Greenwoodmic temperaments: Difference between revisions

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m Spell 1\1 etc. in decimal; note the badness metric being used; misc. cleanup
 
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{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[greenwoodma]], 405/392 ({{monzo| -3 4 1 -2 }}).
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[greenwoodma]], 405/392 ({{monzo| -3 4 1 -2 }}).


Line 7: Line 8:
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Apotome family]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


== Secund ==
== Secund ==
Line 16: Line 17:
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}


{{Multival|legend=1| 5 -6 7 -21 -3 33 }}
: mapping generaators: ~2, ~16/15


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


{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}
Line 33: Line 37:


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


{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}


Badness (Smith): 0.042562
Badness (Smith): 0.042562
Line 47: Line 52:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 137.455
* POTE: ~2 = 1200.000, ~12/11 = 136.937
* POTE: ~2 = 1200.000, ~12/11 = 136.937


{{Optimal ET sequence|legend=0| 9, 26f, 35 }}
{{Optimal ET sequence|legend=0| 9, 35 }}


Badness (Smith): 0.049684
Badness (Smith): 0.049684
Line 61: Line 67:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~12/11 = 137.917
* CTE: ~2 = 1200.000, ~13/12 = 138.583
* POTE: ~2 = 1200.000, ~13/12 = 137.917


{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf, 87bccf }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}


Badness (Smith): 0.026212
Badness (Smith): 0.026212
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[[Comma list]]: 405/392, 35721/32768
[[Comma list]]: 405/392, 35721/32768


{{Mapping|legend=1| 2 2 16 9 | 0 1 -10 -3 }}
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
 
: mapping generators: ~189/128, ~3


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


[[Support]]ing [[ET]]s: {{EDOs| 14c, 30b, 44bd, 58bcd, 72bbccdd }}
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}


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


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 99/98, 43659/40960
Comma list: 45/44, 99/98, 35721/32768


Mapping: {{mapping| 2 2 16 9 16 | 0 1 -10 -3 -8}}
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~189/128 = 600.000, ~3/2 = 681.438
* 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)


Supporting ETs: {{EDOs| 14c, 30b, 44bd, 58bcde, 72bbccddee }}
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}


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


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: {{mapping| 2 2 16 9 16 4 | 0 1 -10 -3 -8 3 }}
Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~189/128 = 600.000, ~3/2 = 681.406
* 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)


Supporting ETs: {{EDOs| 14c, 30b }}
{{Optimal ET sequence|legend=0| 14c, 30b }}


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


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic| ]] <!-- key article -->
[[Category:Greenwoodmic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:26, 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.

This is a collection of rank-2 temperaments tempering out the greenwoodma, 405/392 ([-3 4 1 -2⟩).

Temperaments discussed elsewhere are

Secund

Subgroup: 2.3.5.7

Comma list: 405/392, 525/512

Mapping: [⟨1 1 3 2], ⟨0 5 -6 7]]

mapping generaators: ~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:

  • Smith: 0.333
  • Dirichlet: 8.420

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:

  • Smith: 0.125
  • Dirichlet: 4.148

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:

  • Smith: 0.106
  • Dirichlet: 4.377
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