Greenwoodmic temperaments: Difference between revisions

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__FORCETOC__
{{Technical data page}}
These temper out the greenwoodma, |-3 4 1 -2> = 405/392.
This is a collection of [[rank-2 temperament|rank-2]] '''greenwoodmic temperaments''', which [[tempering out|temper out]] [[405/392]], the [[greenwoodma]].  


=Greenwood=
Temperaments discussed elsewhere are
Commas: 405/392, 1323/1280
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


POTE generator: ~8/7 = 241.490
Considered below are secund and semishallowtone.  


Map: [<7 11 1 12|, <0 0 2 1|]
== Secund ==
Secund tempers out the greendwoodma, the [[avicennma]], and the [[orwellisma]]. It may be described as the {{nowrap| 9 & 26 }} temperament, with a [[ploidacot]] signature of pentacot. It divides the [[3/2|perfect fifth]] into five [[~]][[16/15]] generators, two for [[7/6]] and three for [[9/7]].


Wedgie: <<0 14 7 22 11 -23||
[[Subgroup]]: 2.3.5.7


EDOs: 14c, 21, 35, 84bcd
[[Comma list]]: 405/392, 525/512


Badness: 0.1218
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
: mapping generators: ~2, ~16/15


==11-limit==
[[Optimal tuning]]s:
Commas: 45/44, 99/98, 1323/1280
* [[WE]]: ~2 = 1203.6786{{c}}, ~16/15 = 138.3805{{c}}
: [[error map]]: {{val| +3.679 -6.374 -5.561 +7.195 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 138.0648{{c}}
: error map: {{val| 0.000 -11.631 -14.702 -2.372 }}


POTE generator: ~8/7 = 242.711
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}


Map: [<7 11 1 12 9|, <0 0 2 1 2|]
[[Badness]] (Sintel): 2.27


EDOs: 14c, 21, 35, 49bcde, 84bcde
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0575
Comma list: 45/44, 99/98, 385/384


==13-limit==
Mapping: {{mapping| 1 1 3 2 3 | 0 5 -6 7 4 }}
Commas: 27/26, 45/44, 99/98, 640/637


POTE generaor: ~8/7 = 238.607
Optimal tunings:  
* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}


Map: [<7 11 1 12 9 26|, <0 0 2 1 2 0|]
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}


EDOs: 14c, 21, 35, 91bf
Badness (Sintel): 1.41


Badness: 0.0540
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Secund=
Comma list: 27/26, 45/44, 99/98, 385/384
Commas: 405/392, 525/512


POTE generator: ~16/15 = 137.958
Mapping: {{mapping| 1 1 3 2 3 2 | 0 5 -6 7 4 15 }}


Map: [<1 1 3 2|, <0 5 -6 7|]
Optimal tunings:  
* WE: ~2 = 1202.4465{{c}}, ~12/11 = 137.2165{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.0309{{c}}


Wedgie: <<5 -6 7 -21 -3 33||
{{Optimal ET sequence|legend=0| 9, 35 }}


EDOs: 9, 17, 26, 61bc, 87bc
Badness (Sintel): 2.05


Badness: 0.0898
==== Secundly ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 45/44, 65/64, 78/77, 99/98
Commas: 45/44, 99/98, 385/384


POTE generator: ~12/11 = 137.900
Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}


Map: [<1 1 3 2 3|, <0 5 -6 7 4|]
Optimal tunings:  
* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}


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


Badness: 0.0426
Badness (Sintel): 1.08


==13-limit==
== Semishallowtone ==
Commas: 27/26, 45/44, 99/98, 385/384
{{See also| Shallowtone }}


POTE generator: ~12/11 = 136.937
[[Subgroup]]: 2.3.5.7


Map: [<1 1 3 2 3 2|, <0 5 -6 7 4 15|]
[[Comma list]]: 405/392, 35721/32768


EDOs: 9, 26f, 35
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
: mapping generators: ~189/128, ~3


Badness: 0.0497
[[Optimal tuning]]s:
* [[WE]]: ~189/128 = 603.1653{{c}}, ~3/2 = 686.6365{{c}} (~64/63 = 83.4712{{c}})
: [[error map]]: {{val| +6.331 -8.988 -2.034 -0.248 }}
* [[CWE]]: ~189/128 = 600.0000{{c}}, ~3/2 = 682.6498{{c}} (~64/63 = 82.6498{{c}})
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}
 
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}
 
[[Badness]] (Sintel): 8.42
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 99/98, 35721/32768
 
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}
 
Optimal tunings:
* WE: ~189/128 = 603.0536{{c}}, ~3/2 = 686.6175{{c}} (~64/63 = 83.5640{{c}})
* CWE: ~189/128 = 600.0000{{c}}, ~3/2 = 682.7238{{c}} (~64/63 = 82.7238{{c}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}
 
Optimal tunings:
* WE: ~91/64 = 601.8872{{c}}, ~3/2 = 684.4774{{c}} (~64/63 = 82.5902{{c}})
* CWE: ~91/64 = 600.0000{{c}}, ~3/2 = 682.1676{{c}} (~64/63 = 82.1676{{c}})
 
{{Optimal ET sequence|legend=0| 14c, 30b }}
 
Badness (Sintel): 4.38
 
[[Category:Temperament collections]]
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 11:55, 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

Secund tempers out the greendwoodma, the avicennma, and the orwellisma. It may be described as the 9 & 26 temperament, with a ploidacot signature of pentacot. It divides the perfect fifth into five ~16/15 generators, two for 7/6 and three for 9/7.

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:

  • WE: ~2 = 1203.6786 ¢, ~16/15 = 138.3805 ¢
error map: ⟨+3.679 -6.374 -5.561 +7.195]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 138.0648 ¢
error map: ⟨0.000 -11.631 -14.702 -2.372]

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

Badness (Sintel): 2.27

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:

  • WE: ~2 = 1202.9415 ¢, ~12/11 = 138.2377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 137.9970 ¢

Optimal ET sequence: 9, 17, 26, 61bc

Badness (Sintel): 1.41

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:

  • WE: ~2 = 1202.4465 ¢, ~12/11 = 137.2165 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 137.0309 ¢

Optimal ET sequence: 9, 35

Badness (Sintel): 2.05

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:

  • WE: ~2 = 1203.0195 ¢, ~13/12 = 138.2642 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 138.0317 ¢

Optimal ET sequence: 9, 17, 26, 61bcf

Badness (Sintel): 1.08

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:

  • WE: ~189/128 = 603.1653 ¢, ~3/2 = 686.6365 ¢ (~64/63 = 83.4712 ¢)
error map: ⟨+6.331 -8.988 -2.034 -0.248]
  • CWE: ~189/128 = 600.0000 ¢, ~3/2 = 682.6498 ¢ (~64/63 = 82.6498 ¢)
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:

  • WE: ~189/128 = 603.0536 ¢, ~3/2 = 686.6175 ¢ (~64/63 = 83.5640 ¢)
  • CWE: ~189/128 = 600.0000 ¢, ~3/2 = 682.7238 ¢ (~64/63 = 82.7238 ¢)

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:

  • WE: ~91/64 = 601.8872 ¢, ~3/2 = 684.4774 ¢ (~64/63 = 82.5902 ¢)
  • CWE: ~91/64 = 600.0000 ¢, ~3/2 = 682.1676 ¢ (~64/63 = 82.1676 ¢)

Optimal ET sequence: 14c, 30b

Badness (Sintel): 4.38

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