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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 }}
 
[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]
 
: mapping generators: ~531441/524288, ~5
 
[[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¢
 
{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
 
[[Badness]] (Sintel): 6.670
 
== 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.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 15625/15552, 32805/32768
 
[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}]
 
: mapping generators: ~81/80, ~7
 
[[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¢
 
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
 
[[Badness]] (Sintel): 2.202
 
=== 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.


Comma list: {{monzo| -84 53 }}
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~5/4 = 386.264
Comma list: 385/384, 6250/6237, 19712/19683


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


Mapping generators: ~81/80, ~5/1
Optimal tunings:
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢


{{Multival|legend=1| 0 53 84 }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
Badness (Sintel): 1.800


Badness: 0.2843
==== 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.  


= Schismerc =
Subgroup: 2.3.5.7.11.13
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: 325/324, 385/384, 625/624, 19712/19683


POTE generator: ~225/224 = 5.3666
Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}


Mapping: [{{val| 53 84 123 0 }}, {{val| 0 0 0 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¢


Mapping generators: ~81/80, ~7/1
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


{{Multival|legend=1| 0 0 53 0 84 123 }}
Badness (Sintel): 1.239


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


Badness: 0.0870
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 15625/15552, 32805/32768
 
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¢
 
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}
 
Badness (Sintel): 3.804
 
==== 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.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4096/4095, 13750/13689
 
Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }}
 
Optimal tuning (POTE): ~385/384 = 3.9850
 
Optimal tunings:
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4057¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4008¢
 
{{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 ==
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 Joliet- an extension to the ISO 9660 file system.
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
[[Comma list]]: 225/224, 1728/1715, 3125/3087


POTE generator: ~4125/4096 = 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: ~4125/4096 = 8.1254


Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}]


Mapping generators: ~81/80, ~11/1
Optimal tunings:
* CTE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
* CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.4859¢


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


Badness: 0.0370
Badness (Sintel): 1.528


== Cartography ==
== Iodine ==
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.
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]].  


Comma list: 385/384, 6250/6237, 19712/19683
[[Subgroup]]: 2.3.5.7


POTE generator: ~225/224 = 6.1430
[[Comma list]]: {{monzo| -19 14 -5 3 }}, {{monzo| 8 3 -20 12 }}


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


Mapping generators: ~81/80, ~7/1
: mapping generators: ~3125/3087, ~6075/3584


{{Val list|legend=1| 53, 106d, 159, 212, 371d, 583cde }}
[[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¢


Badness: 0.0545
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}


=== 13-limit ===
[[Badness]] (Sintel): 12.075
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.  


Commas: 325/324, 385/384, 625/624, 19712/19683
=== 11-limit ===
24 periods plus the reduced generator correspond to [[11/8]].


POTE generator: ~225/224 = 6.1430
Subgroup: 2.3.5.7.11


Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}
Comma list: 160083/160000, 820125/819896, 4302592/4296875


Mapping generators: ~81/80, ~7/1
Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]


{{Val list|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}
Optimal tunings:
* CTE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7322¢
* CWE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7345¢


Badness: 0.0300
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}


== Pentacontatritonic ==
Badness (Sintel): 2.893
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.


Comma list: 540/539, 15625/15552, 32805/32768
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~385/384 = 4.1494
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488


Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}]
Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]


Mapping generators: ~81/80, ~7/1
Optimal tunings:
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7115¢
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7126¢


{{Val list|legend=1| 53, 159e, 212e, 265, 318, 583c }}
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}


Badness: 0.1151
Badness (Sintel): 1.967


=== 13-limit ===
=== 17-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: 2.3.5.7.11.13.17


Comma list: 540/539, 729/728, 4096/4095, 13750/13689
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872


POTE generator: ~385/384 = 3.9850
Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]


Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 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, ~7/1
{{Optimal ET sequence|legend=0| 159, 583f, 742 }}


{{Val list|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}
Badness (Sintel): 1.568


Badness: 0.0612
{{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"