Starling temperaments: Difference between revisions

=Starling comma=  

This page discusses some of the rank two temperaments tempering out [[126_125|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 the [[starling tetrad]], the 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 [[12edo]] 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.

=Myna temperament=  

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&31 temperament, or in terms of its wedgie <<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.

=5-limit (Mynic)=  

==7-limit==  

[[POTE tuning|POTE generator]]: 310.146

[[Generator|Generators]]: 2, 5/3

==11-limit==  

[[POTE tuning|POTE generator]]: ~6/5 = 310.144

==13-limit==  

[[POTE tuning|POTE generator]]: ~6/5 = 310.276

==Minah==  

==Maneh==  

==Myno==  

==Coleto==  

//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3|Myna Music]]// by [[Igliashon Jones]]

=Sensi temperament=  

Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&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.

[[POTE tuning|POTE generator]]: ~9/7 = 443.383

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

[[Generator|Generators]]: 2, 14/9

==Sensor==  

[[POTE tuning|POTE generator]]: ~9/7 = 443.294

===13-limit===  

[[POTE tuning|POTE generator]]: ~9/7 = 443.321

==Sensis==  

[[POTE tuning|POTE generator]]: 443.962

===13-limit===  

[[POTE tuning|POTE generator]]: 443.945

==Sensus==  

===13-limit===  

=Valentine temperament=  

[[Valentine]] tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31&46 temperament, and [[77edo]], [[108edo]] or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as <<9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).

Valentine is very closely related to [[Carlos Alpha]], the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in //Beauty in the Beast// suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.

[[Minimax tuning]]:

[[POTE tuning|POTE generator]]: 77.864

Algebraic generator: [[Algebraic number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.

[[Generator|Generators]]: 2, 21/20

==11-limit==  

[[Minimax tuning]]:

[[POTE tuning|POTE generator]]: 77.881

[[Edo|Edos]]: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]]

See also: [[Chords of valentine]]

==Dwynwen==  

==Lupercalia==  

==Valentino==  

==Semivalentine==  

=Alicorn temperament=  

==11-limit==  

==13-limit==  

==Camahueto==  

===13-limit===  

 

=Casablanca temperament=  

==11-limit==  

==Marrakesh==  

===13-limit===  

===Murakuc===  

=Nusecond temperament=  

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70, or in terms of its wedgie as <<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 val|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.

==5-limit==  

==7-limit==  

[[POTE tuning|POTE generator]]: 154.579

[[Generator|Generators]]: 2, 49/45

==11-limit==  

[[POTE tuning|POTE generator]]: ~11/10 = 154.645

Algebraic generator: [[Algebraic number|positive root]] of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.

[[Generator|Generators]]: 2, 11/10

==13-limit==  

=Thuja=  

==11-limit==  

==13-limit==  

==29-limit==  

=Cypress=  

==7-limit==  

==11-limit==  

==13-limit==  

=Bisemidim=  

==11-limit==  

==13-limit==  

=Kumonga=  

==7-limit==  

==11-limit==  

==13-limit==  

Badness: 0.0307</pre></div>

Map: [&amp;lt;1 9 9|, &amp;lt;0 -10 -9|]&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;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 3&lt;br /&gt;

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

Map: [&amp;lt;1 9 9 8 22|, &amp;lt;0 -10 -9 -7 -25|]&lt;br /&gt;

Map: [&amp;lt;1 9 9 8 22 0|, &amp;lt;0 -10 -9 -7 -25 5|]&lt;br /&gt;

Map: [&amp;lt;1 9 9 8 22 20|, &amp;lt;0 -10 -9 -7 -25 -22|]&lt;br /&gt;

Badness: 0.0276&lt;br /&gt;

Map: [&amp;lt;1 9 9 8 22 23|, &amp;lt;0 -10 -9 -7 -25 -26|]&lt;br /&gt;

Map: [&amp;lt;1 9 9 8 -1|, &amp;lt;0 -10 -9 -7 6|]&lt;br /&gt;

Map: [&amp;lt;1 9 9 8 2|, &amp;lt;0 -10 -9 -7 2|]&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;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7&lt;br /&gt;

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

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

Map: [&amp;lt;1 6 8 11 -6|, &amp;lt;0 -7 -9 -13 15|]&lt;br /&gt;

Map: [&amp;lt;1 6 8 11 -6 10|, &amp;lt;0 -7 -9 -13 15 -10|]&lt;br /&gt;

Map: [&amp;lt;1 6 8 11 6|, &amp;lt;0 -7 -9 -13 -4|]&lt;br /&gt;

Map: [&amp;lt;1 6 8 11 6 10|, &amp;lt;0 -7 -9 -13 -4 -10|]&lt;br /&gt;

Map: [&amp;lt;1 6 8 11 23|, &amp;lt;0 -7 -9 -13 -31|]&lt;br /&gt;

Map: [&amp;lt;1 6 8 11 23 10|, &amp;lt;0 -7 -9 -13 -31 -10|]&lt;br /&gt;

EDOs: 8, 19, 27, 46, 303&lt;br /&gt;

&lt;a class="wiki_link" href="/Valentine"&gt;Valentine&lt;/a&gt; tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice of 7-limit tetrads&lt;/a&gt;. Valentine can also be described as the 31&amp;amp;46 temperament, and &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108edo&lt;/a&gt; or &lt;a class="wiki_link" href="/185edo"&gt;185edo&lt;/a&gt; make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as &amp;lt;&amp;lt;9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).&lt;br /&gt;

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

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

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

&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/7&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Valentine temperament-Dwynwen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Dwynwen&lt;/h2&gt;

Map: [&amp;lt;1 1 2 3 3 2|, &amp;lt;0 9 5 -3 7 26|]&lt;br /&gt;

Map: [&amp;lt;1 1 2 3 3 3|, &amp;lt;0 9 5 -3 7 11|]&lt;br /&gt;

Map: [&amp;lt;1 1 2 3 3 5|, &amp;lt;0 9 5 -3 7 -20|]&lt;br /&gt;

Map: [&amp;lt;2 2 4 6 6 7|, &amp;lt;0 9 5 -3 7 3|]&lt;br /&gt;

EDOs: 16, 30, 46, 62, 108ef&lt;br /&gt;