Starling family: Difference between revisions

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

Erato (+81/80) → Didymus rank-3 family
Sensigh (+245/243) → Sensamagic family
Oxpecker (+121/120) → Biyatismic clan
Cuckoo (+243/242) → Rastmic rank-3 clan

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:

7- and 9-odd-limit: 3 and 7 just, 5 1/3-comma sharp

[[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:

7- and 9-odd-limit

[[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:

11-odd-limit

[[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

Starling family: Difference between revisions

Latest revision as of 00:32, 24 June 2025

Contents

Starling

Undecimal starling

Thrush

13-limit

Bluebird

Nightingale

Veery

Thrasher

13-limit

Mockingbird

Catbird

Aplonis

13-limit

Treecreeper

13-limit

Navigation menu

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Technical data page}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+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).
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-17 12:34:03 UTC</tt>.<br>
-: The original revision id was <tt>302802398</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
-The head of the starling [[family]] is starling, which tempers out [[126_125|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 &lt;135 214 314 379|.
+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)^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.
-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 temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
+Temperaments discussed elsewhere include
+* ''[[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]]
-=7-limit starling=
+Considered below are starling, thrush, thrasher, aplonis, and treecreeper.
-[[Comma]] c = 126/125
-- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
+== Starling ==
-[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]
+{{Main| Starling and thrush }}
-Eigenmonzos: 2, 8/7, 4/3
-Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206
+[[Subgroup]]: 2.3.5.7
-Angle(6/5, 5/4) = 100.364 degrees
-Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]
-Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
+[[Comma list]]: [[126/125]]
-Generators: 2, 3, 5
-[[edo|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
+{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
-Badness: 0.0000699
+: mapping generators: ~2, ~3, ~5
+[[Mapping 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 degrees
+[[Minimax tuning]]:
+* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
+: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
+{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
+[[Badness]]: 0.0699 × 10<sup>-3</sup>
 [[Projection pair]]: 7 125/18
-Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
+{{Databox|[[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|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]]: 0.677 × 10<sup>-3</sup>
+== Thrush ==
+{{Main| Starling and thrush }}
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 126/125, 176/175
+{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
+Mapping 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 degrees
+[[Minimax tuning]]:
+* 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 }}]
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
+{{Optimal ET sequence|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
-==[[Minkowski blocks]]==
+[[Associated temperament]]: [[myna]]
-: 25/24, 81/80
-: 16/15, 648/625
-: 27/25, 128/125
-: 16/15, 15625/15552
-: 128/125, 628/625
-: 128/125, 250/243
-: 648/625, 3125/3072
-: 25/24, 20480/19683
-: 81/80, 3125/3072
-: 128/125, 78732/78125
-: 648/625, 16875/16384
-: 81/80, 1990656/1953125
-: 15625/15552, 2048/2025
-=Thrush=
+=== 13-limit ===
-[[Comma|Commas]]: 126/125, 176/175
+Subgroup: 2.3.5.7.11.13
-Associated linear temperament: [[Starling temperaments|myna]]
-and 9 limit minimax
+Comma list: 126/125, 176/175, 196/195
-[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
-|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
-Lattice basis 5/4 length 0.8576 6/5 length 0.9314
+Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
-Angle(5/4, 6/5) = 74.6239 degrees
-Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
-Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
+{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
-[[Generator|Generators]]: 2, 3, 5
-EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
-Badness: 0.000353
-Projection pairs: 7 125/18 11 3125/288
+Badness: 0.677 × 10<sup>-3</sup>
-Scales: [[thrush12]]
+=== Bluebird ===
+Subgroup: 2.3.5.7.11.13
-==13-limit==
+Comma list: 126/125, 144/143, 176/175
-Commas: 126/125, 176/175, 196/195
-Map: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;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 }}
-EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf
-Badness: 0.000677
-==Bluebird==
+{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
-Commas 126/125, 176/175, 144/143
-Map: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]
+Badness: 0.915 × 10<sup>-3</sup>
-EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef
-Badness: 0.000915
 Projection pairs: 7 125/18 11 3125/288 13 41472/3125
-==Nightingale==
+=== Nightingale ===
-Commas: 126/125, 176/175, 66/65
+Subgroup: 2.3.5.7.11.13
+Comma list: 66/65, 126/125, 176/175
+Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
+{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}
+Badness: 0.837 × 10<sup>-3</sup>
+=== Veery ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 91/90, 126/125, 176/175
+Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
+{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}
+Badness: 0.991 × 10<sup>-3</sup>
+== Thrasher ==
+{{See also| Ptolemismic clan #Thrasher }}
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 56/55, 100/99
+{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
+Mapping 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
+[[Minimax tuning]]:
+* [[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 }}]
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
+{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
+<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
+[[Badness]]: 0.480 × 10<sup>-3</sup>
+Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 56/55, 91/90, 100/99
+Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
+{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
+<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
+Badness: 0.876 × 10<sup>-3</sup>
+=== Mockingbird ===
+Subgroup: 2.3.5.7.11.13
+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 }}
+{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
+Badness: 0.859 × 10<sup>-3</sup>
+=== Catbird ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 78/77, 100/99, 126/125
-Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
+Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
-[[edo|EDOs]]: [[15edo|15]], [[19edo}19f]], [[21edo|21e]], [[22edo|22f]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]], [[108edo|108ef]]
-Badness: 0.000837
-=Thrasher=
+{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
-[[Comma|Commas]]: 56/55, 100/99
--limit minimax
+Badness: 0.905 × 10<sup>-3</sup>
-[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;,
-|1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;,
-|2 -1/2 0 1/2 1/4&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9
-Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
+== Aplonis ==
-Angle(6/5, 5/4) = 98.8447
+[[Subgroup]]: 2.3.5.7.11
-Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
-Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
+[[Comma list]]: 126/125, 540/539
-[[Generator|Generators]]: 2, 3, 5
-EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
-Badness: 0.000480
-==13-limit==
+{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
-Commas: 126/125, 100/99, 91/90
-Map: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
+{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
-EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
-Badness: 0.000876
-==Mockingbird==
+[[Badness]]: 0.648 × 10<sup>-3</sup>
-Commas: 126/125, 100/99, 40/39
-Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
+=== 13-limit ===
-EDOs: 12f, 15, 27ef
+Subgroup: 2.3.5.7.11.13
-Badness: 0.000859
-==Catbird==
+Comma list: 126/125, 144/143, 196/195
-Commas: 126/125, 100/99, 78/77
-Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
+Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
-[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
-Badness: 0.000905
-=Aplonis=
+{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
-Commas: 126/125, 540/539
-Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
+Badness: 0.821 × 10<sup>-3</sup>
-EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
-Badness: 0.000648
-==13-limit==
+== Treecreeper ==
-Commas: 126/125, 144/143, 196/195
+[[Subgroup]]: 2.3.5.7.11
-Map: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
+[[Comma list]]: 126/125, 1232/1215
-EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
-Badness: 0.000821
-=Oxpecker=
+{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
-Commas: 121/120, 126/125
-Map: [&lt;1 0 1 2 2|,  &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
+{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
-EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
-Badness: 0.000699
-==Woodpecker==
+[[Badness]]: 1.585 × 10<sup>-3</sup>
-Commas: 66/65, 121/120, 126/125
-Map: [&lt;1 0 1 2 2 2|,  &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]
+=== 13-limit ===
-EDOs: 15, 31, 46f, 70f, 101f
+Subgroup: 2.3.5.7.11.13
-Badness: 0.001093
-=Treecreeper=
+Comma list: 91/90, 126/125, 352/351
-Commas: 126/125, 1232/1215
-Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
+Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
-EDOs: 34, 46, 119c, 165c
-Badness: 0.001585
-==13-limit==
+{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}
-Commas: 91/90, 126/125, 352/351
-Map: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
+Badness: 1.588 × 10<sup>-3</sup>
-EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
-Badness: 0.001588
-</pre></div>
+[[Category:Temperament families]]
-<h4>Original HTML content:</h4>
+[[Category:Pages with mostly numerical content]]
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Starling family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:32:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;a href="#x7-limit starling"&gt;7-limit starling&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Thrush"&gt;Thrush&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt; | &lt;a href="#Thrasher"&gt;Thrasher&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt; | &lt;a href="#Aplonis"&gt;Aplonis&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt; | &lt;a href="#Oxpecker"&gt;Oxpecker&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt; | &lt;a href="#Treecreeper"&gt;Treecreeper&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;
