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Mercator family: Difference between revisions

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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.
{{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>


= Mercator =
The '''mercator family''' of [[regular temperament|temperaments]] [[tempering out|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.


Comma list: {{monzo| -84 53 }}
== Mercator ==
[[Subgroup]]: 2.3.5


[[POTE generator]]: ~5/4 = 386.264
[[Comma list]]: {{monzo| -84 53 }}


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


Mapping generators: ~81/80, ~5/1
[[Optimal tuning]]s:
* [[WE]]: ~531441/524288 = 22.6419{{c}}, ~5/4 = 386.2707{{c}} (~32805/32768 = 1.3581{{c}})
: [[error map]]: {{val| +0.022 -0.034 -0.000 }}
* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}} (~32805/32768 = 1.3747{{c}})
: error map: {{val| 0.000 -0.068 -0.033 }}


{{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): 6.67


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 11-limit extensions are cartography, pentacontatritonic, boiler, and auric.


= 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, and Pentacontatritonic.


Comma list: 15625/15552, 32805/32768
[[Comma list]]: 15625/15552, 32805/32768


POTE generator: ~225/224 = 5.3666
{{Mapping|legend=1| 53 84 123 0 | 0 0 0 1 }}
: mapping generators: ~81/80, ~7


Mapping: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]
[[Optimal tuning]]s:
* [[WE]]: ~81/80 = 22.6456{{c}}, ~7/4 = 968.3928{{c}} (~225/224 = 5.3675{{c}})
: [[error map]]: {{val| +0.216 +0.275 -0.906 -0.001 }}
* [[CWE]]: ~81/80 = 22.6415{{c}}, ~7/4 = 968.3701{{c}} (~225/224 = 5.2148{{c}})
: error map: {{val| 0.000 -0.068 -1.408 -0.456 }}


Mapping generators: ~81/80, ~7/1
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc, 1113ccd }}


{{Multival|legend=1| 0 0 53 0 84 123 }}
[[Badness]] (Sintel): 2.20


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


Badness: 0.0870
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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 6250/6237, 19712/19683
 
{{Mapping|legend=0| 53 84 123 0 332 | 0 0 0 1 -1 }}
 
Optimal tunings:
* WE: ~81/80 = 22.6478{{c}}, ~7/4 = 967.7312{{c}} (~225/224 = 6.1221{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 967.4822{{c}} (~225/224 = 6.1027{{c}})
 
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}
 
Badness (Sintel): 1.80
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 625/624, 19712/19683
 
{{Mapping|legend=0| 53 84 123 0 332 196 | 0 0 0 1 -1 0 }}
 
Optimal tunings:
* WE: ~81/80 = 22.6491{{c}}, ~7/4 = 967.7656{{c}} (~225/224 = 6.1450{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 967.4603{{c}} (~225/224 = 6.1246{{c}})
 
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}
 
Badness (Sintel): 1.24
 
=== Pentacontatritonic ===
First proposed by [[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.
 
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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 15625/15552, 32805/32768
 
{{Mapping|legend=0| 53 84 123 0 481 | 0 0 0 1 -2 }}
 
Optimal tunings:
* WE: ~81/80 = 22.6427{{c}}, ~7/4 = 969.4874{{c}} (~385/384 = 4.1496{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 969.4303{{c}} (~385/384 = 4.1546{{c}})
 
{{Optimal ET sequence|legend=0| 53, 212e, 265, 318 }}
 
Badness (Sintel): 3.80
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4096/4095, 13750/13689
 
{{Mapping|legend=0| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}
 
Optimal tunings:
* WE: ~81/80 = 22.6416{{c}}, ~7/4 = 969.6046{{c}} (~385/384 = 3.9850{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 969.5992{{c}} (~385/384 = 3.9858{{c}})
 
{{Optimal ET sequence|legend=0| 53, 212ef, 265, 318 }}
 
Badness (Sintel): 2.53
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 15625/15552, 32805/32768
 
{{Mapping|legend=0| 106 168 246 0 69 | 0 0 0 1 1 }}
: mapping generators: ~2835/2816, ~7
 
Optimal tunings:
* WE: ~2835/2816 = 11.3229{{c}}, ~7/4 = 968.8469{{c}} (~225/224 = 4.9241{{c}})
* CWE: ~2835/2816 = 11.3208{{c}}, ~7/4 = 968.8633{{c}} (~225/224 = 4.7483{{c}})
 
