Greenwoodmic temperaments: Difference between revisions

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{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
: mapping generators: ~2, ~16/15
: mapping generators: ~2, ~16/15


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~16/15 = 138.454
* [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 138.454{{c}}
: [[error map]]: {{val| 0.000 -9.684 -17.038 +0.353 }}
: [[error map]]: {{val| 0.000 -9.684 -17.038 +0.353 }}
* [[POTE]]: ~2 = 1200.000, ~16/15 = 137.958
* [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 137.958{{c}}
: error map: {{val| 0.000 -12.167 -14.059 -3.123 }}
: error map: {{val| 0.000 -12.167 -14.059 -3.123 }}


Line 39: Line 38:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 138.420
* CTE: ~2 = 1200.000{{c}}, ~12/11 = 138.420{{c}}
* POTE: ~2 = 1200.000, ~12/11 = 137.900
* POTE: ~2 = 1200.000{{c}}, ~12/11 = 137.900{{c}}


{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 137.455
* CTE: ~2 = 1200.000{{c}}, ~12/11 = 137.455{{c}}
* POTE: ~2 = 1200.000, ~12/11 = 136.937
* POTE: ~2 = 1200.000{{c}}, ~12/11 = 136.937{{c}}


{{Optimal ET sequence|legend=0| 9, 35 }}
{{Optimal ET sequence|legend=0| 9, 35 }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.583
* CTE: ~2 = 1200.000{{c}}, ~13/12 = 138.583{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 137.917
* POTE: ~2 = 1200.000{{c}}, ~13/12 = 137.917{{c}}


{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}
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{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
: mapping generators: ~189/128, ~3
: mapping generators: ~189/128, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~189/128 = 600.000, ~3/2 = 681.532 (~64/63 = 81.532)
* [[CTE]]: ~189/128 = 600.000{{c}}, ~3/2 = 681.532{{c}} (~64/63 = 81.532{{c}})
: [[error map]]: {{val| 0.000 -20.423 -1.629 -13.421 }}
: [[error map]]: {{val| 0.000 -20.423 -1.629 -13.421 }}
* [[CWE]]: ~189/128 = 600.000, ~3/2 = 682.650 (~64/63 = 82.650)
* [[CWE]]: ~189/128 = 600.000{{c}}, ~3/2 = 682.650{{c}} (~64/63 = 82.650{{c}})
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}


{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}


[[Badness]]:
[[Badness]] (Sintel): 8.42
* Smith: 0.333
* Dirichlet: 8.420


=== 11-limit ===
=== 11-limit ===
Line 107: Line 103:


Optimal tunings:  
Optimal tunings:  
* CTE: ~189/128 = 600.000, ~3/2 = 681.438 (~64/63 = 81.438)
* CTE: ~189/128 = 600.000{{c}}, ~3/2 = 681.438{{c}} (~64/63 = 81.438{{c}})
* CWE: ~189/128 = 600.000, ~3/2 = 682.724 (~64/63 = 82.724)
* CWE: ~189/128 = 600.000{{c}}, ~3/2 = 682.724{{c}} (~64/63 = 82.724{{c}})


{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}


Badness:
Badness (Sintel): 4.15
* Smith: 0.125
* Dirichlet: 4.148


=== 13-limit ===
=== 13-limit ===
Line 124: Line 118:


Optimal tunings:  
Optimal tunings:  
* CTE: ~91/64 = 600.000, ~3/2 = 681.406 (~64/63 = 81.406)
* CTE: ~91/64 = 600.000{{c}}, ~3/2 = 681.406{{c}} (~64/63 = 81.406{{c}})
* CWE: ~91/64 = 600.000, ~3/2 = 682.168 (~64/63 = 82.168)
* CWE: ~91/64 = 600.000{{c}}, ~3/2 = 682.168{{c}} (~64/63 = 82.168{{c}})


{{Optimal ET sequence|legend=0| 14c, 30b }}
{{Optimal ET sequence|legend=0| 14c, 30b }}


Badness:
Badness (Sintel): 4.38
* Smith: 0.106
* Dirichlet: 4.377


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 11:27, 4 July 2026

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

This is a collection of rank-2 greenwoodmic temperaments, which temper out 405/392, the greenwoodma.

Temperaments discussed elsewhere are

Considered below are secund and semishallowtone.

Secund

Subgroup: 2.3.5.7

Comma list: 405/392, 525/512

Mapping: [⟨1 1 3 2], ⟨0 5 -6 7]]

mapping generators: ~2, ~16/15

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~16/15 = 138.454 ¢
error map: ⟨0.000 -9.684 -17.038 +0.353]
  • POTE: ~2 = 1200.000 ¢, ~16/15 = 137.958 ¢
error map: ⟨0.000 -12.167 -14.059 -3.123]

Optimal ET sequence: 9, 17, 26, 61bc, 87bcc

Badness (Smith): 0.089840

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 385/384

Mapping: [⟨1 1 3 2 3], ⟨0 5 -6 7 4]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~12/11 = 138.420 ¢
  • POTE: ~2 = 1200.000 ¢, ~12/11 = 137.900 ¢

Optimal ET sequence: 9, 17, 26, 61bc

Badness (Smith): 0.042562

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 385/384

Mapping: [⟨1 1 3 2 3 2], ⟨0 5 -6 7 4 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~12/11 = 137.455 ¢
  • POTE: ~2 = 1200.000 ¢, ~12/11 = 136.937 ¢

Optimal ET sequence: 9, 35

Badness (Smith): 0.049684

Secundly

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 99/98

Mapping: [⟨1 1 3 2 3 3], ⟨0 5 -6 7 4 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~13/12 = 138.583 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/12 = 137.917 ¢

Optimal ET sequence: 9, 17, 26, 61bcf

Badness (Smith): 0.026212

Semishallowtone

See also: Shallowtone

Subgroup: 2.3.5.7

Comma list: 405/392, 35721/32768

Mapping: [⟨2 0 36 15], ⟨0 1 -10 -3]]

mapping generators: ~189/128, ~3

Optimal tunings:

  • CTE: ~189/128 = 600.000 ¢, ~3/2 = 681.532 ¢ (~64/63 = 81.532 ¢)
error map: ⟨0.000 -20.423 -1.629 -13.421]
  • CWE: ~189/128 = 600.000 ¢, ~3/2 = 682.650 ¢ (~64/63 = 82.650 ¢)
error map: ⟨0.000 -19.305 -12.811 -16.775]

Optimal ET sequence: 14c, 44bd, 58bcd

Badness (Sintel): 8.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 35721/32768

Mapping: [⟨2 0 36 15 32], ⟨0 1 -10 -3 -8]]

Optimal tunings:

  • CTE: ~189/128 = 600.000 ¢, ~3/2 = 681.438 ¢ (~64/63 = 81.438 ¢)
  • CWE: ~189/128 = 600.000 ¢, ~3/2 = 682.724 ¢ (~64/63 = 82.724 ¢)

Optimal ET sequence: 14c, 44bd, 58bcde, 72bbccddee

Badness (Sintel): 4.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 35721/32768

Mapping: [⟨2 0 36 15 32 -2], ⟨0 1 -10 -3 -8 3]]

Optimal tunings:

  • CTE: ~91/64 = 600.000 ¢, ~3/2 = 681.406 ¢ (~64/63 = 81.406 ¢)
  • CWE: ~91/64 = 600.000 ¢, ~3/2 = 682.168 ¢ (~64/63 = 82.168 ¢)

Optimal ET sequence: 14c, 30b

Badness (Sintel): 4.38

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