Starling family: Difference between revisions

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The head of the '''starling family''' is starling, which tempers out [[126/125]], the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and the nonpatent [[135edo]] val {{val| 135 214 314 379 }}.
{{Technical data page}}


In starling, (6/5)<sup>3</sup> = 126/125 × 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. 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.
The '''starling family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[starling comma]] ([[ratio]]: 126/125, {{monzo|legend=1| 1 2 -3 1 }}), also known as ''septimal semicomma''.  


Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[Starling temperaments #Valentine|valentine temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
== Starling ==
{{Main| Starling and thrush }}


== Starling ==
The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list of {2, 3, 5}; hence a [[5-limit]] scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and [[135edo]] in the nonpatent 135c val {{val| 135 214 '''314''' 379 }}.
Subgroup: 2.3.5.7
 
Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[126/125]]
[[Comma list]]: [[126/125]]


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5


Mapping generators: ~2, ~3, ~5
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]


Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
[[Minkowski lattice basis]]:  
: ~6/5 length = 1.068, ~5/4 length = 1.206
: angle (~6/5, ~5/4) = 100.364°


Minkowski lattice basis:  
[[Optimal tuning]]s:  
: 6/5 length = 1.068, 5/4 length = 1.206
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
: Angle (6/5, 5/4) = 100.364 degrees
: [[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]]:  
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3c sharp
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
: [{{val| 1 0 0 0 }}, {{val| 0 1 0 0 }}, {{val| 1/3 2/3 0 1/3 }}, {{val| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
: Eigenmonzos: 2, 8/7, 4/3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


{{Val list|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]]: 0.0699 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.308


[[Projection pair]]: 7 125/18
[[Projection pair]]: <code>7 125/18</code>
 
Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
 
; Music
* [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by [[Jake Freivald]]. The melody depends on tempering out 126/125.
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">


{{Databox|[[Minkowski blocks]]|
* 7: 25/24, 81/80
* 7: 25/24, 81/80
* 8: 16/15, 648/625
* 8: 16/15, 648/625
Line 53: Line 54:
* 31: 81/80, 1990656/1953125
* 31: 81/80, 1990656/1953125
* 34: 15625/15552, 2048/2025
* 34: 15625/15552, 2048/2025
}}


</div></div>
=== Overview to extensions ===
Temperaments discussed elsewhere include
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]


== Undecimal starling ==
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.  
Subgroup: 2.3.5.7.11


[[Comma list]]: 126/125, 385/384
== Thrush ==
 
{{Main| Starling and thrush }}
[[Mapping]]: [{{val| 1 0 0 -1 8 }}, {{val| 0 1 0 -2 3 }}, {{val| 0 0 1 3 -4]]


Mapping generators: ~2, ~3, ~5
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]].


{{Val list|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
[[Subgroup]]: 2.3.5.7.11
 
[[Badness]]: 0.677 × 10<sup>-3</sup>
 
== Thrush ==
Subgroup: 2.3.5.7.11


[[Comma list]]: 126/125, 176/175
[[Comma list]]: 126/125, 176/175


[[Mapping]]: [{{val| 1 0 0 -1 -5 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 5 }}]
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}


Mapping generators: ~2, ~3, ~5
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]


Map to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
Lattice basis:  
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239°


Lattice basis:  
[[Optimal tuning]]s:  
: 5/4 length = 0.8576, 6/5 length = 0.9314
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
: Angle(5/4, 6/5) = 74.6239 degrees
: [[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]]:  
* 7- and [[9-odd-limit]]
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
: [[Eigenmonzo]]s: 2, 7/6, 4/3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
 
{{Val list|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
 
[[Badness]]: 0.353 × 10<sup>-3</sup>


[[Projection pair]]s: 7 125/18 11 3125/288
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}


[[Associated temperament]]: [[Starling temperaments #Myna|myna]]
[[Badness]] (Sintel): 0.424


Scales: [[thrush12]]
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>


=== 13-limit ===
=== 13-limit ===
Line 104: Line 104:
Comma list: 126/125, 176/175, 196/195
Comma list: 126/125, 176/175, 196/195


Mapping: [{{val| 1 0 0 -1 -5 0 }}, {{val| 0 1 0 -2 -2 -5 }}, {{val| 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}}


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


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


=== Bluebird ===
=== Bluebird ===
{{Distinguish| Bluebirds }}
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 144/143, 176/175
Comma list: 126/125, 144/143, 176/175


Mapping: [{{val| 1 0 0 -1 -5 9 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 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 }}


Vals: {{Val list| 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}}


Badness: 0.915 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}


Projection pairs: 7 125/18 11 3125/288 13 41472/3125
Badness (Sintel): 0.856
 
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>


=== Nightingale ===
=== Nightingale ===
Line 128: Line 138:
Comma list: 66/65, 126/125, 176/175
Comma list: 66/65, 126/125, 176/175