{{Optimal ET sequence|legend=0| 106, 212, 530c, 742ce }}
 
Badness (Sintel): 3.62
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 15625/15552, 32805/32768
 
{{Mapping|legend=0| 159 252 369 0 550 | 0 0 0 1 0 }}
: mappping generators: ~243/242, ~7
 
Optimal tunings:
* WE: ~243/242 = 7.5484{{c}}, ~7/4 = 968.4469{{c}} (~3024/3025 = 2.2569{{c}})
* CWE: ~243/242 = 7.5472{{c}}, ~7/4 = 968.4005{{c}} (~3024/3025 = 2.3627{{c}})
 
{{Optimal ET sequence|legend=0| 159, 318, 477c }}
 
Badness (Sintel): 5.23


== Joliet ==
== Joliet ==
Joliet can be characterized as the 53 & 106 temperament, adding the [[kalisma]] to Schismerc's list of tempered commas. The name for this temperament is a reference to 106 being the maximum number of characters in Joliet- an extension to the ISO 9660 file system.
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:  
[[Subgroup]]: 2.3.5.7.11


POTE generator:  
[[Comma list]]: 225/224, 1728/1715, 3125/3087


Mapping:  
{{Mapping|legend=1| 53 84 123 149 0 | 0 0 0 0 1 }}
: mapping generators: ~81/80, ~11


Mapping generators:  
[[Optimal tuning]]s:  
* [[WE]]: ~81/80 = 22.6365{{c}}, ~11/8 = 551.1028{{c}} (~176/175 = 8.8263{{c}})
: [[error map]]: {{val| -0.264 -0.487 -2.022 +4.015 -0.009 }}
* [[CWE]]: ~81/80 = 22.6415{{c}}, ~11/8 = 552.0415{{c}} (~176/175 = 8.6452{{c}})
: error map: {{val| 0.000 -0.068 -1.408 +4.759 +0.724 }}


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


Badness:
[[Badness]] (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 325/324, 640/637
{{Mapping|legend=0| 53 84 123 149 0 196 | 0 0 0 0 1 0 }}


Comma list:  
Optimal tunings:  
* WE: ~81/80 = 22.6403{{c}}, ~11/8 = 551.4931{{c}} (~176/175 = 8.1250{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~11/8 = 551.4859{{c}} (~176/175 = 8.0897{{c}})


POTE generator:
{{Optimal ET sequence|legend=0| 53, 106 }}


Mapping:  
Badness (Sintel): 1.53


Mapping generators:
== 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. 3 generators plus 2 periods reach the [[5/1|5th harmonic]]. 5 generators reach the [[14/1|14th harmonic]].


Vals:
In the 11-limit, the [[~]][[11/8]] can serve as a generator, being 16 periods less than the canonical generator.


Badness:
[[Subgroup]]: 2.3.5.7


== Cartography ==
[[Comma list]]: {{monzo| -19 14 -5 3 }}, {{monzo| 8 3 -20 12 }}
Cartography nails down the 7-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
{{Mapping|legend=1| 53 84 2 -53 | 0 0 3 5 }}
: mapping generators: ~3125/3087, ~6075/3584
 
[[Optimal tuning]]s:
* [[WE]]: ~3125/3087 = 22.6417{{c}}, ~6075/3584 = 913.7361{{c}} (~225/224 = 8.0675{{c}})
: [[error map]]: {{val| +0.011 -0.051 +0.178 -0.156 }}
* [[CWE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7301{{c}} (~225/224 = 8.0697{{c}})
: error map: {{val| 0.000 -0.068 +0.159 -0.176 }}
 
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
 
[[Badness]] (Sintel): 12.1
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~225/224 = 6.1430
Comma list: 160083/160000, 820125/819896, 4302592/4296875


Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}]
{{Mapping|legend=0| 53 84 2 -53 143 | 0 0 3 5 1 }}


Mapping generators: ~81/80, ~7/1
Optimal tunings:
* WE: ~1815/1792 = 22.6414{{c}}, ~6075/3584 = 913.7318{{c}} (~225/224 = 8.0750{{c}})
* CWE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7345{{c}} (~225/224 = 8.0742{{c}})


{{Val list|legend=1| 53, 106d, 159, 212, 371d, 583cde }}
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742 }}


Badness: 0.0545
Badness (Sintel): 2.89


=== 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.  
Subgroup: 2.3.5.7.11.13
 
