Mercator family

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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: ~531441/524288, ~5/1

Wedgie⟨⟨0 53 84]]

Vals53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650

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, pentacontatritonic and boiler.

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]]

Vals53, 159, 212, 689c, 901cc

Badness: 0.0870

Cartography

Cartography 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

POTE generator: ~225/224 = 6.1204

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

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

Vals53, 106d, 159, 212, 371d, 583cde

Badness: 0.0545

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

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

Vals53, 106d, 159, 212, 371df, 583cdeff

Badness: 0.0300

Pentacontatritonic

First proposed by Xenllium, this temperament nails down both the 7-limit and the 11-limit by tempering out 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

Vals53, 159e, 212e, 265, 318, 583c

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

Vals53, 159ef, 212ef, 265, 318, 583cf

Badness: 0.0612

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 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, as well as to a number of mechanical devices that boil water for various purposes.

Comma list: 9801/9800, 15625/15552, 32805/32768

POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232

Mapping: [106 168 246 0 69], 0 0 0 1 1]]

Mapping generators: ~2835/2816, ~7

Vals106, 212

Badness: 0.1096

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

Vals53, 106, 159d

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

Vals53, 106, 159d

Badness: 0.0370