Werckismic temperaments: Difference between revisions

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→Varuna: +13-limit (according to the temp finder)
-aphrodite (addressed in biyatismic temperaments)
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This is a collection of rank-3 temperaments tempering out [[441/440]], but we can begin by looking at the rank-4 temperament.  
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:  
Temperaments discussed elsewhere are:  
* ''[[festival]]'', {45/44, 50/49} → [[Jubilismic family #Festival]]
* ''[[Festival]]'', {45/44, 50/49} → [[Jubilismic family #Festival|Jubilismic family]]
* ''[[vulcan]]'', {56/55, 64/63} → [[Archytas family #Vulcan]]
* ''[[Vulcan]]'', {56/55, 64/63} → [[Archytas family #Vulcan|Archytas family]]
* ''[[euterpe]]'', {81/80, 99/98} → [[Didymus rank three family #Euterpe]]
* ''[[Euterpe]]'', {81/80, 99/98} → [[Didymus rank three family #Euterpe|Didymus rank-3 family]]
* ''[[nakika]]'', {100/99, 245/242} → [[Octagar family #Nakika]]
* ''[[Nakika]]'', {100/99, 245/242} → [[Octagar family #Nakika|Octagar family]]
* ''[[thrush]]'', {126/125, 441/440} → [[Starling family #Thrush]]
* ''[[Aphrodite]]'', {121/120, 441/440} → [[Biyatismic temperaments #Aphrodite|Biyatismic temperaments]]
* ''[[prodigy]]'', {225/224, 441/440} → [[Marvel family #Prodigy]]
* ''[[Thrush]]'', {126/125, 441/440} → [[Starling family #Thrush|Starling family]]
* ''[[jove]]'', {243/242, 441/440} → [[Breed family #Jove]]
* ''[[Prodigy]]'', {225/224, 441/440} → [[Marvel family #Prodigy|Marvel family]]
* ''[[sensawer]]'', {245/243, 441/440} → [[Sensamagic family #Sensawer]]
* ''[[Jove]]'', {243/242, 441/440} → [[Breed family #Jove|Breed family]]
* ''[[portent]]'', {385/384, 441/440} → [[Gamelismic family #Portent]]
* ''[[Sensawer]]'', {245/243, 441/440} → [[Sensamagic family #Sensawer|Sensamagic family]]
* ''[[demeter]]'', {441/440, 686/675} → [[Sengic family #Demeter]]
* ''[[Portent]]'', {385/384, 441/440} → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele]]
* ''[[Demeter]]'', {441/440, 686/675} → [[Sengic family #Demeter|Sengic family]]
* ''[[tiberius]]'', {441/440, 2048/2025} → [[Diaschismic rank three family #Tiberius]]
* ''[[Pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[belobog]]'', {441/440, 3136/3125} → [[Hemimean family #Belobog]]
* ''[[Tiberius]]'', {441/440, 2048/2025} → [[Diaschismic rank three family #Tiberius|Diaschismic rank-3 family]]
* ''[[turan]]'', {441/440, 10976/10935} → [[Hemimage family #Turan]]
* ''[[Belobog]]'', {441/440, 3136/3125} → [[Hemimean family #Belobog|Hemimean family]]
* ''[[snape]]'', {441/440, 19683/19600} → [[Cataharry family #Snape]]
* ''[[Turan]]'', {441/440, 10976/10935} → [[Hemimage family #Turan|Hemimage family]]
* ''[[Snape]]'', {441/440, 19683/19600} → [[Cataharry family #Snape|Cataharry family]]


== Werckismic ==
== Werckismic ==
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Badness: 1.478 × 10<sup>-3</sup>
Badness: 1.478 × 10<sup>-3</sup>
== Aphrodite ==
Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)<sup>3</sup> with 8/7.
=== 7-limit (squalentine) ===
Subgroup: 2.3.5.7
[[Comma list]]: 64827/64000
[[Mapping]]: [{{val| 1 0 1 3 }}, {{val| 0 1 1 0 }}, {{val| 0 0 4 3 }}]
Mapping generators: ~2, ~3, ~21/20
{{Val list|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
[[Badness]]: 0.943 × 10<sup>-3</sup>
[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Comma list]]: 121/120, 441/440
[[Mapping]]: [{{val| 1 0 1 3 2 }}, {{val| 0 1 1 0 1 }}, {{val| 0 0 4 3 2 }}]
Mapping generators: ~2, ~3, ~22/21
{{Val list|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
[[Badness]]: 0.583 × 10<sup>-3</sup>
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 351/350, 441/440
Mapping: [{{val| 1 0 1 3 2 6 }}, {{val| 0 1 1 0 1 -1 }}, {{val| 0 0 4 3 2 11 }}]
Vals: {{Val list| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
Badness: 1.456 × 10<sup>-3</sup>
==== Eros ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 196/195, 352/351
Mapping: [{{val| 1 0 1 3 2 7 }}, {{val| 0 1 1 0 1 -2 }}, {{val| 0 0 4 3 2 2 }}]
Vals: {{Val list| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
Badness: 1.150 × 10<sup>-3</sup>
==== Inanna ====
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 121/120, 275/273
Mapping: [{{val| 1 0 1 3 2 1 }}, {{val| 0 1 1 0 1 2 }}, {{val| 0 0 4 3 2 7 }}]
Vals: {{Val list| 14cf, 15, 29, 31, 45ef, 60e }}
Badness: 1.077 × 10<sup>-3</sup>
==== Ishtar ====
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 441/440
Mapping: [{{val| 1 0 1 3 2 -1 }}, {{val| 0 1 1 0 1 3 }}, {{val| 0 0 4 3 2 1 }}]
Vals: {{Val list| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
Badness: 1.151 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 15:53, 8 December 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:

Werckismic

Subgroup: 2.3.5.7.11

Comma list: 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 0 0 2], ⟨0 0 1 0 -1], ⟨0 0 0 1 2]]

Minimax tuning:

[[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

Template:Val list

History

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993

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

Mapping generators: ~2, ~21/20, ~5

Minimax tuning:

[[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

Template:Val list

Badness: 0.604 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675

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

7-limit

Subgroup: 2.3.5.7

Comma list: 321489/320000

Mapping: [⟨2 0 0 9], ⟨0 1 0 -4], ⟨0 0 1 2]]

Mapping generators: ~567/400, ~3, ~5

Template:Val list

Badness: 0.501 × 10-3

11-limit

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]]

Template:Val list

Badness: 0.419 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440

Mapping: [⟨2 0 0 9 12 11], ⟨0 1 0 -4 -6 -7], ⟨0 0 1 2 3 4]]

Vals: Template:Val list

Badness: 0.636 × 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

Template:Val list *

* optimal patent val: 294

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]]

Template:Val list

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

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