Catalog of rank-4 temperaments: Difference between revisions

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+ancient from keenanismic family, +comma list for each temp
Edo lists reviewed (1/2)
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Generators: ~2, ~3, ~5, ~7
Generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 12, 31, 41, 46, 58, 72, 87, 118, 130 }}
{{Val list|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}


=== Commas 364/363, 441/440 ===
=== Commas 364/363, 441/440 ===
Line 27: Line 27:
* [[13-odd-limit]]
* [[13-odd-limit]]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/3 -1/3 -1/3 1/3 }}, {{monzo| 1/6 0 5/6 2/3 -5/6 1/3 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 1/6 0 -1/6 2/3 1/6 1/3 }}, {{monzo| 0 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/3 -1/3 -1/3 1/3 }}, {{monzo| 1/6 0 5/6 2/3 -5/6 1/3 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 1/6 0 -1/6 2/3 1/6 1/3 }}, {{monzo| 0 0 0 0 0 1 }}]
: Eigenmonzos: 2, 11/10, 8/7, 16/13
: [[Eigenmonzo]]s: 2, 11/10, 8/7, 16/13
* [[15-odd-limit]]
* [[15-odd-limit]]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 1/4 1/4 -1/4 -1/4 1/4 }}, {{monzo| 5/4 -3/4 5/4 -1/4 -1/4 1/4 }}, {{monzo| 17/8 -11/8 5/8 -1/8 3/8 1/8 }}, {{monzo| 5/2 -3/2 1/2 -1/2 1/2 1/2 }}, {{monzo| 17/8 -11/8 5/8 -9/8 3/8 9/8 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 1/4 1/4 -1/4 -1/4 1/4 }}, {{monzo| 5/4 -3/4 5/4 -1/4 -1/4 1/4 }}, {{monzo| 17/8 -11/8 5/8 -1/8 3/8 1/8 }}, {{monzo| 5/2 -3/2 1/2 -1/2 1/2 1/2 }}, {{monzo| 17/8 -11/8 5/8 -9/8 3/8 9/8 }}]
: Eigenmonzos: 2, 14/13, 6/5, 11/9
: [[Eigenmonzo]]s: 2, 14/13, 6/5, 11/9


{{Val list|legend=1| 15, 23, 26, 29, 41, 43, 46, 58, 113, 130, 159, 217, 289, 393 }}
{{Val list|legend=1| 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289 }}


=== Commas 351/350, 441/440 ===
=== Commas 351/350, 441/440 ===
Line 43: Line 43:
Generators: ~2, ~3, ~5, ~7
Generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 26, 31, 46, 58, 72, 103, 130 }}
{{Val list|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}


=== Commas 196/195, 352/351 ===
=== Commas 196/195, 352/351 ===
Line 54: Line 54:
Generators: ~2, ~3, ~5, ~7
Generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 29, 31, 41, 46, 58, 77, 87 }}
{{Val list|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}


=== Commas 441/440, 847/845 ===
=== Commas 441/440, 847/845 ===
Line 65: Line 65:
Generators: ~2, ~3, ~5, ~13/11
Generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 29, 41, 46, 58, 87, 103, 149 }}
{{Val list|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}


== Keenanismic (385/384) ==
== Keenanismic (385/384) ==
Line 85: Line 85:
: Eigenmonzos: 2, 3, 7/5, 11/5
: Eigenmonzos: 2, 3, 7/5, 11/5


{{Val list|legend=1| 15, 22, 31, 41, 46, 53, 72, 87, 94, 118 }}
{{Val list|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}


[[Badness]]: 15.159 × 10<sup>-9</sup>
[[Badness]]: 15.159 × 10<sup>-9</sup>
Line 108: Line 108:
: Eigenmonzos: 2, 14/11, 13/10, 4/3
: Eigenmonzos: 2, 14/11, 13/10, 4/3


{{Val list|legend=1| 7, 8, 15, 19, 26, 34, 41, 46, 53, 68, 72, 79, 87, 94, 99, 113, 140, 159, 212, 253 }}
{{Val list|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}


[[Badness]]: 2.206 × 10<sup>-6</sup>
[[Badness]]: 2.206 × 10<sup>-6</sup>
Line 121: Line 121:
[[Transpose]]: [2 3 5 7 385/35 625/48]
[[Transpose]]: [2 3 5 7 385/35 625/48]


{{Val list|legend=1| 6, 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190 }}
{{Val list|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}


[[Badness]]: 2.573 × 10<sup>-6</sup>
[[Badness]]: 2.573 × 10<sup>-6</sup>
Line 134: Line 134:
Generators: ~2, ~3, ~5, ~7
Generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 19, 22, 26, 31, 46, 50, 53, 72, 77, 103 }}
{{Val list|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}