+[[Category:Starling family| ]] <!-- main article -->
-&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;br /&gt;
+[[Category:Starling| ]] <!-- key article -->
-The head of the starling &lt;a class="wiki_link" href="/family"&gt;family&lt;/a&gt; is starling, which tempers out &lt;a class="wiki_link" href="/126_125"&gt;126/125&lt;/a&gt;, the &lt;a class="wiki_link" href="/starling%20comma"&gt;starling comma&lt;/a&gt; or &lt;a class="wiki_link" href="/septimal%20semicomma"&gt;septimal semicomma&lt;/a&gt;. 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 &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;.  Other possible tunings are &lt;a class="wiki_link" href="/108edo"&gt;108edo&lt;/a&gt; and &lt;a class="wiki_link" href="/185edo"&gt;185edo&lt;/a&gt;, and the nonpatent &lt;a class="wiki_link" href="/135edo"&gt;135edo&lt;/a&gt; val &amp;lt;135 214 314 379|.&lt;br /&gt;
+[[Category:Rank 3]]
-&lt;br /&gt;
-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.&lt;br /&gt;
-&lt;br /&gt;
-Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;valentine temperament&lt;/a&gt;, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x7-limit starling"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;7-limit starling&lt;/h1&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 126/125&lt;br /&gt;
-&lt;br /&gt;
-- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;
-[&amp;lt;1 0 0 0|, &amp;lt;0 1 0 0|, &amp;lt;1/3 2/3 0 1/3|, &amp;lt;0 0 0 1|]&lt;br /&gt;
-Eigenmonzos: 2, 8/7, 4/3&lt;br /&gt;
-&lt;br /&gt;
-Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206&lt;br /&gt;
-Angle(6/5, 5/4) = 100.364 degrees&lt;br /&gt;
-Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]&lt;br /&gt;
-Generators: 2, 3, 5&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, &lt;a class="wiki_link" href="/77edo"&gt;77&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108&lt;/a&gt;, &lt;a class="wiki_link" href="/166edo"&gt;166c&lt;/a&gt;, &lt;a class="wiki_link" href="/185edo"&gt;185&lt;/a&gt;, &lt;a class="wiki_link" href="/243edo"&gt;243c&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0000699&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 125/18&lt;br /&gt;
-&lt;br /&gt;
-Scales: &lt;a class="wiki_link" href="/starling7"&gt;starling7&lt;/a&gt;, &lt;a class="wiki_link" href="/starling8"&gt;starling8&lt;/a&gt;, &lt;a class="wiki_link" href="/starling9"&gt;starling9&lt;/a&gt;, &lt;a class="wiki_link" href="/starling11"&gt;starling11&lt;/a&gt;, &lt;a class="wiki_link" href="/starling12"&gt;starling12&lt;/a&gt;, &lt;a class="wiki_link" href="/starling15"&gt;starling15&lt;/a&gt;, &lt;a class="wiki_link" href="/starling16"&gt;starling16&lt;/a&gt;, &lt;a class="wiki_link" href="/starling17"&gt;starling17&lt;/a&gt;, &lt;a class="wiki_link" href="/starling19"&gt;starling19&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x7-limit starling-Minkowski blocks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/Minkowski%20blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;/h2&gt;
-: 25/24, 81/80&lt;br /&gt;
-: 16/15, 648/625&lt;br /&gt;
-: 27/25, 128/125&lt;br /&gt;
-: 16/15, 15625/15552&lt;br /&gt;
-: 128/125, 628/625&lt;br /&gt;
-: 128/125, 250/243&lt;br /&gt;
-: 648/625, 3125/3072&lt;br /&gt;
-: 25/24, 20480/19683&lt;br /&gt;
-: 81/80, 3125/3072&lt;br /&gt;
-: 128/125, 78732/78125&lt;br /&gt;
-: 648/625, 16875/16384&lt;br /&gt;
-: 81/80, 1990656/1953125&lt;br /&gt;
-: 15625/15552, 2048/2025&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Thrush&lt;/h1&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;
-Associated linear temperament: &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;myna&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-and 9 limit minimax&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |0 1 0 0 0&amp;gt;, |1/3 2/3 0 1/3 0&amp;gt;, &lt;br /&gt;
-|0 0 0 1 0&amp;gt;, |-10/3 4/3 0 5/3 0&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
-&lt;br /&gt;
-Lattice basis 5/4 length 0.8576 6/5 length 0.9314&lt;br /&gt;
-Angle(5/4, 6/5) = 74.6239 degrees&lt;br /&gt;
-Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 -5|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 5|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
-EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c&lt;br /&gt;
-Badness: 0.000353&lt;br /&gt;
-&lt;br /&gt;
-Projection pairs: 7 125/18 11 3125/288 &lt;br /&gt;
-&lt;br /&gt;
-Scales: &lt;a class="wiki_link" href="/thrush12"&gt;thrush12&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Thrush-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
-Commas: 126/125, 176/175, 196/195&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 -5 0|, &amp;lt;0 1 0 -2 -2 -5|, &amp;lt;0 0 1 3 5 5|]&lt;br /&gt;
-EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf&lt;br /&gt;
-Badness: 0.000677&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Thrush-Bluebird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Bluebird&lt;/h2&gt;
-Commas 126/125, 176/175, 144/143&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 -5 9|, &amp;lt;0 1 0 -2 -2 4|, &amp;lt;0 0 1 3 5 -5|]&lt;br /&gt;
-EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef&lt;br /&gt;
