Catalog of rank-4 temperaments: Difference between revisions

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==Werckismic (441/440)==
== Werckismic (441/440) ==
Map: [<1 0 0 0 -3|, <0 1 0 0 2|, <0 0 1 0 -1|, <0 0 0 1 2|]
[[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
Generators: ~2, ~3, ~5, ~7


EDOs: 12, 31, 41, 46, 58, 72, 87, 118, 130
{{Val list|legend=1| 12, 31, 41, 46, 58, 72, 87, 118, 130 }}


===Commas 441/440, 364/363===
=== Commas 441/440, 364/363 ===
13-limit minimax


[|1 0 0 0 0 0>, |5/3 0 1/3 -1/3 -1/3 1/3>,
[[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|]
|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
Generators: ~2, ~3, ~5, ~7


15-limit minimax
Map to lattice: [<0 1 1 -1 -1 0|, <0 0 1 0 -1 -2|, <0 0 1 1 1 1|]


[|1 0 0 0 0 0>, |5/4 1/4 1/4 -1/4 -1/4 1/4>,
Lattice basis:
|5/4 -3/4 5/4 -1/4 -1/4 1/4>,  
: 3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657
|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
[[Minimax tuning]]:
* [[13-odd-limit]]
: [|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
* [[15-odd-limit]]
: [|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


Lattice basis: 3/2 1.2263 14/11 1.4629 21/16 1.4657
{{Val list|legend=1| 15, 23, 26, 29, 41, 43, 46, 58, 113, 130, 159, 217, 289, 393 }}


Map to lattice: [<0 1 1 -1 -1 0|,
=== Commas 441/440, 351/350 ===
[[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|]


<0 0 1 0 -1 -2|, <0 0 1 1 1 1|]
Generators: ~2, ~3, ~5, ~7


Map: [<1 0 0 0 -3 -8|, <0 1 0 0 2 5|, <0 0 1 0 -1 -2|, <0 0 0 1 2 3|]
{{Val list|legend=1| 26, 31, 46, 58, 72, 103, 130 }}


Generators: 2, 3, 5, 7
=== Commas 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|]


EDOs: 15, 23, 26, 29, 41, 43, 46, 58, 113, 130, 159, 217, 289, 393
Generators: ~2, ~3, ~5, ~7


===Commas 441/440, 351/350===
{{Val list|legend=1| 29, 31, 41, 46, 58, 77, 87 }}
Map: [<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
=== Commas 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|]


EDOs: 26, 31, 46, 58, 72, 103, 130
Generators: ~2, ~3, ~5, ~13/11


===Commas 196/195, 352/351===
{{Val list|legend=1| 29, 41, 46, 58, 87, 103, 149 }}
Map: [<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
== Keenanismic (385/384) ==
{{Main| Keenanismic family }}


EDOs: 29, 31, 41, 46, 58, 77, 87
[[Mapping]]: [<1 0 0 0 7|, <0 1 0 0 1|, <0 0 1 0 -1|, <0 0 0 1 -1|]


===Commas 441/440, 847/845===
Generators: ~2, ~3, ~5, ~7
Map: [<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
{{Val list|legend=1| 15, 22, 31, 41, 46, 53, 72, 87, 94, 118 }}


EDOs: 29, 41, 46, 58, 87, 103, 149
=== Commas 385/384, 325/324 ===


==Keenanismic (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|]
Map: [<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
Generators: ~2, ~3, ~5, ~7


EDOs: 15, 22, 31, 41, 46, 53, 72, 87, 94, 118
Lattice basis:  
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247


===Commas 385/384, 325/324===
[[Minimax tuning]]:
* [[13-odd-limit|13-]] and [[15-odd-limit]]
: [<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


13 and 15 limit minimax
{{Val list|legend=1| 7, 8, 15, 19, 26, 34, 41, 46, 53, 68, 72, 79, 87, 94, 99, 113, 140, 159, 212, 253 }}


[<1 0 0 0 0 0|, <0 1 0 0 0 0|,
=== Commas 385/384, 351/350 ===
[[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|]


<2/3 4/3 1/3 0 0 -1/3|,  
Generators: ~2, ~3, ~5, ~7


<19/6 -1/6 -1/6 1/2 -1/2 1/6|,  
{{Val list|legend=1| 19, 22, 26, 31, 46, 50, 53, 72, 77, 103 }}


<19/6 -1/6 -1/6 -1/2 1/2 1/6|,
=== Commas 385/384, 352/351 ===
[[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|]


<2/3 4/3 -2/3 0 0 2/3|]
Generators: ~2, ~3, ~5, ~7


Eigenmonzos: 2, 14/11, 13/10, 4/3
{{Val list|legend=1| 10, 22, 31, 41, 46, 53, 63, 77, 87, 94 }}


Lattice basis: 4/3 1.0820 6/5 1.3935 10/9 1.6247
=== Commas 385/384, 364/363 ===
[[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|]


Map: [<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


Generators: 2, 3, 5, 7
{{Val list|legend=1| 9, 15, 22, 26, 41, 46, 50, 72, 87 }}


EDOs: 7, 8, 15, 19, 26, 34, 41, 46, 53, 68, 72, 79, 87, 94, 99, 113, 140, 159, 212, 253
=== Commas 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|]


===Commas 385/384, 351/350===
Generators: ~2, ~3, ~5, ~13/11
Map: [<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
{{Val list|legend=1| 7, 9, 41, 46, 50, 53, 87, 94, 103, 140 }}


