Starling family: Difference between revisions

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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 [[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  ==
== Starling  ==
Line 10: Line 10:
[[Comma list]]: [[126/125]]
[[Comma list]]: [[126/125]]


[[Mapping]]: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]


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


Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]


Minkowski lattice basis:  
Minkowski lattice basis:  
Line 22: Line 22:
[[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 1/3c sharp
: [&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]
: [{{val| 1 0 0 0 }}, {{val| 0 1 0 0 }}, {{val| 1/3 2/3 0 1/3 }}, {{val| 0 0 0 1 }}]
: Eigenmonzos: 2, 8/7, 4/3
: Eigenmonzos: 2, 8/7, 4/3


Line 61: Line 61:
[[Comma list]]: 126/125, 385/384
[[Comma list]]: 126/125, 385/384


[[Mapping]]: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
[[Mapping]]: [{{val| 1 0 0 -1 8 }}, {{val| 0 1 0 -2 3 }}, {{val| 0 0 1 3 -4]]


Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5
Line 74: Line 74:
[[Comma list]]: 126/125, 176/175
[[Comma list]]: 126/125, 176/175


[[Mapping]]: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
[[Mapping]]: [{{val| 1 0 0 -1 -5 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 5 }}]


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


[[Minimax tuning]]:
Map to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
* 7- and [[9-odd-limit]]
: [|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]]s: 2, 7/6, 4/3
 
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]


Lattice basis:  
Lattice basis:  
: 5/4 length = 0.8576, 6/5 length = 0.9314
: 5/4 length = 0.8576, 6/5 length = 0.9314
: Angle(5/4, 6/5) = 74.6239 degrees
: Angle(5/4, 6/5) = 74.6239 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]]s: 2, 7/6, 4/3


{{Val list|legend=1| 12, 15, 31, 46, 58, 89, 135c, 224c }}
{{Val list|legend=1| 12, 15, 31, 46, 58, 89, 135c, 224c }}
Line 104: Line 104:
Comma list: 126/125, 176/175, 196/195
Comma list: 126/125, 176/175, 196/195


Mapping: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]
Mapping: [{{val| 1 0 0 -1 -5 0 }}, {{val| 0 1 0 -2 -2 -5 }}, {{val| 0 0 1 3 5 5 }}]


Vals: {{Val list| 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf }}
Vals: {{Val list| 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf }}
Line 115: Line 115:
Comma list: 126/125, 176/175, 144/143
Comma list: 126/125, 176/175, 144/143


Mapping: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]
Mapping: [{{val| 1 0 0 -1 -5 9 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 5 -5 }}]


Vals: {{Val list| 12, 15, 31, 43, 48c, 58, 147cf, 205cef }}
Vals: {{Val list| 12, 15, 31, 43, 48c, 58, 147cf, 205cef }}
Line 128: Line 128:
Comma list: 126/125, 176/175, 66/65
Comma list: 126/125, 176/175, 66/65


Mapping: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
Mapping: [{{val| 1 0 0 -1 -5 -4 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 0 0 1 3 5 4 }}]


Vals: {{Val list| 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef }}
Vals: {{Val list| 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef }}
Line 139: Line 139:
Comma list: 91/90, 126/125, 176/175
Comma list: 91/90, 126/125, 176/175


Mapping: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]
Mapping: [{{val| 1 0 0 -1 -5 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 5 -2 }}]


Vals: {{Val list| 12, 15, 46 }}
Vals: {{Val list| 12, 15, 46 }}
Line 150: Line 150:
[[Comma list]]: 56/55, 100/99
[[Comma list]]: 56/55, 100/99


[[Mapping]]: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
[[Mapping]]: [{{val| 1 0 0 -1 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 3 2 }}]


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


Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
Map to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]


Lattice basis:  
Lattice basis:  
Line 162: Line 162:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|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;]
: [{{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]]s: 2, 8/7, 11/9


Line 174: Line 174:
Comma list: 126/125, 100/99, 91/90
Comma list: 126/125, 100/99, 91/90


Mapping: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
Mapping: [{{val| 1 0 0 -1 2 2 }}, {{val| 0 1 0 -2 -2 4 }}, {{val| 0 0 1 3 2 -2 }}]


Vals: {{Val list| 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e }}
Vals: {{Val list| 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e }}
Line 185: Line 185:
Comma list: 40/39, 100/99, 126/125
Comma list: 40/39, 100/99, 126/125


Mapping: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;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 }}]


Vals: {{Val list| 12f, 15, 27ef }}
Vals: {{Val list| 12f, 15, 27ef }}
Line 196: Line 196:
Comma list: 78/77, 100/99, 126/125
Comma list: 78/77, 100/99, 126/125


Mapping: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;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 }}]