-Badness: 0.000915&lt;br /&gt;
-&lt;br /&gt;
-Projection pairs: 7 125/18 11 3125/288 13 41472/3125&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Thrush-Nightingale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Nightingale&lt;/h2&gt;
-Commas: 126/125, 176/175, 66/65&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 -5 -4|, &amp;lt;0 1 0 -2 -2 -1|, &amp;lt;0 0 1 3 5 4|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, [[19edo}19f]], &lt;a class="wiki_link" href="/21edo"&gt;21e&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22f&lt;/a&gt;, &lt;a class="wiki_link" href="/28edo"&gt;28&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/89edo"&gt;89&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108ef&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000837&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Thrasher&lt;/h1&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 56/55, 100/99&lt;br /&gt;
-&lt;br /&gt;
--limit minimax&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;, &lt;br /&gt;
-|1 1/2 0 1/2 -1/4&amp;gt;, |0 0 0 1 0&amp;gt;, &lt;br /&gt;
-|2 -1/2 0 1/2 1/4&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 11/9&lt;br /&gt;
-&lt;br /&gt;
-Lattice basis: 6/5 length 0.9089 5/4 length 1.2007&lt;br /&gt;
-Angle(6/5, 5/4) = 98.8447&lt;br /&gt;
-Map to lattice: [&amp;lt;0 1 0 -2 -2|, &amp;lt;0 1 1 1 0|]&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 2|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 2|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
-EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e&lt;br /&gt;
-Badness: 0.000480&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Thrasher-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
-Commas: 126/125, 100/99, 91/90&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 2 2|, &amp;lt;0 1 0 -2 -2 4|, &amp;lt;0 0 1 3 2 -2|]&lt;br /&gt;
-EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e&lt;br /&gt;
-Badness: 0.000876&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Thrasher-Mockingbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Mockingbird&lt;/h2&gt;
-Commas: 126/125, 100/99, 40/39&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 2 3|, &amp;lt;0 1 0 -2 -2 -1|, &amp;lt;0 0 1 3 2 1|]&lt;br /&gt;
-EDOs: 12f, 15, 27ef&lt;br /&gt;
-Badness: 0.000859&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Thrasher-Catbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Catbird&lt;/h2&gt;
-Commas: 126/125, 100/99, 78/77&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 2 0|, &amp;lt;0 1 0 -2 -2 -5|, &amp;lt;0 0 1 3 2 5|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17c&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22ef&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27e&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58e&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000905&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Aplonis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Aplonis&lt;/h1&gt;
-Commas: 126/125, 540/539&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 4|, &amp;lt;0 1 0 -2 7|, &amp;lt;0 0 1 3 -5|]&lt;br /&gt;
-EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce&lt;br /&gt;
-Badness: 0.000648&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Aplonis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h2&gt;
-Commas: 126/125, 144/143, 196/195&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 4 0|, &amp;lt;0 1 0 -2 7 -5|, &amp;lt;0 0 1 3 -5 5|]&lt;br /&gt;
-EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef&lt;br /&gt;
-Badness: 0.000821&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Oxpecker"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Oxpecker&lt;/h1&gt;
-Commas: 121/120, 126/125&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 1 2 2|,  &amp;lt;0 1 1 1 1|, &amp;lt;0 0 -2 -6 -1|]&lt;br /&gt;
-EDOs: 8d, 15, 31, 46, 77, 185e, 262cde&lt;br /&gt;
-Badness: 0.000699&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Oxpecker-Woodpecker"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Woodpecker&lt;/h2&gt;
-Commas: 66/65, 121/120, 126/125&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 1 2 2 2|,  &amp;lt;0 1 1 1 1 1|, &amp;lt;0 0 -2 -6 -1 -1|]&lt;br /&gt;
-EDOs: 15, 31, 46f, 70f, 101f &lt;br /&gt;
-Badness: 0.001093&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Treecreeper"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Treecreeper&lt;/h1&gt;
-Commas: 126/125, 1232/1215&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 -3|, &amp;lt;0 1 0 -2 7|, &amp;lt;0 0 1 3 -2|]&lt;br /&gt;
-EDOs: 34, 46, 119c, 165c&lt;br /&gt;
-Badness: 0.001585&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Treecreeper-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;13-limit&lt;/h2&gt;
-Commas: 91/90, 126/125, 352/351&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 -1 -3 2|, &amp;lt;0 1 0 -2 7 4|, &amp;lt;0 0 1 3 -2 -2|]&lt;br /&gt;
-EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf&lt;br /&gt;
-Badness: 0.001588&lt;/body&gt;&lt;/html&gt;</pre></div>

Starling family: Difference between revisions

Latest revision as of 00:32, 24 June 2025

Starling

Undecimal starling

Thrush

13-limit

Bluebird

Nightingale

Veery

Thrasher

13-limit

Mockingbird

Catbird

Aplonis

13-limit

Treecreeper

13-limit

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