Mapping: [{{val| 1 0 0 -1 -5 -4 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 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}}


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


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


=== Veery ===
=== Veery ===
Line 139: Line 153:
Comma list: 91/90, 126/125, 176/175
Comma list: 91/90, 126/125, 176/175


Mapping: [{{val| 1 0 0 -1 -5 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 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 }}
 
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


Vals: {{Val list| 12, 15, 19e, 27e, 31f, 46 }}
Comma list: 126/125, 196/195, 385/384


Badness: 0.991 × 10<sup>-3</sup>
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 ==
Subgroup: 2.3.5.7.11
{{See also| Ptolemismic clan }}
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 56/55, 100/99
[[Comma list]]: 56/55, 100/99


[[Mapping]]: [{{val| 1 0 0 -1 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 2 }}]
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}


Mapping generators: ~2, ~3, ~5
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]


Map to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
Lattice basis:  
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
: angle (~6/5, ~5/4) = 98.8447°


Lattice basis:  
[[Optimal tuning]]s:  
: 6/5 length = 0.9089, 5/4 length = 1.2007
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
: Angle (6/5, 5/4) = 98.8447
: [[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]]s: 2, 8/7, 11/9
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9


{{Val list|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *


<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]


[[Badness]]: 0.480 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.577
 
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 100/99, 126/125
Comma list: 56/55, 91/90, 100/99


Mapping: [{{val| 1 0 0 -1 2 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 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 }}


Vals: {{Val list| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
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 }} *


<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]


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


=== Mockingbird ===
=== Mockingbird ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 100/99, 126/125
Comma list: 40/39, 56/55, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}


Mapping: [{{val| 1 0 0 -1 2 3 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 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}}


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


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


=== Catbird ===
=== Catbird ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 100/99, 126/125
Comma list: 56/55, 78/77, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}


Mapping: [{{val| 1 0 0 -1 2 0 }}, {{val| 0 1 0 -2 -2 -5 }}, {{val| 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}}


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


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


== Aplonis ==
== Aplonis ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 540/539
[[Comma list]]: 126/125, 540/539


[[Mapping]]: [{{val| 1 0 0 -1 4 }}, {{val| 0 1 0 -2 7 }}, {{val| 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 }}


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


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


=== 13-limit ===
=== 13-limit ===
Line 222: Line 304:
Comma list: 126/125, 144/143, 196/195
Comma list: 126/125, 144/143, 196/195


Mapping: [{{val| 1 0 0 -1 4 0 }}, {{val| 0 1 0 -2 7 -5 }}, {{val| 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 }}
 
Vals: {{Val list| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
 
Badness: 0.821 × 10<sup>-3</sup>
 
== Oxpecker ==
Subgroup: 2.3.5.7.11


[[Comma list]]: 121/120, 126/125
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}}


[[Mapping]]: [{{val| 1 0 1 2 2 }}, {{val| 0 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 }}]
{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}


{{Val list|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
Badness (Sintel): 0.768
 
[[Badness]]: 0.699 × 10<sup>-3</sup>
 
=== Woodpecker ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 121/120, 126/125
 
Mapping: [{{val| 1 0 1 2 2 2 }}, {{val| 0 1 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 -1 }}]
 
Vals: {{Val list| 7d, 8d, 15, 23de, 24d, 31 }}
 
Badness: 1.093 × 10<sup>-3</sup>


== Treecreeper ==
== Treecreeper ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 1232/1215
[[Comma list]]: 126/125, 1232/1215


[[Mapping]]: [{{val| 1 0 0 -1 -3 }}, {{val| 0 1 0 -2 7 }}, {{val| 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 }}


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


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


=== 13-limit ===
=== 13-limit ===
Line 266: Line 336:
Comma list: 91/90, 126/125, 352/351
Comma list: 91/90, 126/125, 352/351


Mapping: [{{val| 1 0 0 -1 -3 2 }}, {{val| 0 1 0 -2 7 4 }}, {{val| 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 }}
 
Vals: {{Val list| 7d, 12e, 19e, 27e, 46 }}
 
Badness: 1.588 × 10<sup>-3</sup>
 
== Cuckoo ==
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 126/125, 243/242
 
[[Mapping]]: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 0 -4 5 }}, {{val| 0 0 1 3 0 }}]
 
{{Val list|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
 
[[Badness]]: 0.933 × 10<sup>-3</sup>
 
=== 13-limit  ===
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 196/195, 243/242
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}}


Mapping: [{{val| 1 1 0 -3 2 -5 }}, {{val| 0 2 0 -4 5 -10 }}, {{val| 0 0 1 3 0 5 }}]
{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}


Vals: {{Val list| 27e, 31, 58, 96d, 154 }}
Badness (Sintel): 1.48


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Starling family| ]] <!-- main article -->
[[Category:Thrush]]
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]

Latest revision as of 12:46, 4 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The starling family of rank-3 temperaments tempers out the starling comma (ratio: 126/125, monzo: [1 2 -3 1⟩), also known as septimal semicomma.