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488
 
{{Mapping|legend=0| 53 84 2 -53 143 -46 | 0 0 3 5 1 6 }}


Commas: 325/324, 385/384, 625/624, 19712/19683
Optimal tunings:  
* WE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7113{{c}} (~225/224 = 8.0526{{c}})
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7126{{c}} (~225/224 = 8.0522{{c}})


POTE generator: ~225/224 = 6.1430
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}


Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}
Badness (Sintel): 1.97


Mapping generators: ~81/80, ~7/1
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872


Badness: 0.0300
{{Mapping|legend=0| 53 84 2 -53 143 -46 257 | 0 0 3 5 1 6 -1 }}


== Pentacontatritonic ==
Optimal tunings:
First proposed by [[User:Xenllium|Xenllium]], this temperament differs from Cartography in that it tempers out a different 11-limit comma in order to nail down the 7-limit – specifically, the swetisma.
* WE: ~78/77 = 22.6412{{c}}, ~441/260 = 913.7109{{c}} (~225/224 = 8.0623{{c}})
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7208{{c}} (~225/224 = 8.0605{{c}})


Comma list: 540/539, 15625/15552, 32805/32768
{{Optimal ET sequence|legend=0| 159, 583f, 742 }}
 
Badness (Sintel): 1.57
 
== Aemilic ==
''Aemilic'' tempers out the [[landscape comma]] alongisde the mercator comma. It may be described as the 159 & 954 temperament. It was named after the minor planet {{w|159 Aemilia}}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| -84 53 }}
 
{{Mapping|legend=1| 159 252 0 -292 | 0 0 1 2 }}
: mapping generators: ~225/224, ~5
 
[[Optimal tuning]]s
* [[WE]]: ~225/224 = 7.5473{{c}}, ~5/4 = 386.2754{{c}} (~32805/32768 = 1.3629{{c}})
: [[error map]]: {{val| +0.021 -0.035 +0.004 -0.003 }}
* [[CWE]]: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2798{{c}} (~32805/32768 = 1.3741{{c}})
: error map: {{val| 0.000 -0.068 -0.034 -0.040 }}
 
{{Optimal ET sequence|legend=1| 159, …, 795, 954, 1749, 6201, 7950, 9699bd, 11448bd }}
 
[[Badness]] (Sintel): 4.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~385/384 = 4.1494
Comma list: 3025/3024, 102487/102400, {{monzo| 34 19 8 4 -1 }}


Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]
{{Mapping|legend=0| 159 252 0 -292 550 | 0 0 1 2 0 }}


Mapping generators: ~81/80, ~7/1
Optimal tunings:
* WE: ~225/224 = 7.5475{{c}}, ~5/4 = 386.2382{{c}} (~32805/32768 = 1.3168{{c}})
* CWE: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2421{{c}} (~32805/32768 = 1.3365{{c}})


{{Val list|legend=1| 53, 159e, 212e, 265, 318, 583c }}
{{Optimal ET sequence|legend=0| 159, …, 795, 954, 1749e, 2703e, 3657ee }}


Badness: 0.1151
Badness (Sintel): 2.79


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


Comma list: 540/539, 729/728, 4096/4095, 13750/13689
Comma list: 3025/3024, 102487/102400, 123201/123200, 4100625/4100096
 
{{Mapping|legend=0| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}
 
Optimal tunings:
* WE: ~225/224 = 7.5475{{c}}, ~5/4 = 386.2350{{c}} (~32805/32768 = 1.3136{{c}})
* CWE: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2395{{c}} (~32805/32768 = 1.3338{{c}})
 
{{Optimal ET sequence|legend=0| 159, …, 795, 954, 2703e, 3657eef }}
 
Badness (Sintel): 1.42
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2431/2430, 3025/3024, 57375/57344, 102487/102400, 123201/123200
 
{{Mapping|legend=0| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}


POTE generator: ~385/384 = 3.9850
Optimal tunings:  
* WE: ~225/224 = 7.5476{{c}}, ~5/4 = 386.1835{{c}} (~32805/32768 = 1.2536{{c}})
* CWE: ~225/224 = 7.5472{{c}}, ~5/4 = 386.1765{{c}} (~32805/32768 = 1.2708{{c}})


Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}
{{Optimal ET sequence|legend=0| 159, 795g, 954 }}


Mapping generators: ~81/80, ~7/1
Badness (Sintel): 1.64


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


Badness: 0.0612
[[Category:Mercator family]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]


[[Category:Theory]]
<!-- as 53rd-octave temperaments page redirects here -->
[[Category:Temperament family]]
[[Category:53edo]]
[[Category:Mercator]]
[[Category:Fractional-octave temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"