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 | ||
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264 | [[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264 | ||
{{ | {{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 [[ | 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 | ||
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Mapping generators: ~81/80, ~7/1 | Mapping generators: ~81/80, ~7/1 | ||
[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666 | [[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666 | ||
{{ | {{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 | {{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 | {{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 | {{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 | {{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 | {{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 | ||
{{ | {{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 | {{Optimal ET sequence|legend=1| 53, 106, 159d }} | ||
Badness: 0.036988 | Badness: 0.036988 | ||
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[[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347 | [[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347 | ||
{{ | {{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }} | ||
[[Badness]]: 0.477 | [[Badness]]: 0.477 | ||
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Optimal tuning (CTE): ~6075/3584 = 913.7322 | Optimal tuning (CTE): ~6075/3584 = 913.7322 | ||
Optimal | {{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 | {{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 | {{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]] | ||