Catalog of rank-4 temperaments

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

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:

• 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

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

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

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

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

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:

• 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

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

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

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

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

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

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

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

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

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

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

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:

• 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