Starling temperaments: Difference between revisions

{{main|Valentine}}

{{see also|Gamelismic clan #Valentine}}

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]]:

7-limit: [|1 0 0 0>, |5/2 3/4 0 -3/4>,

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

[[Eigenmonzo]]s: 2, 7/6

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;<nowiki>]</nowiki>

[[Eigenmonzo]]s: 2, 9/7

[[POTE_tuning|POTE generator]]: ~21/20 = 77.864

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

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

Mapping [[generator]]s: 2, 21/20

EDOs: {{EDOs|15, 31, 46, 77, 185, 262cd}}

Badness: 0.0311

[[Comma]]s: 121/120, 126/125, 176/175

[|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;<nowiki>]</nowiki>

[[Eigenmonzo]]s: 2, 11/7

Minimax generator: (11/7)^(1/10) = 78.249

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|]

[[EDO]]s: {{EDOs|15, 31, 46, 77, 262cdee, 339cdeee}}