Werckismic temperaments: Difference between revisions
+links to temperaments discussed elsewhere |
+trimyna moved from trimyna family |
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* ''[[pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele]] | * ''[[pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele]] | ||
* ''[[belobog]]'', {441/440, 3136/3125} → [[Hemimean family #Belobog]] | * ''[[belobog]]'', {441/440, 3136/3125} → [[Hemimean family #Belobog]] | ||
* ''[[turan]]'', {441/440, 10976/10935} → [[Hemimage family #Turan]] | * ''[[turan]]'', {441/440, 10976/10935} → [[Hemimage family #Turan]] | ||
* ''[[snape]]'', {441/440, 19683/19600} → [[Cataharry family #Snape]] | * ''[[snape]]'', {441/440, 19683/19600} → [[Cataharry family #Snape]] | ||
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[[Badness]]: 0.419 × 10<sup>-3</sup> | [[Badness]]: 0.419 × 10<sup>-3</sup> | ||
== Trimyna == | |||
=== 7-limit === | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 50421/50000 | |||
[[Mapping]]: [{{val| 1 4 0 0 }}, {{val| 0 -5 0 1 }}, {{val| 0 0 1 1 }}] | |||
Mapping generators: ~2, ~7/5, ~5 | |||
{{Val list|legend=1| 27, 31, 58, 87, 118, 267d, 385, 412d }} <nowiki>*</nowiki> | |||
<nowiki>*</nowiki> [[optimal patent val]]: [[294edo|294]] | |||
[[Badness]]: 0.870 × 10<sup>-3</sup> | |||
[[Projection pair]]: 3 50000/16807 to 2.5.7 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
[[Comma list]]: 441/440, 3388/3375 | |||
[[Mapping]]: [{{val| 1 4 0 0 5 }}, {{val| 0 -5 0 1 -8 }}, {{val| 0 0 1 1 1 }}] | |||
{{Val list|legend=1| 27e, 31, 58, 87, 118, 294, 412d }} | |||
[[Badness]]: 0.512 × 10<sup>-3</sup> | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 196/195, 352/351, 1001/1000 | |||
Mapping: [{{val| 1 4 0 0 5 -2 }}, {{val| 0 -5 0 1 -8 7 }}, {{val| 0 0 1 1 1 1 }}] | |||
Vals: {{Val list| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }} | |||
Badness: 0.814 × 10<sup>-3</sup> | |||
==== Trimy ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 441/440, 676/675, 847/845 | |||
Mapping: [{{val| 1 4 0 0 5 5 }}, {{val| 0 -10 0 2 -16 -15 }}, {{val| 0 0 1 1 1 1 }}] | |||
Mapping generators: ~2, ~13/11, ~5 | |||
Vals: {{Val list| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }} | |||
Badness: 1.478 × 10<sup>-3</sup> | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
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[[Category:Rank 3]] | [[Category:Rank 3]] | ||
[[Category:Varuna]] | [[Category:Varuna]] | ||
[[Category:Trimyna]] | |||
Revision as of 15:06, 6 June 2021
This is a collection of rank-3 temperaments tempering out 441/440, but we can begin by looking at the rank-4 temperament.
Temperaments discussed elsewhere are:
- vulcan, {56/55, 64/63} → Archytas family #Vulcan
- euterpe, {81/80, 99/98} → Didymus rank three family #Euterpe
- nakika, {100/99, 245/242} → Octagar family #Nakika
- thrush, {126/125, 441/440} → Starling family #Thrush
- prodigy, {225/224, 441/440} → Marvel family #Prodigy
- jove, {243/242, 441/440} → Breed family #Jove
- sensawer, {245/243, 441/440} → Sensamagic family #Sensawer
- portent, {385/384, 441/440} → Gamelismic family #Portent
- pele, {441/440, 896/891} → Hemifamity family #Pele
- belobog, {441/440, 3136/3125} → Hemimean family #Belobog
- turan, {441/440, 10976/10935} → Hemimage family #Turan
- snape, {441/440, 19683/19600} → Cataharry family #Snape
Werckismic
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 -3], ⟨0 1 0 0 2], ⟨0 0 1 0 -1], ⟨0 0 0 1 2]]
- [[1 0 0 0 0⟩, [1/2 2/3 1/6 -1/3 1/6⟩, [0 0 1 0 0⟩, [1 -2/3 1/3 1/3 1/3⟩, [0 0 0 0 1⟩]
- Eigenmonzos: 2, 5, 11, 9/7
History
Subgroup: 2.3.5.7.11
Comma list: 441/400, 4000/3993
Mapping: [⟨1 2 0 0 1], ⟨0 -6 0 7 2], ⟨0 0 1 1 1]]
Mapping generators: ~2, ~21/20, ~5
- 11-odd-limit
- [[1 0 0 0 0⟩, [4/17 12/17 0 -6/17 6/17⟩, [-27/17 4/17 0 -2/17 19/17⟩, [8/17 -10/17 0 5/17 12/17⟩, [0 0 0 0 1⟩]
- Eigenmonzos: 2, 11, 9/7
Badness: 0.604 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 1001/1000
Mapping: [⟨1 2 0 0 1 2], ⟨0 -6 0 7 2 -9], ⟨0 0 1 1 1 1]]
Mapping generators: ~2, ~21/20, ~5
Minimax tuning:
- 13-odd-limit
- [[1 0 0 0 0 0⟩, [5/4 0 0 -3/8 0 3/8⟩, [0 0 1 0 0 0⟩, [7/8 0 1 7/16 0 -7/16⟩, [5/4 0 1 1/8 0 -1/8⟩, [7/8 0 1 -9/16 0 9/16⟩]
- Eigenmonzos: 2, 5, 13/7
Vals: Template:Val list
Badness: 0.540 × 10-3
Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
Varuna
Subgroup: 2.3.5.7.11
Comma list: 441/440, 8019/8000
Mapping: [⟨2 0 0 9 12], ⟨0 1 0 -4 -6], ⟨0 0 1 2 3]]
Mapping generators: ~99/70, ~3, ~5
Badness: 0.419 × 10-3
Trimyna
7-limit
Subgroup: 2.3.5.7
Comma list: 50421/50000
Mapping: [⟨1 4 0 0], ⟨0 -5 0 1], ⟨0 0 1 1]]
Mapping generators: ~2, ~7/5, ~5
Badness: 0.870 × 10-3
Projection pair: 3 50000/16807 to 2.5.7
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3388/3375
Mapping: [⟨1 4 0 0 5], ⟨0 -5 0 1 -8], ⟨0 0 1 1 1]]
Badness: 0.512 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 1001/1000
Mapping: [⟨1 4 0 0 5 -2], ⟨0 -5 0 1 -8 7], ⟨0 0 1 1 1 1]]
Vals: Template:Val list
Badness: 0.814 × 10-3
Trimy
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 676/675, 847/845
Mapping: [⟨1 4 0 0 5 5], ⟨0 -10 0 2 -16 -15], ⟨0 0 1 1 1 1]]
Mapping generators: ~2, ~13/11, ~5
Vals: Template:Val list
Badness: 1.478 × 10-3