Biyatismic clan: Difference between revisions

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Badness: 1.456 × 10<sup>-3</sup>
Badness: 1.456 × 10<sup>-3</sup>


==== (13-limit) [[#Eros|Eros]] ====
==== Eros ====
This is the [[13-limit]] restriction of [[#Eros|23-limit Eros]], documented for readability in a separate section of this page.
Eros fairs impressively into the 23-limit as a rank 3 temperament<sup>[''how?'']</sup>.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 230: Line 230:
Badness: 1.150 × 10<sup>-3</sup>
Badness: 1.150 × 10<sup>-3</sup>


==== Inanna ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13
Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.


Comma list: 105/104, 121/120, 275/273
Subgroup: 2.3.5.7.11.13.17


Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
Comma list: 121/120, 154/153, 196/195, 352/351


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}


{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
* CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203


Badness: 1.077 × 10<sup>-3</sup>
Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}


==== Ishtar ====
Badness:
Subgroup: 2.3.5.7.11.13
* Smith: 0.979 × 10<sup>-3</sup>
* Dirichlet: 0.931


Comma list: 91/90, 121/120, 441/440
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
Comma list: 121/120, 154/153, 196/195, 286/285, 352/351


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}


{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479


Badness: 1.151 × 10<sup>-3</sup>
Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


== Eros ==
Badness:
This temperament is an extension of [[#Aphrodite]], specifically [[#(13-limit) Eros|(13-limit) Eros]], into the 23-limit, in which it fairs impressively as a rank 3 temperament.
* Smith: 1.13 × 10<sup>-3</sup>
* Dirichlet: 1.159


Due to having natural extensions to such a high prime limit and due to its unusual efficiency, it is documented here in its own section to keep other extensions of Aphrodite more readable.
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


=== 13-limit ===
Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351
Subgroup: [[13-limit]]


Comma list: 121/120, 441/440, 352/351
Mapping: {{mapping| 1 0 1 3 2 7 6 9 3 | 0 1 1 0 1 -2 -1 -3 1 | 0 0 -4 -3 -2 -2 -5 0 -1 }}


Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
Optimal tunings:  
* CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
* CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188


Optimal tuning ([[CTE]]): 2, ~3 = 1901.722, ~22/21 = 78.110
Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


[[Patent val]] EDO tunings: [[29edo|29]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[92edo|92]]
Badness:  
* Smith: 0.939 × 10<sup>-3</sup>
* Dirichlet: 1.084


Dirichlet badness: 1.075
==== Inanna ====
Subgroup: 2.3.5.7.11.13


=== 17-limit ===
Comma list: 105/104, 121/120, 275/273
Note that there is one more patent val EDO tuning in the 17-limit than in the 19- and 23-limit corresponding to using 46 EDO with a better mapping for 19. 29 disappears as it requires the 29g val.


Subgroup: [[17-limit]]
Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}


Comma list: 121/120, 441/440, 352/351, 154/153
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096


Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}


Optimal tuning ([[CTE]]): 2, ~3 = 1901.930, ~22/21 = 78.254
Badness: 1.077 × 10<sup>-3</sup>


[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]], [[92edo|92]]
==== Ishtar ====
Subgroup: 2.3.5.7.11.13


Dirichlet badness: 0.931
Comma list: 91/90, 121/120, 441/440


=== 19-limit ===
Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
Subgroup: [[19-limit]]


Comma list: 441/440, 1617/1600, 196/195, 154/153, 286/285
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578


Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}
{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}


Optimal tuning ([[CTE]]): 2, ~3/2 = 1901.564, ~22/21 = 78.235
Badness: 1.151 × 10<sup>-3</sup>
 
[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]]
 
Dirichlet badness: 1.159
 
=== 23-limit ===
Subgroup: [[23-limit]]
 
Comma list: 441/440, 1617/1600, 196/195, 154/153, 286/285, 161/160
 
Mapping: {{mapping| 1 0 1 3 2 7 6 9 3 | 0 1 1 0 1 -2 -1 -3 1 | 0 0 -4 -3 -2 -2 -5 0 -1 }}
 
Optimal tuning ([[CTE]]): 2, ~3 = 1901.711, ~23/22 = 78.205
 
[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]]
 
Dirichlet badness: 1.084


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 08:55, 21 April 2024

The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120 = [-3 -1 -1 0 2⟩.

Temperaments discussed elsewhere are:

Considered below are zeus, artemis, oxpecker, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see Rank-4 temperament #Biyatismic (121/120).

Protomere

Subgroup: 2.3.5.11

Comma list: 121/120

Sval mapping: [⟨1 0 1 2], ⟨0 1 1 1], ⟨0 0 -2 -1]]

Mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466

Optimal ET sequence: 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee

Badness: 0.0297 × 10-3

Zeus

Main article: Zeus
See also: Porwell family § Zeus

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175

Mapping: [⟨1 0 1 4 2], ⟨0 1 1 -1 1], ⟨0 0 -2 3 1]]

Mapping to lattice: [⟨0 1 -1 2 0], ⟨0 1 1 -1 1]]

Lattice basis:

11/10, 11/8
Angle (11/10, 11/8) = 87.464 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881

Minimax tuning:

[[1 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0⟩, [22/9 2/9 1/3 -4/9 0⟩, [22/9 2/9 -2/3 5/9 0⟩, [10/3 2/3 0 -1/3 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5.9/7

