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, ~5/1
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 Joliet, Boiler, 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, ~7/1
Wedgie: ⟨⟨ 0 0 53 0 84 123 ]]
Badness: 0.0870
Joliet
Joliet can be characterized as the 53 & 106 temperament, having 7-limit representation akin to 53edo with the addition of harmonic 11 represented by an independent generator. The name for this temperament is a reference to 106 being the maximum number of characters in the Joliet extension to the ISO 9660 file system.
Comma list: 225/224, 1728/1715, 3125/3087
POTE generator: ~176/175 = 8.8283
Mapping: [⟨53 84 123 149 0], ⟨0 0 0 0 1]]
Mapping generators: ~81/80, ~11/1
Badness: 0.0633
13-limit
Comma list: 169/168, 225/224, 325/324, 640/637
POTE generator: ~176/175 = 8.1254
Mapping: [⟨53 84 123 149 0 196], ⟨0 0 0 0 1 0]]
Mapping generators: ~81/80, ~11/1
Badness: 0.0370
Boiler
Compared to Joliet, Boiler nails down the 7-limit by adding the kalisma to Schismerc's list of tempered commas, though unlike with the other extensions of Schismerc, this temperament lacks a good 13-limit extension. The name for this temperament is a reference to how 212 degrees Fahrenheit is the boiling point of water.
Comma list:
POTE generator:
Mapping:
Mapping generators:
Vals:
Badness:
Cartography
Cartography nails down the 7-limit by adding the symbiotic comma to Schismerc's list of tempered commas. The name for this temperament comes from how good the mappings are, and also from the idea of "Mercator" being a dual reference to both Nicolas Mercator and Gerardus Mercator.
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, ~7/1
Badness: 0.0545
13-limit
13-limit Cartography adds the island comma to the list of tempered commas- a development which fits will with the ideas of mapmaking and geography. 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, ~7/1
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, ~7/1
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, ~7/1
Badness: 0.0612