Greenwoodmic temperaments: Difference between revisions

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


Temperaments discussed elsewhere include [[Augmented family #August|august]], [[Meantone family #Injera|injera]], [[Archytas clan #Schism|schism]], [[Dicot family #Sidi|sidi]], and [[Pelogic family #Superpelog|superpelog]].
Temperaments discussed elsewhere are
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[Ripple]]'' (+126/125) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Wesley]]'' (+875/864) → [[Wesley family #Septimal wesley|Wesley family]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


== Greenwood ==
Considered below are secund and semishallowtone.
{{see also|Apotome family #Greenwood}}


Subgroup: 2.3.5.7
== 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]]. Related temperaments include [[progression]] and [[jerome]], which only differ by the mapping of 5.  


[[Comma list]]: 405/392, 1323/1280
The fourth generator step (representing [[45/32]]~[[49/36]]) comes close to [[11/8]]. This allows a simple [[11-limit]] extension in which the generator doubles as [[12/11]], tempering out [[45/44]] and [[99/98]] as is natural for greenwoodmic temperaments. From here, it only makes sense to let the generator take on the additional role of [[13/12]]~[[14/13]], resulting in a full [[13-limit]] temperament.


[[Mapping]]: [{{val|7 11 1 12}}, {{val|0 0 2 1}}]
Possible tunings include [[26edo]], [[35edo]] in the 35f val, or anything in between.


{{Multival|legend=1|0 14 7 22 11 -23}}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~8/7 = 241.490
[[Comma list]]: 405/392, 525/512


{{Val list|legend=1| 14c, 21, 35 }}
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
: mapping generators: ~2, ~16/15


[[Badness]]: 0.121752
[[Optimal tuning]]s:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}
 
[[Badness]] (Sintel): 2.27


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


Comma list: 45/44, 99/98, 1323/1280
Comma list: 45/44, 99/98, 385/384


Mapping: [{{val|7 11 1 12 9}}, {{val|0 0 2 1 2}}]
Mapping: {{mapping| 1 1 3 2 3 | 0 5 -6 7 4 }}


POTE generator: ~8/7 = 242.711
Optimal tunings:  
* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}


Vals: {{Val list| 14c, 21, 35, 49bcde, 84bbccde }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}


Badness: 0.057471
Badness (Sintel): 1.41


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


Comma list: 27/26, 45/44, 99/98, 640/637
Comma list: 45/44, 65/64, 78/77, 99/98
 
Mapping: [{{val|7 11 1 12 9 26}}, {{val|0 0 2 1 2 0}}]


POTE generator: ~8/7 = 238.607
Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}


Vals: {{Val list| 14c, 21, 35 }}
Optimal tunings:  
* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}


Badness: 0.054009
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}


== Secund ==
Badness (Sintel): 1.08
{{see also|Avicennmic temperaments #Secund}}


Subgroup: 2.3.5.7
== Semishallowtone ==
{{See also| Shallowtone }}


[[Comma list]]: 405/392, 525/512
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 1 3 2}}, {{val|0 5 -6 7}}]
[[Comma list]]: 405/392, 35721/32768


{{Multival|legend=1|5 -6 7 -21 -3 33}}
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
: mapping generators: ~189/128, ~3


[[POTE generator]]: ~16/15 = 137.958
[[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 }}


{{Val list|legend=1| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}


[[Badness]]: 0.089840
[[Badness]] (Sintel): 8.42


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


Comma list: 45/44, 99/98, 385/384
Comma list: 45/44, 99/98, 35721/32768


Mapping: [{{val|1 1 3 2 3}}, {{val|0 5 -6 7 4}}]
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}


POTE generator: ~12/11 = 137.900
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}})


Vals: {{Val list| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}


Badness: 0.042562
Badness (Sintel): 4.15


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


Comma list: 27/26, 45/44, 99/98, 385/384
Comma list: 27/26, 45/44, 99/98, 35721/32768
 
Mapping: [{{val|1 1 3 2 3 2}}, {{val|0 5 -6 7 4 15}}]
 
POTE generator: ~12/11 = 136.937
 
Vals: {{Val list| 9, 26f, 35 }}
 
Badness: 0.049684
 
=== Secundly ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 65/64, 78/77, 99/98


Mapping: [{{val|1 1 3 2 3 3}}, {{val|0 5 -6 7 4 6}}]
Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}


POTE generator: ~12/11 = 137.917
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}})


Vals: {{Val list| 9, 17, 26, 61bcf, 87bccf }}
{{Optimal ET sequence|legend=0| 14c, 30b }}


Badness: 0.026212
Badness (Sintel): 4.38


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

Latest revision as of 14:27, 11 September 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. Related temperaments include progression and jerome, which only differ by the mapping of 5.

The fourth generator step (representing 45/32~49/36) comes close to 11/8. This allows a simple 11-limit extension in which the generator doubles as 12/11, tempering out 45/44 and 99/98 as is natural for greenwoodmic temperaments. From here, it only makes sense to let the generator take on the additional role of 13/12~14/13, resulting in a full 13-limit temperament.

Possible tunings include 26edo, 35edo in the 35f val, or anything in between.

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: 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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