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Mercator family: Difference between revisions

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{{Technical data page}}
<div class="toccolours" style="float: right">
<center>'''[[Fractional-octave temperaments]]'''</center>
----
<small>← [[{{Ordinal|{{#expr:53-1}}}}-octave temperaments]]</small> 53rd-octave temperaments <small>[[{{Ordinal|{{#expr:53+1}}}}-octave temperaments]] →</small>
</div>
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]
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.  
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.  


= Mercator =
Discussed elsewhere are:
 
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]
 
== Mercator ==
[[Subgroup]]: 2.3.5


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


[[POTE generator]]: ~5/4 = 386.264
[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]


Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
: mapping generators: ~531441/524288, ~5


Mapping generators: ~531441/524288, ~5/1
[[Optimal tuning]]s:
* [[CTE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.3137¢
* [[CWE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.2804¢


{{Multival|legend=1| 0 53 84 }}
{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}


{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
[[Badness]] (Sintel): 6.670


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.


= Schismerc =
Subgroup: 2.3.5.7
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
[[Comma list]]: 15625/15552, 32805/32768


POTE generator: ~225/224 = 5.3666
[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]


Mapping: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]
: mapping generators: ~81/80, ~7


Mapping generators: ~81/80, ~7/1
[[Optimal tuning]]s:
* [[CTE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.1741¢
* [[CWE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.6299¢


{{Multival|legend=1| 0 0 53 0 84 123 }}
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}


{{Val list|legend=1| 53, 159, 212, 689c, 901cc }}
[[Badness]] (Sintel): 2.202


Badness: 0.0870
=== Cartography ===
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.


== Cartography ==
Subgroup: 2.3.5.7.11
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
Comma list: 385/384, 6250/6237, 19712/19683
POTE generator: ~225/224 = 6.1204


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


Mapping generators: ~81/80, ~7/1
Optimal tunings:
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢


{{Val list|legend=1| 53, 106d, 159, 212, 371d, 583cde }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


Badness: 0.0545
Badness (Sintel): 1.800


=== 13-limit ===
==== 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.  
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
Subgroup: 2.3.5.7.11.13


POTE generator: ~225/224 = 6.1430
Comma list: 325/324, 385/384, 625/624, 19712/19683


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


Mapping generators: ~81/80, ~7/1
Optimal tunings:
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢
 
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


{{Val list|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}
Badness (Sintel): 1.239


Badness: 0.0300
=== Pentacontatritonic ===
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.


== Pentacontatritonic ==
Subgroup: 2.3.5.7.11
First proposed by [[User:Xenllium|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
Comma list: 540/539, 15625/15552, 32805/32768


POTE generator: ~385/384 = 4.1494
Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]


Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5956¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5697¢


Mapping generators: ~81/80, ~7/1
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}


{{Val list|legend=1| 53, 159e, 212e, 265, 318, 583c }}
Badness (Sintel): 3.804


Badness: 0.1151
==== 13-limit ====
13-limit pentacontatritonic adds the minisma to the list of commas being tempered out – in this extension the harmonic 13 is connected to the generator.


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
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
Comma list: 540/539, 729/728, 4096/4095, 13750/13689


POTE generator: ~385/384 = 3.9850
Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}
 
Optimal tuning (POTE): ~385/384 = 3.9850


Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4057¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4008¢


Mapping generators: ~81/80, ~7/1
{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}


{{Val list|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}
Badness (Sintel): 2.527


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 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.


== Boiler ==
Subgroup: 2.3.5.7.11
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.


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


POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232
Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]
 
: mapping generators: ~2835/2816, ~7


Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]
Optimal tunings:  
* CTE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 230.6341¢
* CWE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 231.1634¢


Mapping generators: ~2835/2816, ~7
{{Optimal ET sequence|legend=0| 106, 212 }}


{{Val list|legend=1| 106, 212 }}
Badness (Sintel): 3.625


Badness: 0.1096
== Joliet ==
Joliet can be characterized as the 53 &amp; 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.


= Joliet =
Subgroup: 2.3.5.7.11
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
[[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 }}]


Mapping: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}]
: mapping generators: ~50/49, ~11


Mapping generators: ~81/80, ~11/1
[[Optimal tuning]]s:
* [[CTE]]: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
* [[CWE]]: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 552.0415¢


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


Badness: 0.0633
[[Badness]] (Sintel): 2.091


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


POTE generator: ~176/175 = 8.1254
Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]
 
Optimal tunings:
* CTE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
* CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.4859¢
 
{{Optimal ET sequence|legend=0| 53, 106, 159d }}
 
Badness (Sintel): 1.528
 
== Iodine ==
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 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -19 14 -5 3 }}, {{monzo| 8 3 -20 12 }}
 
[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]
 
: mapping generators: ~3125/3087, ~6075/3584
 
[[Optimal tuning]]s:
* [[CTE]]: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7347¢
* [[CWE]]: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7301¢
 
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
 
[[Badness]] (Sintel): 12.075
 
=== 11-limit ===
24 periods plus the reduced generator correspond to [[11/8]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 160083/160000, 820125/819896, 4302592/4296875
 
Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]
 
Optimal tunings:
* CTE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7322¢
* CWE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7345¢
 
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}
 
Badness (Sintel): 2.893
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488
 
Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]
 
Optimal tunings:
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7115¢
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7126¢
 
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}
 
Badness (Sintel): 1.967
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872
 
Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]
 
Optimal tunings:
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7131¢
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7208¢


Mapping generators: ~81/80, ~11/1
{{Optimal ET sequence|legend=0| 159, 583f, 742 }}


{{Val list|legend=1| 53, 106, 159d }}
Badness (Sintel): 1.568


Badness: 0.0370
{{Navbox fractional-octave|53}}


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Mercator]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"