Starling temperaments: Difference between revisions

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== 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|77EDO]], [[108edo|108EDO]] or [[185edo|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)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46EDO has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.

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[[Minimax tuning]]:

* [[7-odd-limit]]

* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}

: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]

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

* [[9-odd-limit]]

* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}

: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]

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

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Minimax tuning:

* 11-odd-limit

* [[11-odd-limit]]: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]

: Eigenmonzos: 2, 11/7

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[[Minimax tuning]]:

* [[7-odd-limit]]

* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}

: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]

: [[Eigenmonzo]]s: 2, 5

* [[9-odd-limit]]

* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}

: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]

: [[Eigenmonzo]]s: 2, 3

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Minimax tuning:

* 11-odd-limit

* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]

: Eigenmonzos: 2, 11/9

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Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [{{val| 1 -4 0 7 3 -7 12 1 5 3 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 4 }}]

@@ Line 142: / Line 142: @@
 == Valentine ==
 {{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&amp;46 temperament, and [[77edo|77EDO]], [[108edo|108EDO]] or [[185edo|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)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46EDO has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
@@ Line 172: / Line 171: @@
 [[Minimax tuning]]:
-* [[7-odd-limit]]
+* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
 : [[Eigenmonzo]]s: 2, 7/6
-* [[9-odd-limit]]
+* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
 : [[Eigenmonzo]]s: 2, 9/7
@@ Line 197: / Line 196: @@
 Minimax tuning:
-* 11-odd-limit
+* [[11-odd-limit]]: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
 : Eigenmonzos: 2, 11/7
@@ Line 359: / Line 358: @@
 [[Minimax tuning]]:
-* [[7-odd-limit]]
+* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]
 : [[Eigenmonzo]]s: 2, 5
-* [[9-odd-limit]]
+* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
 : [[Eigenmonzo]]s: 2, 3
@@ Line 382: / Line 381: @@
 Minimax tuning:
-* 11-odd-limit
+* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
 : Eigenmonzos: 2, 11/9
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 Subgroup: 2.3.5.7.11.13.17.19.23.29
+Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
 Mapping: [{{val| 1 -4 0 7 3 -7 12 1 5 3 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 4 }}]