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[[Minkowski lattice basis]]:  
[[Minkowski lattice basis]]:  
: ~6/5 length = 1.068, ~5/4 length = 1.206
: ~6/5 length = 1.068, ~5/4 length = 1.206
: angle (~6/5, ~5/4) = 100.364 degrees
: angle (~6/5, ~5/4) = 100.364°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}
: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 30: Line 36:
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}


[[Badness]] (Smith): 0.0699 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.308


[[Projection pair]]: <code>7 125/18</code>
[[Projection pair]]: <code>7 125/18</code>
Line 53: Line 59:
Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic family]]
* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]


Considered below are starling, thrush, thrasher, aplonis, and treecreeper.  
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.  
 
== Undecimal starling ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 385/384
 
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
 
[[Badness]] (Smith): 0.677 × 10<sup>-3</sup>


== Thrush ==
== Thrush ==
{{Main| Starling and thrush }}
{{Main| Starling and thrush }}
Thrush is the main extension of starling into the [[11-limit]]. It may be described as 12 & 15 & 31, and is loosely [[associated temperament|associated]] with [[myna]].


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 83: Line 80:
Lattice basis:  
Lattice basis:  
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239 degrees
: angle(~5/4, ~6/5) = 74.6239°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}
: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 92: Line 95:
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}


[[Badness]] (Smith): 0.353 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.424


[[Projection pair]]s: <code>7 125/18 11 3125/288</code>
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
 
[[Associated temperament]]: [[myna]]


=== 13-limit ===
=== 13-limit ===
Line 104: Line 105:


Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
Optimal tunings:
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}


{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}


Badness (Smith): 0.677 × 10<sup>-3</sup>
Badness (Sintel): 0.633


=== Bluebird ===
=== Bluebird ===
Line 118: Line 123:
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}


{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
Optimal tunings:
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}
 
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}


Badness (Smith): 0.915 × 10<sup>-3</sup>
Badness (Sintel): 0.856


Projection pairs: 7 125/18 11 3125/288 13 41472/3125
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>


=== Nightingale ===
=== Nightingale ===
Line 130: Line 139:


Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
Optimal tunings:
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}


{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}


Badness (Smith): 0.837 × 10<sup>-3</sup>
Badness (Sintel): 0.783


=== Veery ===
=== Veery ===
Line 141: Line 154:


Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
Optimal tunings:
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}


{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}


Badness (Smith): 0.991 × 10<sup>-3</sup>
Badness (Sintel): 0.927
 
== Starnova ==
Starnova tempers out [[385/384]]. This temperament used to be known as ''undecimal starling'', but was decanonicalized in 2026. The name ''starnova'' was proposed by [[Flora Canou]] after [[Gene Ward Smith]]'s [[Star]] and [[Nova]] scales.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 385/384
 
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
 
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
 
[[Badness]] (Sintel): 0.813
 
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
 
Scales: [[Star and Nova]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 385/384
 
Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
 
Badness (Sintel): 1.07


== Thrasher ==
== Thrasher ==
{{See also| Ptolemismic clan #Thrasher }}
{{See also| Ptolemismic clan }}


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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Lattice basis:  
Lattice basis:  
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
: angle (~6/5, ~5/4) = 98.8447
: angle (~6/5, ~5/4) = 98.8447°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9


{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
Line 170: Line 231:
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]


[[Badness]] (Smith): 0.480 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.577


Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
Line 180: Line 241:


Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
Optimal tunings:
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}


{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
Line 185: Line 250:
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]


Badness (Smith): 0.876 × 10<sup>-3</sup>
Badness (Sintel): 0.820


=== Mockingbird ===
=== Mockingbird ===
Line 193: Line 258:


Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
Optimal tunings:
* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}


{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}
{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}


Badness (Smith): 0.859 × 10<sup>-3</sup>
Badness (Sintel): 0.804


=== Catbird ===
=== Catbird ===
Line 204: Line 273:


Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
Optimal tunings:
* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}


{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}


Badness (Smith): 0.905 × 10<sup>-3</sup>
Badness (Sintel): 0.846


== Aplonis ==
== Aplonis ==
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{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}
: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}


{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}


[[Badness]] (Smith): 0.648 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.778


=== 13-limit ===
=== 13-limit ===
Line 226: Line 305:


Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
Optimal tunings:
* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}


{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}


Badness (Smith): 0.821 × 10<sup>-3</sup>
Badness (Sintel): 0.768


== Treecreeper ==
== Treecreeper ==
Line 237: Line 320:


{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}


{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}


[[Badness]] (Smith): 1.585 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.91


=== 13-limit ===
=== 13-limit ===
Line 248: Line 337:


Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
Optimal tunings:
* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}


{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}


Badness (Smith): 1.588 × 10<sup>-3</sup>
Badness (Sintel): 1.48


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Starling family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Starling_family"