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

Schismerc temperament

As per the name, Schismerc 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. 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. Among the known 11-limit extensions are Cartography, and Pentacontatritonic.

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 ]]

Badness: 0.0870

Cartography

Cartography nails down the 7-limit by adding the symbiotic comma to Schismerc's 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:

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:

Badness: 0.0300

Pentacontatritonic

First proposed by Xenllium, this temperament differs from Cartography in that it tempers out a different 11-limit comma in order 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:

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:

Badness: 0.0612