Mercator family: Difference between revisions

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

* [[CTE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.3137{{c}}

* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}}

[[Badness]] (Sintel): 6.670

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.

* [[CTE]]: ~81/80 = 22.6415{{c}}, ~8/7 = 231.1741{{c}}

* [[CWE]]: ~81/80 = 22.641{{c}}, ~8/7 = 231.6299{{c}}

{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}

[[Badness]] (Sintel): 2.202

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: {{mapping| 53 84 123 0 332 | 0 0 0 1 -1 }}

* CTE: ~81/80 = 22.641{{c}}, ~8/7 = 232.4299{{c}}

* CWE: ~81/80 = 22.641{{c}}, ~8/7 = 232.5178{{c}}

Badness (Sintel): 1.800

Mapping: {{mapping| 53 84 123 0 332 196 | 0 0 0 1 -1 0 }}

* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.4299{{c}}

* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.5397{{c}}

Badness (Sintel): 1.239

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: {{mapping| 53 84 123 0 481 | 0 0 0 1 -2 }}

* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5956{{c}}

* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5697{{c}}

{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}

Badness (Sintel): 3.804

Mapping: {{mapping| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}

 

* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4057{{c}}

* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4008{{c}}

{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}

Badness (Sintel): 2.527

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: {{mapping| 106 168 246 0 69 | 0 0 0 1 1 }}

* CTE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 230.6341{{c}}

* CWE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 231.1634{{c}}

{{Optimal ET sequence|legend=0| 106, 212 }}

Badness (Sintel): 3.625

== Auric ==

''Auric'', named after [[Aura]], is defined by setting the 2.3.5.11 mapping to that of 159edo and the generator for the prime 7 being independent.

Mapping: {{mapping| 159 252 369 0 550 | 0 0 0 1 0 }}

: mappping generators: ~243/242 = 1\159, ~7

Optimal tuning (CTE): ~7/4 = 968.726{{c}}

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 generators: ~50/49, ~11

* [[CTE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}

* [[CWE]]: ~50/49 = 22.641{{c}}, ~11/8 = 552.0415{{c}}

[[Badness]] (Sintel): 2.091

Mapping: {{mapping| 53 84 123 149 0 196 | 0 0 0 0 1 0 }}

* CTE: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}

* CWE: ~50/49 = 22.6415{{c}}, ~11/8 = 551.4859{{c}}

{{Optimal ET sequence|legend=0| 53, 106, 159d }}

Badness (Sintel): 1.528

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]].  

* [[CTE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7347{{c}}

* [[CWE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7301{{c}}

[[Badness]] (Sintel): 12.075

Mapping: {{mapping| 53 84 2 -53 143 | 0 0 3 5 1 }}

* CTE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7322{{c}}

* CWE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7345{{c}}

{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}

Badness (Sintel): 2.893

Mapping: {{mapping| 53 84 2 -53 143 -46 | 0 0 3 5 1 6 }}

* CTE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7115{{c}}

* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7126{{c}}

Badness (Sintel): 1.967

Mapping: {{mapping| 53 84 2 -53 143 -46 257 | 0 0 3 5 1 6 -1 }}

* CTE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7131{{c}}

* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7208{{c}}

Badness (Sintel): 1.568

''Aemilic'' is described as the 159 & 954 temperament in the 17-limit and is named after the minor planet {{w|159 Aemilia}}.

 

In the 7-limit, aemilic tempers out the [[landscape comma]] alongisde the [[Mercator comma]], and also {{monzo| -76 41 12 -6 }}.

: mapping generators: ~225/224 = 1\159, ~5

[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.303{{c}}

[[Support]]ing [[ET]]s: {{EDOs| 159, 795, 954, 1749, 2544, 2703 }}

Comma list: 3025/3024, 160083/160000, {{monzo| 34 19 8 4 -1 }}

Mapping: {{mapping| 159 252 0 -292 550 | 0 0 1 2 0 }}

Optimal tuning (CTE): ~5/4 = 386.303{{c}}

Supporting ETs: {{EDOs| 159, 795, 954 }}

Comma list: 3025/3024, 123201/123200, 160083/160000, 4100625/4100096

Mapping: {{mapping| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}

Optimal tuning (CTE): ~5/4 = 386.303{{c}}

Supporting ETs: {{EDOs| 159, 795, 954 }}

Comma list: 2431/2430, 3025/3024, 57375/57344, 123201/123200, 160083/160000

Mapping: {{mapping| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}

Optimal tuning (CTE): ~5/4 = 386.263{{c}}

Supporting ETs: {{EDOs| 159, 795g, 954 }}