Vals: {{Val list| 12, 15, 17c, 19, 22ef], 27e, 34, 58e }}
Vals: {{Val list| 12, 15, 17c, 19, 22ef], 27e, 34, 58e }}
Line 207: Line 207:
[[Comma list]]: 126/125, 540/539
[[Comma list]]: 126/125, 540/539


[[Mapping]]: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
[[Mapping]]: [{{val| 1 0 0 -1 4 }}, {{val| 0 1 0 -2 7 }}, {{val| 0 0 1 3 -5 }}]


{{Val list|legend=1| 19, 31, 58, 67, 89, 197c, 286ce, 375ce }}
{{Val list|legend=1| 19, 31, 58, 67, 89, 197c, 286ce, 375ce }}
Line 218: Line 218:
Comma list: 126/125, 144/143, 196/195
Comma list: 126/125, 144/143, 196/195


Mapping: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
Mapping: [{{val| 1 0 0 -1 4 0 }}, {{val| 0 1 0 -2 7 -5 }}, {{val| 0 0 1 3 -5 5 }}]


Vals: {{Val list| 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef }}
Vals: {{Val list| 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef }}
Line 229: Line 229:
[[Comma list]]: 121/120, 126/125
[[Comma list]]: 121/120, 126/125


[[Mapping]]: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
[[Mapping]]: [{{val| 1 0 1 2 2 }}, {{val| 0 1 1 1 1 }}, {{val| 0 0 -2 -6 -1 }}]


{{Val list|legend=1| 8d, 15, 31, 46, 77, 185e, 262cde }}
{{Val list|legend=1| 8d, 15, 31, 46, 77, 185e, 262cde }}
Line 240: Line 240:
Comma list: 66/65, 121/120, 126/125
Comma list: 66/65, 121/120, 126/125


Mapping: [&lt;1 0 1 2 2 2|, &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]
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| 15, 31, 46f, 70f, 101f }}
Vals: {{Val list| 15, 31, 46f, 70f, 101f }}
Line 251: Line 251:
[[Comma list]]: 126/125, 1232/1215
[[Comma list]]: 126/125, 1232/1215


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


{{Val list|legend=1| 34, 46, 119c, 165c }}
{{Val list|legend=1| 34, 46, 119c, 165c }}
Line 262: Line 262:
Comma list: 91/90, 126/125, 352/351
Comma list: 91/90, 126/125, 352/351


Mapping: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
Mapping: [{{val| 1 0 0 -1 -3 2 }}, {{val| 0 1 0 -2 7 4 }}, {{val| 0 0 1 3 -2 -2 }}]


Vals: {{Val list| 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf }}
Vals: {{Val list| 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf }}
Line 273: Line 273:
[[Comma list]]: 126/125, 243/242
[[Comma list]]: 126/125, 243/242


[[Mapping]]: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]
[[Mapping]]: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 0 -4 5 }}, {{val| 0 0 1 3 0 }}]


{{Val list|legend=1| 31, 34, 58, 65, 89, 154, 185, 216 }}
{{Val list|legend=1| 31, 34, 58, 65, 89, 154, 185, 216 }}
Line 284: Line 284:
Comma list: 126/125, 196/195, 243/242
Comma list: 126/125, 196/195, 243/242


Mapping: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|]
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| 31, 58, 65f, 89f, 92e, 154, 185, 216f }}
Vals: {{Val list| 31, 58, 65f, 89f, 92e, 154, 185, 216f }}

Revision as of 09:18, 8 June 2021

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].

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 temperament, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.

Starling

Subgroup: 2.3.5.7

Comma list: 126/125

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

Mapping generators: ~2, ~3, ~5

Map 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]]
Eigenmonzos: 2, 8/7, 4/3

Template:Val list

Badness: 0.0699 × 10-3

Projection pair: 7 125/18

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

Music
  • 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], {{val| 0 0 1 3 -4]]

Mapping generators: ~2, ~3, ~5

Template:Val list

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 generators: ~2, ~3, ~5

Map 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⟩]
Eigenmonzos: 2, 7/6, 4/3

Template:Val list

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]]

Vals: Template:Val list

Badness: 0.677 × 10-3

Bluebird

Subgroup: 2.3.5.7.11.13

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

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

Vals: Template:Val list

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

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

Vals: Template:Val list

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]]

Vals: Template:Val list

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 generators: ~2, ~3, ~5

Map 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⟩]
Eigenmonzos: 2, 8/7, 11/9

Template:Val list

Badness: 0.480 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Vals: Template:Val list

Badness: 0.876 × 10-3

Mockingbird

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 100/99, 126/125

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

Vals: Template:Val list

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]]

Vals: Template:Val list

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]]

Template:Val list

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]]

Vals: Template:Val list

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]]

Template:Val list

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]]

Vals: Template:Val list

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]]

Template:Val list

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]]

Vals: Template:Val list

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]]

Template:Val list

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]]

Vals: Template:Val list

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