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__FORCETOC__
{{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>


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''' 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.  


[[POTE_tuning|POTE generator]]: ~5/4 = 386.264
== Mercator ==
[[Subgroup]]: 2.3.5


Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
[[Comma list]]: {{monzo| -84 53 }}


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


Wedgie: {{wedgie| 0 53 84 }}
[[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 }}


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


Badness: 0.2843
[[Badness]] (Sintel): 6.67


=Cartography temperament=
== Schismerc ==
In terms of the normal comma list, Cartography 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 is exactly the same as the 5-limit of 53edo. Cartography can also be characterized as the 53&amp;159 temperament, with [[212edo]] being a possible tuning.  It should be noted that the 7-limit is somewhat independent for this temperament and is only really fully nailed down in one way or another by extending to the 11-limit.
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.


Commas: 32805/32768
[[Subgroup]]: 2.3.5.7


POTE generator: ~225/224 = 5.3666
[[Comma list]]: 15625/15552, 32805/32768


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


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


Wedgie: << 0 0 53 0 84 123 ]]
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc, 1113ccd }}


EDOs: {{EDOs| 53, 159, 212, 689c, 901cc }}
[[Badness]] (Sintel): 2.20


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.


==11-limit Cartography==
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.  
11-limit Cartography nails down the 7-limit by adding the symbiotic comma to the list of tempered commas.


Commas: 19712/19683, 32805/32768
Subgroup: 2.3.5.7.11


POTE generator: ~225/224 = 6.1430
Comma list: 385/384, 6250/6237, 19712/19683


Mapping: [<53 84 123 0 332 196], <0 0 0 1 -1 0]]
{{Mapping|legend=0| 53 84 123 0 332 | 0 0 0 1 -1 }}


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


Wedgie:
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


EDOs: {{EDOs| 53, 106d, 159, 212, 371d, 583cde }}
Badness (Sintel): 1.80


Badness: 0.0545
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 325/324, 385/384, 625/624, 19712/19683
13-limit Cartography adds the island comma to the list of tempered commas, and while this extension is connected to the 5-limit, it is independent of the 11-limit and 7-limit, so it can just as easily be added by itself to make a no-sevens no-elevens version of Cartography.


Commas: 676/675, 19712/19683, 32805/32768
{{Mapping|legend=0| 53 84 123 0 332 196 | 0 0 0 1 -1 0 }}


POTE generator: ~225/224 = 6.1430
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}})


Mapping: [<53 84 123 0 332 196], <0 0 0 1 -1 0]]
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


Mapping generators: ~81/80, ~7/1
Badness (Sintel): 1.24


Wedgie:
=== 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.


EDOs: {{EDOs| 53, 106d, 159, 212, 371df, 583cdeff }}
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.


Badness: 0.0300
Subgroup: 2.3.5.7.11


==Pentacontatritonic==
Comma list: 540/539, 15625/15552, 32805/32768
This temperament differs from Cartography in that it uses a different 11-limit extension to nail down the 7-limit- specifically, the swetisma.


Commas: 540/539, 32805/32768
{{Mapping|legend=0| 53 84 123 0 481 | 0 0 0 1 -2 }}


POTE generator: ~385/384 = 4.1494
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}})


Mapping: [<53 84 123 0 481], <0 0 0 1 -2]]
{{Optimal ET sequence|legend=0| 53, 212e, 265, 318 }}


Mapping generators: ~81/80, ~7/1
Badness (Sintel): 3.80


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


EDOs: {{EDOs| 53, 159e, 212e, 265, 318, 583c }}
Comma list: 540/539, 729/728, 4096/4095, 13750/13689


Badness: 0.1151
{{Mapping|legend=0| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}


===13-limit===
Optimal tunings:
13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out- this extension is connected to the 7-limit.
* 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}})


Commas: 540/539, 4096/4095, 13750/13689
{{Optimal ET sequence|legend=0| 53, 212ef, 265, 318 }}


POTE generator: ~385/384 = 3.9850
Badness (Sintel): 2.53


Mapping: [<53 84 123 0 481 345], <0 0 0 1 -2 1]]
=== 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.


Mapping generators: ~81/80, ~7/1
Subgroup: 2.3.5.7.11


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


EDOs: {{EDOs| 53, 159ef, 212ef, 265, 318, 583cf }}
{{Mapping|legend=0| 106 168 246 0 69 | 0 0 0 1 1 }}
: mapping generators: ~2835/2816, ~7


Badness: 0.0612
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 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
 
[[Comma list]]: 225/224, 1728/1715, 3125/3087
 
{{Mapping|legend=1| 53 84 123 149 0 | 0 0 0 0 1 }}
: mapping generators: ~81/80, ~11
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 53, 106, 159d }}
 
[[Badness]] (Sintel): 2.09
 
=== 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 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 53, 106 }}
 
Badness (Sintel): 1.53
 
== 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]].
 
In the 11-limit, the [[~]][[11/8]] can serve as a generator, being 16 periods less than the canonical generator.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -19 14 -5 3 }}, {{monzo| 8 3 -20 12 }}
 
{{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
 
Comma list: 160083/160000, 820125/819896, 4302592/4296875
 
{{Mapping|legend=0| 53 84 2 -53 143 | 0 0 3 5 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}})
 
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742 }}
 
Badness (Sintel): 2.89
 
=== 13-limit ===
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 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}
 
Badness (Sintel): 1.97
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872
 
{{Mapping|legend=0| 53 84 2 -53 143 -46 257 | 0 0 3 5 1 6 -1 }}
 
Optimal tunings:
* 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}})
 
{{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
 
Comma list: 3025/3024, 102487/102400, {{monzo| 34 19 8 4 -1 }}
 
{{Mapping|legend=0| 159 252 0 -292 550 | 0 0 1 2 0 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 159, …, 795, 954, 1749e, 2703e, 3657ee }}
 
Badness (Sintel): 2.79
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 159, 795g, 954 }}
 
Badness (Sintel): 1.64
 
{{Navbox fractional-octave|53}}
 
[[Category:Mercator family]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
 
<!-- as 53rd-octave temperaments page redirects here -->
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"