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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.
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]


[[POTE_tuning|POTE generator]]: ~5/4 = 386.264
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 }}]
Discussed elsewhere are:


Mapping generators:
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]


Wedgie: {{wedgie| 0 53 84 }}
== Mercator ==
[[Subgroup]]: 2.3.5


{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
[[Comma list]]: {{monzo| -84 53 }}


Badness: 0.2843
[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]


=Cartography temperament=
: mapping generators: ~531441/524288, ~5
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.


Commas: 32805/32768
[[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¢


POTE generator: ~225/224 = 5.3666
{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}


Mapping: [<53 84 123 0], <0 0 0 1]]
[[Badness]] (Sintel): 6.670


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


Wedgie: << 0 0 53 0 84 123 ]]
Subgroup: 2.3.5.7


EDOs: {{EDOs| 53, 159, 212, 689c, 901cc }}
[[Comma list]]: 15625/15552, 32805/32768


Badness: 0.0870
[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]


==Undecimal Cartography==
: mapping generators: ~81/80, ~7
Undecimal Cartography nails down the 7-limit by adding the [[symbiotic comma]] to the list of tempered commas.


Commas: 19712/19683, 32805/32768
[[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¢


POTE generator: ~225/224 = 6.1430
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}


Mapping: [<53 84 123 0 332 196], <0 0 0 1 -1 0]]
[[Badness]] (Sintel): 2.202


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


Wedgie:
Subgroup: 2.3.5.7.11


EDOs: {{EDOs| 53, 106d, 159, 212, 371d, 583cde }}
Comma list: 385/384, 6250/6237, 19712/19683


Badness: 0.0545
Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}]


===13-limit===
Optimal tunings:
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.
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢


Commas: 676/675, 19712/19683, 32805/32768
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


POTE generator: ~225/224 = 6.1430
Badness (Sintel): 1.800


Mapping: [<53 84 123 0 332 196], <0 0 0 1 -1 0]]
==== 13-limit ====
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.


Mapping generators: ~81/80, ~7/1
Subgroup: 2.3.5.7.11.13


Wedgie:
Comma list: 325/324, 385/384, 625/624, 19712/19683


EDOs: {{EDOs| 53, 106d, 159, 212, 371df, 583cdeff }}
Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}


Badness: 0.0300
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢


==Pentacontatritonic==
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}
This temperament differs from Undecimal Cartography in that it tempers out a different 11-limit comma in order to nail down the 7-limit- specifically, the swetisma.


Commas: 540/539, 32805/32768
Badness (Sintel): 1.239


POTE generator: ~385/384 = 4.1494
=== 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.


Mapping: [<53 84 123 0 481], <0 0 0 1 -2]]
Subgroup: 2.3.5.7.11


Mapping generators: ~81/80, ~7/1
Comma list: 540/539, 15625/15552, 32805/32768


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


EDOs: {{EDOs| 53, 159e, 212e, 265, 318, 583c }}
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5956¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5697¢


Badness: 0.1151
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}


===13-limit===
Badness (Sintel): 3.804
13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out- this extension is connected to the 7-limit.


Commas: 540/539, 4096/4095, 13750/13689
==== 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.


POTE generator: ~385/384 = 3.9850
Subgroup: 2.3.5.7.11.13


Mapping: [<53 84 123 0 481 345], <0 0 0 1 -2 1]]
Comma list: 540/539, 729/728, 4096/4095, 13750/13689


Mapping generators: ~81/80, ~7/1
Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}


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


EDOs: {{EDOs| 53, 159ef, 212ef, 265, 318, 583cf }}
Optimal tunings:  
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4057¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4008¢


Badness: 0.0612
{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}
 
Badness (Sintel): 2.527
 
=== 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: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}]
 
: mapping generators: ~2835/2816, ~7
 
Optimal tunings:
* CTE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 230.6341¢
* CWE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 231.1634¢
 
{{Optimal ET sequence|legend=0| 106, 212 }}
 
Badness (Sintel): 3.625
 
== 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.
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 225/224, 1728/1715, 3125/3087
 
[[Mapping]]: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}]
 
: mapping generators: ~50/49, ~11
 
[[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¢
 
{{Optimal ET sequence|legend=1| 53, 106, 159d }}
 
[[Badness]] (Sintel): 2.091
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 325/324, 640/637
 
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]].
 
[[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¢
 
{{Optimal ET sequence|legend=0| 159, 583f, 742 }}
 
Badness (Sintel): 1.568
 
{{Navbox fractional-octave|53}}
 
[[Category:Temperament families]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"