Starling

Main article: Starling and thrush

The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list of {2, 3, 5}; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by 77edo. Other possible tunings are 108edo and 185edo, and 135edo in the nonpatent 135c val ⟨135 214 314 379].

Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to valentine, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.

Subgroup: 2.3.5.7

Comma list: 126/125

Mapping: [⟨1 0 0 -1], ⟨0 1 0 -2], ⟨0 0 1 3]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 0 -2], ⟨0 1 1 1]]

Minkowski lattice basis:

~6/5 length = 1.068, ~5/4 length = 1.206
angle (~6/5, ~5/4) = 100.364°

Optimal tunings:

  • WE: ~2 = 1199.7931 ¢, ~3/2 = 701.1306 ¢, ~5/4 = 390.0298 ¢
error map: ⟨-0.207 -1.031 +3.302 -1.618]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0811 ¢, ~5/4 = 389.8833 ¢
error map: ⟨0.000 -0.874 +3.570 -1.338]

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [1/3 2/3 0 1/3⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c

Badness (Sintel): 0.308

Projection pair: 7 125/18

Minkowski blocks
  • 7: 25/24, 81/80
  • 8: 16/15, 648/625
  • 9: 27/25, 128/125
  • 11: 16/15, 15625/15552
  • 12: 128/125, 628/625
  • 15: 128/125, 250/243
  • 16: 648/625, 3125/3072
  • 17: 25/24, 20480/19683
  • 19: 81/80, 3125/3072
  • 27: 128/125, 78732/78125
  • 28: 648/625, 16875/16384
  • 31: 81/80, 1990656/1953125
  • 34: 15625/15552, 2048/2025

Overview to extensions

Temperaments discussed elsewhere include

Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.

Thrush

Main article: 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 with myna.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175

Mapping: [⟨1 0 0 -1 -5], ⟨0 1 0 -2 -2], ⟨0 0 1 3 5]]

Mapping to lattice: [⟨0 1 1 1 3], ⟨0 1 0 -2 -2]]

Lattice basis:

~5/4 length = 0.8576, ~6/5 length = 0.9314
angle(~5/4, ~6/5) = 74.6239°

Optimal tunings:

  • WE: ~2 = 1199.4742 ¢, ~3/2 = 701.6379 ¢, ~5/4 = 391.0192 ¢
error map: ⟨-0.526 -0.843 +3.654 -0.621 -1.075]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6298 ¢, ~5/4 = 390.8558 ¢
error map: ⟨0.000 -0.325 +4.542 +0.482 -0.298]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [1/3 2/3 0 1/3 0⟩, [0 0 0 1 0⟩, [-10/3 4/3 0 5/3 0⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c

Badness (Sintel): 0.424

Projection pairs: 7 125/18 11 3125/288

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195

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 ¢, ~3/2 = 702.2858 ¢, ~5/4 = 391.3567 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4562 ¢, ~5/4 = 391.2080 ¢

Optimal ET sequence: 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf

Badness (Sintel): 0.633

Bluebird

Not to be confused with Bluebirds.

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175

Mapping: [⟨1 0 0 -1 -5 9], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 5 -5]]

Optimal tunings:

  • WE: ~2 = 1198.7153 ¢, ~3/2 = 701.647 ¢, ~5/4 = 391.9180 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6263 ¢, ~5/4 = 391.7786 ¢

Optimal ET sequence: 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff

Badness (Sintel): 0.856

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

Nightingale

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 126/125, 176/175

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 ¢, ~3/2 = 699.2886 ¢, ~5/4 = 388.2054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.2840 ¢, ~5/4 = 388.2437 ¢

Optimal ET sequence: 12f, 15, 19e, 27eff, 31

Badness (Sintel): 0.783

Veery

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 176/175

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 ¢, ~3/2 = 704.4117 ¢, ~5/4 = 391.7571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.3959 ¢, ~5/4 = 391.6910 ¢

Optimal ET sequence: 12, 15, 19e, 27e, 31f, 46

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: [⟨1 0 0 -1 8], ⟨0 1 0 -2 3], ⟨0 0 1 3 -4]]

Optimal tunings:

  • WE: ~2 = 1200.6133 ¢, ~3/2 = 700.4314 ¢, ~5/4 = 388.3236 ¢
error map: ⟨+0.613 -0.910 +3.237 -2.878 -1.478]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4020 ¢, ~5/4 = 388.3631 ¢
error map: ⟨0.000 -1.553 +2.049 -4.541 -3.564]

Optimal ET sequence: 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de

Badness (Sintel): 0.813

Projection pairs: 7 125/18 11 6912/625

Scales: Star and Nova

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384

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 ¢, ~3/2 = 700.7452 ¢, ~5/4 = 388.4655 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.5708 ¢, ~5/4 = 388.4346 ¢

Optimal ET sequence: 15f, 19, 27, 31, 46, 77, 96d

Badness (Sintel): 1.07

Thrasher

See also: Ptolemismic clan

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99

Mapping: [⟨1 0 0 -1 2], ⟨0 1 0 -2 -2], ⟨0 0 1 3 2]]

Mapping to lattice: [⟨0 1 0 -2 -2], ⟨0 1 1 1 0]]

Lattice basis:

~6/5 length = 0.9089, ~5/4 length = 1.2007
angle (~6/5, ~5/4) = 98.8447°

Optimal tunings:

  • WE: ~2 = 1196.9183 ¢, ~3/2 = 706.1793 ¢, ~5/4 = 394.5223 ¢
error map: ⟨-3.082 +1.143 +2.045 -6.863 +13.041]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.2523 ¢, ~5/4 = 392.7970 ¢
error map: ⟨0.000 +4.297 +6.483 -2.940 +21.771]

Minimax tuning:

[[1 0 0 0 0⟩, [1 3/4 0 1/4 -3/8⟩, [1 1/2 0 1/2 -1/4⟩, [0 0 0 1 0⟩, [2 -1/2 0 1/2 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7.11/9

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e *

* optimal patent val: 34

Badness (Sintel): 0.577

Scales: Thrasher chromatic, Thrasher diatonic

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99

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 ¢, ~3/2 = 707.3847 ¢, ~5/4 = 394.2040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.1440 ¢, ~5/4 = 392.7909 ¢

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e, 69bceef *

* optimal patent val: 34

Badness (Sintel): 0.820

Mockingbird

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 100/99

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 ¢, ~3/2 = 710.0779 ¢, ~5/4 = 400.7221 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.2922 ¢, ~5/4 = 398.0867 ¢

Optimal ET sequence: 7d, 8d, 12f, 15, 27eff

Badness (Sintel): 0.804

Catbird

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 100/99

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 ¢, ~3/2 = 704.8749 ¢, ~5/4 = 394.9837 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9205 ¢, ~5/4 = 393.2290 ¢

Optimal ET sequence: 7df, 8d, 12f, 19, 27e, 66cdeeef

Badness (Sintel): 0.846

Aplonis

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539

Mapping: [⟨1 0 0 -1 4], ⟨0 1 0 -2 7], ⟨0 0 1 3 -5]]

Optimal tunings:

  • WE: ~2 = 1199.6006 ¢, ~3/2 = 700.5746 ¢, ~5/4 = 390.2558 ¢
error map: ⟨-0.399 -1.780 +3.143 -0.406 +1.026]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4342 ¢, ~5/4 = 389.9796 ¢
error map: ⟨0.000 -1.521 +3.666 +0.244 +1.824]

Optimal ET sequence: 12e, 19, 27e, 31, 58, 89, 197c, 228c

Badness (Sintel): 0.778

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195

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 ¢, ~3/2 = 701.2333 ¢, ~5/4 = 390.8385 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0203 ¢, ~5/4 = 390.2413 ¢

Optimal ET sequence: 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff

Badness (Sintel): 0.768

Treecreeper

Subgroup: 2.3.5.7.11

Comma list: 126/125, 1232/1215

Mapping: [⟨1 0 0 -1 -3], ⟨0 1 0 -2 7], ⟨0 0 1 3 -2]]

Optimal tunings:

  • WE: ~2 = 1198.8957 ¢, ~3/2 = 704.9055 ¢, ~5/4 = 392.8725 ¢
error map: ⟨-1.104 +1.846 +4.350 -3.332 -2.725]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.0045 ¢, ~5/4 = 392.3159 ¢
error map: ⟨0.000 +3.050 +6.002 -1.887 -0.918]

Optimal ET sequence: 7d, 12e, 19e, 27e, 39d, 46, 119c

Badness (Sintel): 1.91

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 352/351

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 ¢, ~3/2 = 704.6703 ¢, ~5/4 = 391.3513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7011 ¢, ~5/4 = 391.2846 ¢

Optimal ET sequence: 7d, 12e, 19e, 27e, 46

Badness (Sintel): 1.48

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