=== Commas 352/351, 385/384 ===
=== Commas 352/351, 385/384 ===
Line 145: Line 145:
Generators: ~2, ~3, ~5, ~7
Generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 10, 22, 31, 41, 46, 53, 63, 77, 87, 94 }}
{{Val list|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}


=== Commas 364/363, 385/384 ===
=== Commas 364/363, 385/384 ===
Line 156: Line 156:
Generators: ~2, ~3, ~5, ~7
Generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 9, 15, 22, 26, 41, 46, 50, 72, 87 }}
{{Val list|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}


=== Commas 385/384, 847/845 ===
=== Commas 385/384, 847/845 ===
Line 167: Line 167:
Generators: ~2, ~3, ~5, ~13/11
Generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 7, 9, 41, 46, 50, 53, 87, 94, 103, 140 }}
{{Val list|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}


== Swetismic (540/539) ==
== Swetismic (540/539) ==

Revision as of 09:08, 1 August 2021

Werckismic (441/440)

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]]

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 364/363, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440

Mapping: [⟨1 0 0 0 -3 -8], ⟨0 1 0 0 2 5], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 2 3]]

Generators: ~2, ~3, ~5, ~7

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

Lattice basis:

3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657

Minimax tuning:

[[1 0 0 0 0 0⟩, [5/3 0 1/3 -1/3 -1/3 1/3⟩, [1/6 0 5/6 2/3 -5/6 1/3⟩, [0 0 0 1 0 0⟩, [1/6 0 -1/6 2/3 1/6 1/3⟩, [0 0 0 0 0 1⟩]
Eigenmonzos: 2, 11/10, 8/7, 16/13
[[1 0 0 0 0 0⟩, [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, [17/8 -11/8 5/8 -1/8 3/8 1/8⟩, [5/2 -3/2 1/2 -1/2 1/2 1/2⟩, [17/8 -11/8 5/8 -9/8 3/8 9/8⟩]
Eigenmonzos: 2, 14/13, 6/5, 11/9

Template:Val list

Commas 351/350, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440

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

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351

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

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 441/440, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845

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

Generators: ~2, ~3, ~5, ~13/11

Template:Val list

Keenanismic (385/384)

Main article: Keenanismic family

Subgroup: 2.3.5.7.11

Comma list: 385/384

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

Generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [7/3 1/3 2/3 -1/3 -1/3⟩, [7/3 1/3 -1/3 2/3 -1/3⟩, [7/3 1/3 -1/3 -1/3 2/3⟩]
Eigenmonzos: 2, 3, 7/5, 11/5

Template:Val list

Badness: 15.159 × 10-9

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

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

Generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35 324/25]

Lattice basis:

4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

Minimax tuning: [to be confirmed]

[⟨1 0 0 0 0 0], ⟨0 1 0 0 0 0], ⟨2/3 4/3 1/3 0 0 -1/3], ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6], ⟨19/6 -1/6 -1/6 -1/2 1/2 1/6], ⟨2/3 4/3 -2/3 0 0 2/3]]
Eigenmonzos: 2, 14/11, 13/10, 4/3

Template:Val list

Badness: 2.206 × 10-6

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

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

Transpose: [2 3 5 7 385/35 625/48]

Template:Val list

Badness: 2.573 × 10-6

Commas 351/350, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384

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

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 352/351, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 -2], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 -1 -1]]

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 364/363, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 3], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 -3]]

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 385/384, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 847/845

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

Generators: ~2, ~3, ~5, ~13/11

Template:Val list

Swetismic (540/539)

Subgroup: 2.3.5.7.11

Comma list: 540/539

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

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 540/539, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845

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

Generators: ~2, ~3, ~5, ~13/11

Template:Val list

Commas 540/539, 625/624

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624

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

Generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 540/539, 676/675

Subgroup: 2.3.5.7.11

Comma list: 540/539, 676/675

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

Generators: ~2, ~26/15, ~5, ~7

Template:Val list

Lehmerismic (3025/3024)

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

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

Generators: ~2, ~3, ~5, ~55/36

Template:Val list

Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

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

Generators: ~99/70, ~3, ~5, ~7

Template:Val list

Commas 1716/1715, 2080/2079

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079

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

Generators: ~99/70, ~3, ~5, ~7

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299

Minimax tuning:

[[1 0 0 0 0 0⟩, [7/10 4/5 0 -2/5 0 1/5⟩, [7/10 -1/5 1 -2/5 0 1/5⟩, [7/5 -2/5 0 1/5 0 2/5⟩, [11/5 -11/5 1 3/5 0 1/5⟩, [0 0 0 0 0 1⟩]
Eigenmonzos: 2, 6/5, 16/13, 9/7

Template:Val list Template:IoT

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