Mercator family

The Mercator family tempers out Mercator's comma, [-84 53⟩, and hence the fifths form a closed 53-note circle of fifths, identical to 53edo. While the tuning of the fifth will be that of 53edo, 0.069 cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

POTE generator: ~5/4 = 386.264

Mapping: [⟨53 84 123], ⟨0 0 1]]

Mapping generators:

Wedgie: ⟨⟨0 53 84]]

Badness: 0.2843

Cartography temperament

In terms of the normal comma list, Cartography is characterized by the addition of the schisma, 32805/32768, to Mercator's comma, which completely reduces all commas in the Schismic-Mercator equivalence continuum to the unison, and thus, the 5-limit is exactly the same as the 5-limit of 53edo. Cartography can also be characterized as the 53&159 temperament, with 212edo being a possible tuning. It should be noted that the 7-limit is somewhat independent for this temperament and is only really fully nailed down in one way or another by extending to the 11-limit.

Commas: 32805/32768

POTE generator: ~225/224 = 5.3666

Mapping: [<53 84 123 0], <0 0 0 1]]

Mapping generators: ~81/80, ~7/1

Wedgie: << 0 0 53 0 84 123 ]]

EDOs: 53, 159, 212, 689c, 901cc

Badness: 0.0870

11-limit

11-limit Cartography nails down the 7-limit by adding the symbiotic comma to the list of tempered commas.

Commas: 19712/19683, 32805/32768

POTE generator: ~225/224 = 6.1430

Mapping: [<53 84 123 0 332 196], <0 0 0 1 -1 0]]

Mapping generators: ~81/80, ~7/1

Wedgie:

EDOs: 53, 106d, 159, 212, 371d, 583cde

Badness: 0.0545

13-limit

13-limit Cartography adds the island comma to the list of tempered commas, and while this extension is connected to the 5-limit, it is independent of the 11-limit and 7-limit, so it can just as easily be added by itself to make a no-sevens no-elevens version of Cartography.

Commas: 676/675, 19712/19683, 32805/32768

POTE generator: ~225/224 = 6.1430

Mapping: [<53 84 123 0 332 196], <0 0 0 1 -1 0]]

Mapping generators: ~81/80, ~7/1

Wedgie:

EDOs: 53, 106d, 159, 212, 371df, 583cdeff

Badness: 0.0300

Pentacontatritonic

This temperament differs from Cartography in that it uses a different 11-limit extension to nail down the 7-limit- specifically, the swetisma.

Commas: 540/539, 32805/32768

POTE generator: ~385/384 = 4.1494

Mapping: [<53 84 123 0 481], <0 0 0 1 -2]]

Mapping generators: ~81/80, ~7/1

Wedgie:

EDOs: 53, 159e, 212e, 265, 318, 583c

Badness: 0.1151

13-limit

13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out- this extension is connected to the 7-limit.

Commas: 540/539, 4096/4095, 13750/13689

POTE generator: ~385/384 = 3.9850

Mapping: [<53 84 123 0 481 345], <0 0 0 1 -2 1]]

Mapping generators: ~81/80, ~7/1

Wedgie:

EDOs: 53, 159ef, 212ef, 265, 318, 583cf

Badness: 0.0612