EDOs: 19, 22, 26, 31, 46, 50, 53, 72, 77, 103
== Swetismic (540/539) ==
[[Mapping]]: [<1 0 0 0 2|, <0 1 0 0 3|, <0 0 1 0 1|, <0 0 0 1 -2|]


===Commas 385/384, 352/351===
Generators: ~2, ~3, ~5, ~7
Map: [<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
{{Val list|legend=1| 22, 31, 41, 53, 58, 72, 80, 130, 152 }}


EDOs: 10, 22, 31, 41, 46, 53, 63, 77, 87, 94
=== Commas 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|]


===Commas 385/384, 364/363===
Generators: ~2, ~3, ~5, ~13/11
Map: [<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
{{Val list|legend=1| 9, 41, 50, 53, 58, 94, 103, 111 }}


EDOs: 9, 15, 22, 26, 41, 46, 50, 72, 87
=== Commas 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|]


===Commas 385/384, 847/845===
Generators: ~2, ~3, ~5, ~7
Map: [<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
{{Val list|legend=1| 19, 31, 50, 53, 72, 103, 121, 224 }}


EDOs: 7, 9, 41, 46, 50, 53, 87, 94, 103, 140
=== Commas 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|]


==Swetismic (540/539)==
Generators: ~2, ~26/15, ~5, ~7
Map: [<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
{{Val list|legend=1| 9, 19, 53, 58, 63, 72, 111, 121, 130, 183 }}


EDOs: 22, 31, 41, 53, 58, 72, 80, 130, 152
== Lehmerismic (3025/3024) ==
[[Mapping]]: [<1 0 0 0 2|, <0 1 0 1 2|, <0 0 1 0 -1|, <0 0 0 2 1|]


===Commas 540/539, 847/845===
Generators: ~2, ~3, ~5, ~55/36
Map: [<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
{{Val list|legend=1| 31, 72, 118, 152, 224, 270, 311, 342, 494, 612 }}


EDOs: 9, 41, 50, 53, 58, 94, 103, 111
== Kalismic (9801/9800) ==
[[Mapping]]: [<2 0 0 0 3|, <0 1 0 0 -2|, <0 0 1 0 1|, <0 0 0 1 1|]


===Commas 540/539, 625/624===
Generators: ~99/70, ~3, ~5, ~7
Map: [<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
{{Val list|legend=1| 270, 342, 494, 612, 764, 836, 1106, 1236, 1578, 1848 }}


EDOs: 19, 31, 50, 53, 72, 103, 121, 224
=== Commas 1716/1715, 2080/2079 ===


===Commas 540/539, 676/675===
[[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|]
Map: [<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
Generators: ~99/70, ~3, ~5, ~7


EDOs: 9, 19, 53, 58, 63, 72, 111, 121, 130, 183
Lattice basis:  
: 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299


==Lehmerismic (3025/3024)==
[[Minimax tuning]]:
Map: [<1 0 0 0 2|, <0 1 0 1 2|, <0 0 1 0 -1|, <0 0 0 2 1|]
* [[13-odd-limit|13-]] and [[15-odd-limit]]
: [|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


Generators: 2, 3, 5, 55/36
{{Val list|legend=1| 26, 36, 46, 58, 68, 72, 84, 94, 130, 140, 166, 198, 224, 270, 494, 764, 1186, 1892 }}


EDOs: 31, 72, 118, 152, 224, 270, 311, 342, 494, 612
[[Category:Regular temperament theory]]
 
[[Category:Rank 4| ]] <!-- main article -->
==Kalismic (9801/9800)==
{{IoT}}
Map: [&lt;2 0 0 0 3|, &lt;0 1 0 0 -2|, &lt;0 0 1 0 1|, &lt;0 0 0 1 1|]
{{Todo| add introduction | expand }}
 
Generators: 99/70, 3, 5, 7
 
EDOs: 270, 342, 494, 612, 764, 836, 1106, 1236, 1578, 1848
 
===Commas 2080/2079, 1716/1715===
 
13 and 15 limit minimax
 
[|1 0 0 0 0 0&gt;, |7/10 4/5 0 -2/5 0 1/5&gt;,
|7/10 -1/5 1 -2/5 0 1/5&gt;, |7/5 -2/5 0 1/5 0 2/5&gt;,
|11/5 -11/5 1 3/5 0 1/5&gt;, |0 0 0 0 0 1&gt;]
 
Eigenmonzos: 2, 6/5, 16/13, 9/7
 
Lattice basis: 3/2 length 1.1956 7/4 length 1.4506 14/13 1.8299
 
Map: [&lt;2 0 0 0 3 -7|, &lt;0 1 0 0 -2 1|, &lt;0 0 1 0 1 0|, &lt;0 0 0 1 1 2|]
 
Generators: 99/70, 3, 5, 7
 
EDOs: 26, 36, 46, 58, 68, 72, 84, 94, 130, 140, 166, 198, 224, 270, 494, 764, 1186, 1892
 
[[Category:rank]]
[[Category:rank-four]]
[[Category:temperament]]
[[Category:theory]]
[[category:todo:expand]]

Revision as of 07:17, 1 August 2021

Werckismic (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 441/440, 364/363

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 441/440, 351/350

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

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

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

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

Template:Val list

Commas 385/384, 325/324

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

Lattice basis:

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

Minimax tuning:

[<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

Commas 385/384, 351/350

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 385/384, 352/351

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 385/384, 364/363

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

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)

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

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

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

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)

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)

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

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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