Mercator family: Difference between revisions

Comma list: {{monzo| -84 53 }}

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

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

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

{{Val list|legend=1| 53, 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: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]

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

{{Val list|legend=1| 53, 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.

{{Val list|legend=1| 53, 106d, 159, 212, 371d, 583cde }}

=== 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

{{Val list|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}

== Pentacontatritonic ==

First proposed by [[User:Xenllium|Xenllium]], this temperament nails down both the 7-limit and the 11-limit by tempering out the [[swetisma]].

{{Val list|legend=1| 53, 159e, 212e, 265, 318, 583c }}

Badness: 0.1151

=== 13-limit ===

{{Val list|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}

Badness: 0.0612

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 are boil water for various purposes.

{{Val list|legend=1| 106, 212 }}

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

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

== 13-limit ==

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

[[Category:Theory]]

[[Category:Temperament family]]

[[Category:Mercator]]