0 0&gt;, |0 1 0 0 &gt;, |9/10 9/10 0 0&gt;, |17/10 7/10 0 0&gt;]

Map: [&lt;1 9 9 8|, &lt;0 -10 -9 -7|]

===Sensi temperament===

Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.

7-limit minimax

[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&gt;]

9-limit minimax

[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,  

|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;]

Algebraic generator: Calista, the real root of x^7-2x^2-1, at 340.6467 cents.  

Map: [&lt;1 6 8 11|, &lt;0 -7 -9 -13|]

===Valentine temperament===

7-limit

7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,  

|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;]

9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,  

|47/21 10/21 0 -5/21&gt;, |20/7 -2/7 0 1/7&gt;]

Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]

11-limit

[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,  

|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;]

Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.

Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]

===Casablanca temperament===

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;70, or in terms of its wedgie as &lt;&lt;11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo]] can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.

7-limit minimax

[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]

9-limit minimax

[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]

Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]

====11-limit====

11-limit minimax

|27/10 13/5 0 0 -13/10&gt;, |33/10 17/5 0 0 -17/10&gt;,  

|19/5 12/5 0 0 -6/5&gt;]

Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]

<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;Starling temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.&lt;br /&gt;

In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&amp;amp;31 temperament, or in terms of its wedgie &amp;lt;&amp;lt;10 9 7 -9 -17 -9||. It has 6/5 as a generator, and &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; can be used as a tuning, with &lt;a class="wiki_link" href="/89edo"&gt;89edo&lt;/a&gt; being a better one, and fans of round amounts in cents may like &lt;a class="wiki_link" href="/120edo"&gt;120edo&lt;/a&gt;. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

7 and 9 limit minimax&lt;br /&gt;

[|1 0 0 0&amp;gt;, |0 1 0 0 &amp;gt;, |9/10 9/10 0 0&amp;gt;, |17/10 7/10 0 0&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

Map: [&amp;lt;1 9 9 8|, &amp;lt;0 -10 -9 -7|]&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--Sensi temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Sensi temperament&lt;/h3&gt;

Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt; is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

7-limit minimax&lt;br /&gt;

[|1 0 0 0&amp;gt;, |1/13 0 0 7/13&amp;gt;, |5/13 0 0 9/13&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

9-limit minimax&lt;br /&gt;

[|1 0 0 0&amp;gt;, |2/5 14/5 -7/5 0&amp;gt;, &lt;br /&gt;

|4/5 18/5 -9/5 0&amp;gt;, |3/5 26/5 -13/5 0&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

Algebraic generator: Calista, the real root of x^7-2x^2-1, at 340.6467 cents. &lt;br /&gt;

&lt;br /&gt;

Map: [&amp;lt;1 6 8 11|, &amp;lt;0 -7 -9 -13|]&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x--Valentine temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Valentine temperament&lt;/h3&gt;

&lt;br /&gt;

7-limit&lt;br /&gt;

&lt;br /&gt;

7-limit: [|1 0 0 0&amp;gt;, |5/2 3/4 0 -3/4&amp;gt;, &lt;br /&gt;

|17/6 5/12 0 -5/12&amp;gt;, [5/2 -1/4 0 1/4&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

9-limit: [|1 0 0 0&amp;gt;, |10/7 6/7 0 -3/7&amp;gt;, &lt;br /&gt;

|47/21 10/21 0 -5/21&amp;gt;, |20/7 -2/7 0 1/7&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

Map: [&amp;lt;1 1 2 3|, &amp;lt;0 9 5 -3|]&lt;br /&gt;

&lt;br /&gt;

11-limit&lt;br /&gt;

&lt;br /&gt;

[|1 0 0 0 0&amp;gt;, |1 0 0 -9/10 9/10&amp;gt;, &lt;br /&gt;

|2 0 0 -1/2 1/2&amp;gt;, |3 0 0 3/10 -3/10&amp;gt;, |3 0 0 -7/10 7/10&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.&lt;br /&gt;

&lt;br /&gt;

Map: [&amp;lt;1 1 2 3 3|, &amp;lt;0 9 5 -3 7|]&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x--Casablanca temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Casablanca temperament&lt;/h3&gt;

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;amp;70, or in terms of its wedgie as &amp;lt;&amp;lt;11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

7-limit minimax&lt;br /&gt;

[|1 0 0 0&amp;gt;, |-5/13 0 11/13 0&amp;gt;, |0 0 1 0&amp;gt;, |-3/13 0 17/13 0&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

9-limit minimax&lt;br /&gt;

[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |5/11 13/11 0 0&amp;gt;, |4/11 17/11 0 0&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

Map: [&amp;lt;1 3 4 5|, &amp;lt;0 -11 -13 -17|]&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc5"&gt;&lt;a name="x--Nusecond temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h4&gt;

&lt;br /&gt;

11-limit minimax&lt;br /&gt;

|27/10 13/5 0 0 -13/10&amp;gt;, |33/10 17/5 0 0 -17/10&amp;gt;, &lt;br /&gt;

|19/5 12/5 0 0 -6/5&amp;gt;]&lt;br /&gt;

&lt;br /&gt;

Map: [&amp;lt;1 3 4 5 5|, &amp;lt;0 -11 -13 -17 -12|]&lt;br /&gt;