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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|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>
 
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
 
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.  
 
Discussed elsewhere are:
 
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]


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


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Mapping generators: ~531441/524288, ~5/1
Mapping generators: ~531441/524288, ~5/1


{{Multival|legend=1| 0 53 84 }}
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264


[[POTE generator]]: ~5/4 = 386.264
{{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]]: 0.284323
[[Badness]]: 0.284323


== Schismerc ==
== 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.
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
[[Comma list]]: 15625/15552, 32805/32768
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


{{Multival|legend=1| 0 0 53 0 84 123 }}
[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666


[[POTE generator]]: ~225/224 = 5.3666
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
 
{{Val list|legend=1| 53, 159, 212, 689c, 901cc }}


[[Badness]]: 0.087022
[[Badness]]: 0.087022
Line 35: Line 51:
=== Cartography ===
=== 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 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.
Subgroup: 2.3.5.7.11


Comma list: 385/384, 6250/6237, 19712/19683
Comma list: 385/384, 6250/6237, 19712/19683
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


POTE generator: ~225/224 = 6.1204
Optimal tuning (POTE): ~225/224 = 6.1204


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


Badness: 0.054452
Badness: 0.054452


==== 13-limit ====
==== 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.  
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
 
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 }}
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


POTE generator: ~225/224 = 6.1430
Optimal tuning (POTE): ~225/224 = 6.1430


Vals: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }}
{{Optimal ET sequence|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}


Badness: 0.029980
Badness: 0.029980
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=== Pentacontatritonic ===
=== 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.
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.
Subgroup: 2.3.5.7.11


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


POTE generator: ~385/384 = 4.1494
Optimal tuning (POTE): ~385/384 = 4.1494


Vals: {{Val list| 53, 159e, 212e, 265, 318, 583c }}
{{Optimal ET sequence|legend=1| 53, 159e, 212e, 265, 318, 583c }}


Badness: 0.115066
Badness: 0.115066
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==== 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.
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: 540/539, 729/728, 4096/4095, 13750/13689
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


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


Vals: {{Val list| 53, 159ef, 212ef, 265, 318, 583cf }}
{{Optimal ET sequence|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}


Badness: 0.061158
Badness: 0.061158


=== Boiler ===
=== 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 good 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 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
Comma list: 9801/9800, 15625/15552, 32805/32768
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Mapping generators: ~2835/2816, ~7
Mapping generators: ~2835/2816, ~7


POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232
Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232


Vals: {{Val list| 106, 212 }}
{{Optimal ET sequence|legend=1| 106, 212 }}


Badness: 0.109648
Badness: 0.109648
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== Joliet ==
== 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 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
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Mapping generators: ~81/80, ~11/1
Mapping generators: ~81/80, ~11/1


[[POTE generator]]: ~176/175 = 8.8283
[[Optimal tuning]] ([[POTE]]): ~176/175 = 8.8283


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


[[Badness]]: 0.063254
[[Badness]]: 0.063254


=== 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


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Mapping generators: ~81/80, ~11/1
Mapping generators: ~81/80, ~11/1


POTE generator: ~176/175 = 8.1254
Optimal tuning (POTE): ~176/175 = 8.1254


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


Badness: 0.036988
Badness: 0.036988


[[Category:Theory]]
== Iodine ==
[[Category:Temperament family]]
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]].
[[Category:Mercator]]
 
[[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]] ([[CTE]]): ~6075/3584 = 913.7347
 
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
 
[[Badness]]: 0.477
 
=== 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 tuning (CTE): ~6075/3584 = 913.7322
 
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
 
Badness: 0.0875
 
=== 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 tuning (CTE): ~441/260 = 913.7115
 
{{Optimal ET sequence|legend=1| 159, 424cdff, 583f, 742, 1643 }}
 
Badness: 0.0476
 
=== 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 tuning (CTE): ~441/260 = 913.7131
 
{{Optimal ET sequence|legend=1| 159, 583f, 742 }}
 
Badness: 0.0328
 
{{Navbox fractional-octave|53}}
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"