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]]. 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

 

[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264

[[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.

Subgroup: 2.3.5.7

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

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

[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666

[[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

Optimal tuning (POTE): ~225/224 = 6.1204

 

Badness: 0.054452

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

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

Optimal tuning (POTE): ~225/224 = 6.1430

{{Optimal ET sequence|legend=1| 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 }}]

 

Optimal tuning (POTE): ~385/384 = 4.1494

{{Optimal ET sequence|legend=1| 53, 159e, 212e, 265, 318, 583c }}

Badness: 0.115066

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

 

Optimal tuning (POTE): ~385/384 = 3.9850

{{Optimal ET sequence|legend=1| 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 clear 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

Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232

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.

Subgroup: 2.3.5.7.11

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

 

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

[[Optimal tuning]] ([[POTE]]): ~176/175 = 8.8283

[[Badness]]: 0.063254

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

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

Optimal tuning (POTE): ~176/175 = 8.1254

{{Optimal ET sequence|legend=1| 53, 106, 159d }}

Badness: 0.036988

Proposed by Eliora, the name of ''iodine'' is taken from the convention of naming some fractional-octave temperaments after elements, in this case the 53rd chemical element. It can be expressed as the 159 & 742 temperament. 2 periods + 3 less than 600 cent generators correspond to [[8/5]]. 5 less than 600 cent generators (minus 1 octave) correspond to [[8/7]].  

[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]

 

Mapping generators: ~3125/3087, 6075/3584

[[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347

[[Badness]]: 0.477

Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]

Optimal tuning (CTE): ~6075/3584 = 913.7322

{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}

Badness: 0.0875

Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]

Optimal tuning (CTE): ~441/260 = 913.7115

{{Optimal ET sequence|legend=1| 159, 424cdff, 583f, 742, 1643 }}

Badness: 0.0476

Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]

Optimal tuning (CTE): ~441/260 = 913.7131

{{Optimal ET sequence|legend=1| 159, 583f, 742 }}

Badness: 0.0328

[[Category:Mercator family]] <!-- main article -->

[[Category:Rank 2]]