Mercator family: Difference between revisions

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<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 ==

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

~~{{Multival|legend=1| 0 53 84 }}~~

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

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== 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

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

~~{{Multival|legend=1| 0 0 53 0 84 123 }}~~

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

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Badness: 0.0328

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Mercator family]]

[[Category:Rank 2]]

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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.
+Discussed elsewhere are:
+* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]
 == Mercator ==
@@ Line 9: / Line 25: @@
 Mapping generators: ~531441/524288, ~5/1
-{{Multival|legend=1| 0 53 84 }}
 [[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264
@@ Line 19: / Line 33: @@
 == 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
@@ Line 28: / Line 42: @@
 Mapping generators: ~81/80, ~7/1
-{{Multival|legend=1| 0 0 53 0 84 123 }}
 [[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666
@@ Line 211: / Line 223: @@
 Badness: 0.0328
+{{Navbox fractional-octave|53}}
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Mercator family]] <!-- main article -->
 [[Category:Rank 2]]