Starling temperaments: Difference between revisions

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* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]

* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]

* [[Diminished (temperament)|Diminished]] (+36/35) → [[~~Dimipent~~ family #Diminished~~|Dimipent~~ family]]

* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]

* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]

* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]

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* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]

* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]

* ''[[Gilead]]'' (+343/324) → [[Shibboleth family #Gilead|Shibboleth family]]

* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]

* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]

* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #~~Diaschismic~~|Diaschismic family]]

* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]

* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]

* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]

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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 {{nowrap|27 & 31}} temperament. 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 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]] - [[6/5]] - [[49/40]] - [[5/4]] - [[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). 11-limit myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]), leaving space for a neutral third in between. In that sense, it is opposed to [[keemic temperaments]], where the chroma between the pental thirds is the same as the distance between the pental and septimal thirds.

In terms of commas tempered, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap|27 & 31}} temperament. 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 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~5/3

~~{{Multival|legend=1| 10 9 7 -9 -17 -9 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146

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* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}

: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

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Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}

~~{{Multival|legend=1| 10 9 7 25 -9 -17 5 -9 27 46 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144

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: mapping generators: ~2, ~49/45

~~{{Multival|legend=1| 11 13 17 -5 -4 3 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579

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

: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

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

: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

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* [[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 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11/9

: unchanged-interval (eigenmonzo) basis: 2.11/9

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

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~~{{Multival|legend=1| 17 18 20 -11 -16 -4 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679

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~~{{Multival|legend=1| 13 9 1 -16 -35 -23 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797

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~~{{Multival|legend=1| 12 17 27 -1 9 15 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828

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~~{{Multival|legend=1| 18 22 30 -7 -3 8 }}~~

[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445

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~~{{Multival|legend=1| 19 14 4 -22 -47 -30 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818

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Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}

~~{{Multival|legend=1| 19 14 4 1 -22 -47 -64 -30 -46 -11 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923

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Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}

~~{{Multival|legend=1| 19 14 4 32 -22 -47 -15 -30 26 76 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791

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~~{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094

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Badness: 0.030666

== Gilead ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 343/324

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1\1, ~6/5 = 321.109

* [[POTE]]: ~2 = 1\1, ~6/5 = 321.423

[[Badness]]: 0.115292

== Supersensi ==

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~~{{Multival|legend=1| 15 17 21 -8 -9 1 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568

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[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Starling temperaments| ]]

[[Category:Myna]]

[[Category:Rank 2]]

@@ Line 6: / Line 6: @@
 * ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
 * ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
-* [[Diminished (temperament)|Diminished]] (+36/35) → [[Dimipent family #Diminished|Dimipent family]]
+* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
 * [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
 * [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
@@ Line 12: / Line 12: @@
 * [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
 * [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
-* ''[[Gilead]]'' (+343/324) → [[Shibboleth family #Gilead|Shibboleth family]]
 * [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
 * [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
-* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Diaschismic|Diaschismic family]]
+* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
 * ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
 * ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
@@ Line 33: / Line 32: @@
 {{Main| Myna }}
-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 {{nowrap|27 &amp; 31}} temperament. 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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
+-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]] - [[6/5]] - [[49/40]] - [[5/4]] - [[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). 11-limit myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]), leaving space for a neutral third in between. In that sense, it is opposed to [[keemic temperaments]], where the chroma between the pental thirds is the same as the distance between the pental and septimal thirds.
+In terms of commas tempered, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap|27 &amp; 31}} temperament. 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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
 [[Subgroup]]: 2.3.5.7
@@ Line 42: / Line 43: @@
 : mapping generators: ~2, ~5/3
-{{Multival|legend=1| 10 9 7 -9 -17 -9 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146
@@ Line 50: / Line 49: @@
 * 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
 : {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
@@ Line 62: / Line 61: @@
 Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}
-{{Multival|legend=1| 10 9 7 25 -9 -17 5 -9 27 46 }}
 Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144
@@ Line 148: / Line 145: @@
 : mapping generators: ~2, ~49/45
-{{Multival|legend=1| 11 13 17 -5 -4 3 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579
@@ Line 156: / Line 151: @@
 * [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
 : {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 * [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
 : {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
@@ Line 177: / Line 172: @@
 * [[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 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
@@ Line 207: / Line 202: @@
 {{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}
-{{Multival|legend=1| 17 18 20 -11 -16 -4 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679
@@ Line 291: / Line 284: @@
 {{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}
-{{Multival|legend=1| 13 9 1 -16 -35 -23 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797
@@ Line 334: / Line 325: @@
 {{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}
-{{Multival|legend=1| 12 17 27 -1 9 15 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828
@@ Line 375: / Line 364: @@
 {{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
-{{Multival|legend=1| 18 22 30 -7 -3 8 }}
 [[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445
@@ Line 424: / Line 411: @@
 {{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}
-{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818
@@ Line 439: / Line 424: @@
 Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}
-{{Multival|legend=1| 19 14 4 1 -22 -47 -64 -30 -46 -11 }}
 Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923
@@ Line 465: / Line 448: @@
 Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}
-{{Multival|legend=1| 19 14 4 32 -22 -47 -15 -30 26 76 }}
 Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791
@@ Line 508: / Line 489: @@
 {{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
-{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094
@@ Line 542: / Line 521: @@
 Badness: 0.030666
+== Gilead ==
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 343/324
+{{Mapping|legend=1| 1 4 5 6 | 0 -9 -10 -12 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~6/5 = 321.109
+* [[POTE]]: ~2 = 1\1, ~6/5 = 321.423
+{{Optimal ET sequence|legend=1| 11cd, 15, 41dd, 56dd }}
+[[Badness]]: 0.115292
 == Supersensi ==
@@ Line 551: / Line 545: @@
 {{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
-{{Multival|legend=1| 15 17 21 -8 -9 1 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568
@@ Line 708: / Line 700: @@
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Starling temperaments| ]] <!-- main article -->
 [[Category:Myna]]
 [[Category:Rank 2]]