Optimal ET sequence: 15, 22, 31, 46, 53, 68, 77, 99, 130e

Badness: 0.400 × 10-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5

Zeus11[22] hobbit transversal

33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
11/6, 15/8, 64/33, 2

Zeus11[24] hobbit transversal

33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
11/6, 15/8, 64/33, 2

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350

Mapping: [⟨1 0 1 4 2 7], ⟨0 1 1 -1 1 -2], ⟨0 0 -2 3 -1 -1]]

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

Lattice basis:

11/10 length = 0.7898, 11/8 length = 1.002
Angle (11/10, 11/8) = 106.7439 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0 0⟩, [22/9 2/9 1/3 -4/9 0 0⟩, [22/9 2/9 -2/3 5/9 0 0⟩, [10/3 2/3 0 -1/3 0 0⟩, [14/3 -8/3 1 1/3 0 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
  • 15-odd-limit
[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [11/5 1/5 2/5 -2/5 0 0⟩, [11/5 1/5 -3/5 3/5 0 0⟩, [13/5 3/5 1/5 -1/5 0 0⟩, [38/5 -12/5 1/5 -1/5 0 0⟩]
eigenmonzo (unchanged-interval) basis: 2.3.7/5

Optimal ET sequence: 15, 22, 31, 46, 53, 77, 99, 130e

Badness: 0.934 × 10-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5

Zeus13[22] hobbit transversal

260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2

Tinia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 176/175

Mapping: [⟨1 0 1 4 2 2], ⟨0 1 1 -1 1 1], ⟨0 0 -2 3 -1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666

Optimal ET sequence: 7, 9, 15, 22f, 24, 31

Badness: 0.808 × 10-3

Artemis

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224

Mapping: [⟨1 0 1 -3 2], ⟨0 1 1 4 1], ⟨0 0 -2 -4 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232

Optimal ET sequence: 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195

Mapping: [⟨1 0 1 -3 2 -5], ⟨0 1 1 4 1 6], ⟨0 0 -2 -4 -1 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117

Optimal ET sequence: 9, 20, 22f, 29, 31

Diana

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273

Mapping: [⟨1 0 1 -3 2 7], ⟨0 1 1 4 1 -2], ⟨0 0 -2 -4 -1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048

Optimal ET sequence: 22, 29, 31, 53, 82e, 84e, 113e, 166ee

Oxpecker

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756

Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31, 46, 77

Badness: 0.699 × 10-3

Woodpecker

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 126/125

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652

Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31

Badness: 1.093 × 10-3

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)3 with 8/7.

7-limit (squalentine)

Subgroup: 2.3.5.7

Comma list: 64827/64000

Mapping: [⟨1 0 1 3], ⟨0 1 1 0], ⟨0 0 -4 -3]]

Mapping generators: ~2, ~3, ~21/20

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694

Optimal ET sequence: 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d

Badness: 0.943 × 10-3

Projection pairs: 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: [⟨1 0 1 3 2], ⟨0 1 1 0 1], ⟨0 0 -4 -3 -2]]

Mapping generators: ~2, ~3, ~22/21

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421

Optimal ET sequence: 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee

Badness: 0.583 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 351/350, 441/440

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211

Optimal ET sequence: 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef

Badness: 1.456 × 10-3

Eros

Eros fairs impressively into the 23-limit as a rank 3 temperament[how?].

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351

Mapping: [⟨1 0 1 3 2 7], ⟨0 1 1 0 1 -2], ⟨0 0 -4 -3 -2 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5014, ~21/20 = 78.6143

Optimal ET sequence: 17c, 29, 31, 46, 60e, 77, 106de, 183dee

Badness: 1.150 × 10-3

17-limit

Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 196/195, 352/351

Mapping: [⟨1 0 1 3 2 7 6], ⟨0 1 1 0 1 -2 -1], ⟨0 0 -4 -3 -2 -2 -5]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
  • CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203

Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 77, 106de

Badness:

  • Smith: 0.979 × 10-3
  • Dirichlet: 0.931
19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 154/153, 196/195, 286/285, 352/351

Mapping: [⟨1 0 1 3 2 7 6 9], ⟨0 1 1 0 1 -2 -1 -3], ⟨0 0 -4 -3 -2 -2 -5 0]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
  • CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479

Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de

Badness:

  • Smith: 1.13 × 10-3
  • Dirichlet: 1.159
23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351

Mapping: [⟨1 0 1 3 2 7 6 9 3], ⟨0 1 1 0 1 -2 -1 -3 1], ⟨0 0 -4 -3 -2 -2 -5 0 -1]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
  • CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188

Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de

Badness:

  • Smith: 0.939 × 10-3
  • Dirichlet: 1.084

Inanna

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 275/273

Mapping: [⟨1 0 1 3 2 1], ⟨0 1 1 0 1 2], ⟨0 0 -4 -3 -2 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096

Optimal ET sequence: 14cf, 15, 29, 31, 45ef, 60e

Badness: 1.077 × 10-3

Ishtar

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 441/440

Mapping: [⟨1 0 1 3 2 -1], ⟨0 1 1 0 1 3], ⟨0 0 -4 -3 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578

Optimal ET sequence: 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def

Badness: 1.151 × 10-3

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