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⟩
Mapping: [⟨53 84 123], ⟨0 0 1]]
Mapping generators: ~531441/524288, ~5/1
Wedgie: ⟨⟨0 53 84]]
POTE generator: ~5/4 = 386.264
Optimal GPV sequence: 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650
Badness: 0.284323
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, pentacontatritonic and boiler.
Comma list: 15625/15552, 32805/32768
Mapping: [⟨53 84 123 0], ⟨0 0 0 1]]
Mapping generators: ~81/80, ~7/1
Wedgie: ⟨⟨0 0 53 0 84 123]]
POTE generator: ~225/224 = 5.3666
Optimal GPV sequence: 53, 159, 212, 689c, 901cc
Badness: 0.087022
Cartography
Cartography is a strong extension to Schismerc that nails down both the 7-limit and the 11-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
Mapping: [⟨53 84 123 0 332], ⟨0 0 0 1 -1]]
Mapping generators: ~81/80, ~7/1
POTE generator: ~225/224 = 6.1204
Vals: 53, 106d, 159, 212, 371d, 583cde
Badness: 0.054452
13-limit
13-limit Cartography adds the island comma to the list of tempered commas- a development which fits well 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
Mapping: [⟨53 84 123 0 332 196], ⟨0 0 0 1 -1 0]
Mapping generators: ~81/80, ~7/1
POTE generator: ~225/224 = 6.1430
Vals: 53, 106d, 159, 212, 371df, 583cdeff
Badness: 0.029980
Pentacontatritonic
First proposed by Xenllium, this temperament nails down both the 7-limit and the 11-limit by tempering out the swetisma. Like Cartography, Pentacontatritonic is a strong extension to Schismerc.
Comma list: 540/539, 15625/15552, 32805/32768
Mapping: [⟨53 84 123 0 481], ⟨0 0 0 1 -2]]
Mapping generators: ~81/80, ~7/1
POTE generator: ~385/384 = 4.1494
Vals: 53, 159e, 212e, 265, 318, 583c
Badness: 0.115066
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
Mapping: [⟨53 84 123 0 481 345], ⟨0 0 0 1 -2 1]
Mapping generators: ~81/80, ~7/1
POTE generator: ~385/384 = 3.9850
Vals: 53, 159ef, 212ef, 265, 318, 583cf
Badness: 0.061158
Boiler
Boiler nails down both the 7-limit and the 11-limit by adding the kalisma to Schismerc's list of tempered commas, though unlike with the other extensions of Schismerc, this temperament is not only a weak extension, but lacks a good 13-limit extension of its own. The name for this temperament is a reference to how 212 degrees Fahrenheit is the boiling point of water, as well as to a number of mechanical devices that boil water for various purposes.
Comma list: 9801/9800, 15625/15552, 32805/32768
Mapping: [⟨106 168 246 0 69], ⟨0 0 0 1 1]]
Mapping generators: ~2835/2816, ~7
POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232
Badness: 0.109648
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
Mapping: [⟨53 84 123 149 0], ⟨0 0 0 0 1]]
Mapping generators: ~81/80, ~11/1
POTE generator: ~176/175 = 8.8283
Optimal GPV sequence: 53, 106, 159d
Badness: 0.063254
13-limit
Comma list: 169/168, 225/224, 325/324, 640/637
Mapping: [⟨53 84 123 149 0 196], ⟨0 0 0 0 1 0]]
Mapping generators: ~81/80, ~11/1
POTE generator: ~176/175 = 8.1254
Badness: 0.036988