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.

Mercator

Comma list: [-84 53⟩

POTE generator: ~5/4 = 386.264

Mapping: [⟨53 84 123], ⟨0 0 1]]

Mapping generators: ~81/80, ~5

Wedgie: ⟨⟨ 0 53 84 ]]

Badness: 0.2843

Schismerc

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 part is exactly the same as the 5-limit of 53edo, with the addition of harmonic 7 represented by an independent generator. Among the known 11-limit extensions are Cartography, and Pentacontatritonic.

Comma list: 15625/15552, 32805/32768

POTE generator: ~225/224 = 5.3666

Mapping: [⟨53 84 123 0], ⟨0 0 0 1]]

Mapping generators: ~81/80, ~7

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.

Comma list: 385/384, 6250/6237, 19712/19683

POTE generator: ~225/224 = 6.1430

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

Mapping generators: ~81/80, ~7

Badness: 0.0545

13-limit

13-limit Cartography adds the island comma to the list of tempered commas. The harmonic 13 in this extension is part of the period and independent of the generator for harmonics 7 and 11.

Commas: 325/324, 385/384, 625/624, 19712/19683

POTE generator: ~225/224 = 6.1430

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

Mapping generators: ~81/80, ~7

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.

Comma list: 540/539, 15625/15552, 32805/32768

POTE generator: ~385/384 = 4.1494

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

Mapping generators: ~81/80, ~7

Badness: 0.1151

13-limit

13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out – in this extension the harmonic 13 is connected to the generator.

Comma list: 540/539, 729/728, 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

Badness: 0.0612