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}}
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 }} (135c).


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.
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.


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.
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.


== Starling ==
Temperaments discussed elsewhere include
Subgroup: 2.3.5.7
* ''[[Erato]]'' (+81/80) → [[Didymus rank three family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank three clan #Cuckoo|Rastmic rank-3 clan]]
 
Considered below are starling, thrush, thrasher, aplonis, and treecreeper.
 
== Starling ==
{{Main| Starling and thrush }}
 
[[Subgroup]]: 2.3.5.7


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


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


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 degrees


[[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 (unchanged intervals): 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]]: 0.0699 × 10<sup>-3</sup>
Line 31: Line 42:
[[Projection pair]]: 7 125/18
[[Projection pair]]: 7 125/18


Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
{{Databox|[[Minkowski blocks]]|
 
; 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">
 
* 7: 25/24, 81/80
* 7: 25/24, 81/80
* 8: 16/15, 648/625
* 8: 16/15, 648/625
Line 53: Line 56:
* 31: 81/80, 1990656/1953125
* 31: 81/80, 1990656/1953125
* 34: 15625/15552, 2048/2025
* 34: 15625/15552, 2048/2025
 
}}
</div></div>


== Undecimal starling ==
== Undecimal starling ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 126/125, 385/384
[[Comma list]]: 126/125, 385/384


[[Mapping]]: [{{val| 1 0 0 -1 8 }}, {{val| 0 1 0 -2 3 }}, {{val| 0 0 1 3 -4 }}]
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
Mapping generators: ~2, ~3, ~5


{{Val list|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}


[[Badness]]: 0.677 × 10<sup>-3</sup>
[[Badness]]: 0.677 × 10<sup>-3</sup>


== Thrush ==
== Thrush ==
Subgroup: 2.3.5.7.11
{{Main| Starling and 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 }}]
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
Line 87: Line 87:
* 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 (unchanged intervals): 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 }}
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}


[[Badness]]: 0.353 × 10<sup>-3</sup>
[[Badness]]: 0.353 × 10<sup>-3</sup>
Line 95: Line 95:
[[Projection pair]]s: 7 125/18 11 3125/288
[[Projection pair]]s: 7 125/18 11 3125/288


[[Associated temperament]]: [[Starling temperaments #Myna|myna]]
[[Associated temperament]]: [[myna]]
 
Scales: [[thrush12]]


=== 13-limit ===
=== 13-limit ===
Line 104: Line 102:
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 }}


Vals: {{Val list| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}


Badness: 0.677 × 10<sup>-3</sup>
Badness: 0.677 × 10<sup>-3</sup>
Line 115: Line 113:
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 ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}


Badness: 0.915 × 10<sup>-3</sup>
Badness: 0.915 × 10<sup>-3</sup>
Line 128: Line 126:
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 }}


Vals: {{Val list| 12f, 15, 19e, 27eff, 31 }}
{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}


Badness: 0.837 × 10<sup>-3</sup>
Badness: 0.837 × 10<sup>-3</sup>
Line 139: Line 137:
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 }}


Vals: {{Val list| 12, 15, 19e, 27e, 31f, 46 }}
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}


Badness: 0.991 × 10<sup>-3</sup>
Badness: 0.991 × 10<sup>-3</sup>


== Thrasher ==
== Thrasher ==
Subgroup: 2.3.5.7.11
{{See also| Ptolemismic clan #Thrasher }}
 
[[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 }}]
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
Line 163: Line 161:
* [[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 (unchanged intervals): 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>


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


[[Badness]]: 0.480 × 10<sup>-3</sup>
[[Badness]]: 0.480 × 10<sup>-3</sup>
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 ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>


<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
Line 187: Line 187:
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: [{{val| 1 0 0 -1 2 3 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 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 }}


Vals: {{Val list| 7d, 8d, 12f, 15, 27eff }}
{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}


Badness: 0.859 × 10<sup>-3</sup>
Badness: 0.859 × 10<sup>-3</sup>
Line 200: Line 200:
Comma list: 78/77, 100/99, 126/125
Comma list: 78/77, 100/99, 126/125


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


Vals: {{Val list| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}


Badness: 0.905 × 10<sup>-3</sup>
Badness: 0.905 × 10<sup>-3</sup>


== 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 }}


{{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]]: 0.648 × 10<sup>-3</sup>
Line 222: Line 222:
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 }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}


Badness: 0.821 × 10<sup>-3</sup>
Badness: 0.821 × 10<sup>-3</sup>
== Oxpecker ==
Subgroup: 2.3.5.7.11
[[Comma list]]: 121/120, 126/125
[[Mapping]]: [{{val| 1 0 1 2 2 }}, {{val| 0 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 }}]
{{Val list|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
[[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 }}


{{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]]: 1.585 × 10<sup>-3</sup>
Line 266: Line 244:
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 }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}


Badness: 1.588 × 10<sup>-3</sup>
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
Mapping: [{{val| 1 1 0 -3 2 -5 }}, {{val| 0 2 0 -4 5 -10 }}, {{val| 0 0 1 3 0 5 }}]
Vals: {{Val list| 27e, 31, 58, 96d, 154 }}
[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Starling family| ]] <!-- main article -->
[[Category:Thrush]]
[[Category:Starling| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]

Latest revision as of 00:32, 24 June 2025

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 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 ⟨135 214 314 379] (135c).

In starling, (6/5)3 = 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.

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.

Temperaments discussed elsewhere include

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

Starling

Main article: Starling and thrush

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 degrees

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: 0.0699 × 10-3

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

Undecimal starling

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

Badness: 0.677 × 10-3

Thrush

Main article: Starling and thrush

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 degrees

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: 0.353 × 10-3

Projection pairs: 7 125/18 11 3125/288

Associated temperament: myna

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

Badness: 0.677 × 10-3

Bluebird

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 ET sequence: 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff

Badness: 0.915 × 10-3

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 ET sequence: 12f, 15, 19e, 27eff, 31

Badness: 0.837 × 10-3

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

Badness: 0.991 × 10-3

Thrasher

See also: Ptolemismic clan § Thrasher

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

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: 0.480 × 10-3

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

* optimal patent val: 34

Badness: 0.876 × 10-3

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 ET sequence: 7d, 8d, 12f, 15, 27eff

Badness: 0.859 × 10-3

Catbird

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 126/125

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

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

Badness: 0.905 × 10-3

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

Badness: 0.648 × 10-3

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

Badness: 0.821 × 10-3

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

Badness: 1.585 × 10-3

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 ET sequence: 7d, 12e, 19e, 27e, 46

Badness: 1.588 × 10-3

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