Mercator family: Difference between revisions

The '''Mercator family''' tempers out [[Mercator's comma]], {{monzo| -84 53 }}, and hence the fifths form a closed 53-note circle of fifths, identical to [[53edo|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.  

[[Mapping]]: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]

Mapping generators: ~531441/524288, ~5/1

{{Multival|legend=1| 0 53 84 }}

[[POTE generator]]: ~5/4 = 386.264

{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}

[[Badness]]: 0.284323

 

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.

[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]

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

{{Multival|legend=1| 0 0 53 0 84 123 }}

[[POTE generator]]: ~225/224 = 5.3666

{{Val list|legend=1| 53, 159, 212, 689c, 901cc }}

[[Badness]]: 0.087022

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.

Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}]

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

POTE generator: ~225/224 = 6.1204

Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371d, 583cde }}

Badness: 0.054452

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.

Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}

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

POTE generator: ~225/224 = 6.1430

Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }}

Badness: 0.029980

First proposed by [[User:Xenllium|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.

Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]

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

POTE generator: ~385/384 = 4.1494

 

Badness: 0.115066

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.

Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}

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

POTE generator: ~385/384 = 3.9850

Optimal GPV sequence: {{Val list| 53, 159ef, 212ef, 265, 318, 583cf }}

Badness: 0.061158

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.

Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]

Mapping generators: ~2835/2816, ~7

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

Optimal GPV sequence: {{Val list| 106, 212 }}

Badness: 0.109648

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.

[[Mapping]]: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}]

Mapping generators: ~81/80, ~11/1

[[POTE generator]]: ~176/175 = 8.8283

{{Val list|legend=1| 53, 106, 159d }}

[[Badness]]: 0.063254

Comma list: 169/168, 225/224, 325/324, 640/637

Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]

Mapping generators: ~81/80, ~11/1

POTE generator: ~176/175 = 8.1254

Optimal GPV sequence: {{Val list| 53, 106, 159d }}

[[Category:Mercator]]

[[Category:Rank 2]]