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Mercator family: Difference between revisions

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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]]
[[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.  
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 ==
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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
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264


{{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]]: 0.284323
[[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
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
[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666


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


[[Badness]]: 0.087022
[[Badness]]: 0.087022
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Optimal tuning (POTE): ~225/224 = 6.1204
Optimal tuning (POTE): ~225/224 = 6.1204


Optimal GPV sequence: {{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
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Optimal tuning (POTE): ~225/224 = 6.1430
Optimal tuning (POTE): ~225/224 = 6.1430


Optimal GPV sequence: {{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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Optimal tuning (POTE): ~385/384 = 4.1494
Optimal tuning (POTE): ~385/384 = 4.1494


Optimal GPV sequence: {{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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Optimal tuning (POTE): ~385/384 = 3.9850
Optimal tuning (POTE): ~385/384 = 3.9850


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


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


Badness: 0.109648
Badness: 0.109648
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[[Optimal tuning]] ([[POTE]]): ~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
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Optimal tuning (POTE): ~176/175 = 8.1254
Optimal tuning (POTE): ~176/175 = 8.1254


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


Badness: 0.036988
Badness: 0.036988


== Iodine ==
== 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.  
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
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347
[[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347


{{Val list|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}


[[Badness]]: 0.477
[[Badness]]: 0.477


=== 11-limit ===
=== 11-limit ===
24 periods plus the reduced generator correspond to [[11/8]].
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Optimal tuning (CTE): ~6075/3584 = 913.7322
Optimal tuning (CTE): ~6075/3584 = 913.7322


Optimal GPV sequence: {{Val list| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}


Badness: 0.0875
Badness: 0.0875
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Optimal tuning (CTE): ~441/260 = 913.7115
Optimal tuning (CTE): ~441/260 = 913.7115


Optimal GPV sequence: {{Val list| 159, 424cdff, 583f, 742, 1643 }}
{{Optimal ET sequence|legend=1| 159, 424cdff, 583f, 742, 1643 }}


Badness: 0.0476
Badness: 0.0476
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Optimal tuning (CTE): ~441/260 = 913.7131
Optimal tuning (CTE): ~441/260 = 913.7131


Optimal GPV sequence: {{Val list| 159, 583f, 742 }}
{{Optimal ET sequence|legend=1| 159, 583f, 742 }}


Badness: 0.0328
Badness: 0.0328
{{Navbox fractional-octave|53}}


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Mercator family]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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