Starling family: Difference between revisions

Revision as of 11:28, 12 August 2023

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 (→ Didymus rank-3 family) and sensigh (→ Sensamagic family). Considered below are starling, thrush, thrasher, aplonis, oxpecker, treecreeper, and cuckoo.

Starling

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⟩]

eigenmonzo (unchanged-interval) 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

Scales: starling7, starling8, starling9, starling11, starling12, starling15, starling16, starling17, starling19

Music

A Seed Planted by Jake Freivald. The melody depends on tempering out 126/125.

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

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⟩]

eigenmonzo (unchanged-interval) 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

Scales: thrush12

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

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⟩]

eigenmonzo (unchanged-interval) basis: 2.7.11/9

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

* optimal patent val: 34

Badness: 0.480 × 10^-3

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

Oxpecker

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125

Mapping: [⟨1 0 1 2 2], ⟨0 1 1 1 1], ⟨0 0 -2 -6 -1]]

mapping generators: ~2, ~3, ~35/32

Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31, 46, 77

Badness: 0.699 × 10^-3

Woodpecker

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 126/125

Mapping: [⟨1 0 1 2 2 2], ⟨0 1 1 1 1 1], ⟨0 0 -2 -6 -1 -1]]

Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31

Badness: 1.093 × 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

Cuckoo

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

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

mapping generators: ~2, ~11/9, ~5

Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness: 0.933 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Optimal ET sequence: 27e, 31, 58, 96d, 154

Starling family: Difference between revisions

Revision as of 11:28, 12 August 2023

Contents

Starling

Undecimal starling

Thrush

13-limit

Bluebird

Nightingale

Veery

Thrasher

13-limit

Mockingbird

Catbird

Aplonis

13-limit

Oxpecker

Woodpecker

Treecreeper

13-limit

Cuckoo

13-limit

Navigation menu

@@ Line 1: / Line 1: @@
-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 }}.
+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.
-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.
 Temperaments discussed elsewhere include [[erato]] (→ [[Didymus rank three family #Erato|Didymus rank-3 family]]) and [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered below are starling, thrush, thrasher, aplonis, oxpecker, treecreeper, and cuckoo.
 == Starling ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}]
-Minkowski lattice basis:
+[[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 1/3c 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|eigenmonzo (unchanged-interval) basis]]: 2.3.7
 {{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
@@ Line 38: / Line 38: @@
 * [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;">
+{{Databox|[[Minkowski blocks]]|
-<div style="line-height:1.6;">[[Minkowski blocks]]</div>
-<div class="mw-collapsible-content">
 * 7: 25/24, 81/80
 * 8: 16/15, 648/625
@@ Line 55: / Line 52: @@
 * 31: 81/80, 1990656/1953125
 * 34: 15625/15552, 2048/2025
+}}
-</div></div>
 == Undecimal starling ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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
 {{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
@@ Line 72: / Line 66: @@
 == Thrush ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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 }}]
@@ Line 89: / Line 81: @@
 * 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]]s (unchanged-intervals): 2, 7/6, 4/3
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.7
 {{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
@@ Line 97: / Line 89: @@
 [[Projection pair]]s: 7 125/18 11 3125/288
-[[Associated temperament]]: [[Starling temperaments #Myna|myna]]
+[[Associated temperament]]: [[myna]]
 Scales: [[thrush12]]
@@ Line 106: / Line 98: @@
 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 ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
@@ Line 117: / Line 109: @@
 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 }}
 {{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
@@ Line 130: / Line 122: @@
 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 ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}
@@ Line 141: / Line 133: @@
 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 ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}
@@ Line 148: / Line 140: @@
 == Thrasher ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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 }}]
@@ Line 165: / Line 155: @@
 * [[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]]s (unchanged-intervals): 2, 8/7, 11/9
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7.11/9
 {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
@@ Line 176: / Line 166: @@
 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 }}
 {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
@@ Line 189: / Line 179: @@
 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 }}
 {{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
@@ Line 202: / Line 192: @@
 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 }}
 {{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
@@ Line 209: / Line 199: @@
 == Aplonis ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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 ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
@@ Line 224: / Line 214: @@
 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 }}
 {{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
@@ Line 235: / Line 225: @@
 [[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 }}]
+{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
+: mapping generators: ~2, ~3, ~35/32
 {{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
@@ Line 246: / Line 238: @@
 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 }}]
+Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 -1 }}
 {{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
@@ Line 253: / Line 245: @@
 == Treecreeper ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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 ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
@@ Line 268: / Line 260: @@
 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 }}
 {{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}
@@ Line 275: / Line 267: @@
 == Cuckoo ==
-Subgroup: 2.3.5.7.11
+[[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 }}]
+{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
+: mapping generators: ~2, ~11/9, ~5
 {{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
@@ Line 290: / Line 284: @@
 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 }}]
+Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}
 {{Optimal ET sequence|legend=1| 27e, 31, 58, 96d, 154 }}

Starling family: Difference between revisions

Revision as of 11:28, 12 August 2023

Starling

Undecimal starling

Thrush

13-limit

Bluebird

Nightingale

Veery

Thrasher

13-limit

Mockingbird

Catbird

Aplonis

13-limit

Oxpecker

Woodpecker

Treecreeper

13-limit

Cuckoo

13-limit

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