Mercator family: Difference between revisions
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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]]. 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 = | |||
[[ | Comma list: {{monzo| -84 53 }} | ||
[[POTE generator]]: ~5/4 = 386.264 | |||
Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}] | Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}] | ||
Mapping generators: | Mapping generators: ~81/80, ~5 | ||
{{Multival|legend=1| 0 53 84 }} | |||
{{Val list|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 }} | ||
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Badness: 0.2843 | Badness: 0.2843 | ||
=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 is exactly the same as the 5-limit of 53edo. | 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 is exactly the same as the 5-limit of 53edo. It should be noted that the 7-limit is somewhat independent for this temperament and is only really fully nailed down in one way or another by extending to the 11-limit. Among the known 11-limit extensions are Cartography, and Pentacontatritonic. | ||
Comma list: 15625/15552, 32805/32768 | |||
POTE generator: ~225/224 = 5.3666 | POTE generator: ~225/224 = 5.3666 | ||
Mapping: [ | Mapping: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}] | ||
Mapping generators: ~81/80, ~7 | Mapping generators: ~81/80, ~7 | ||
{{Multival|legend=1| 0 0 53 0 84 123 }} | |||
{{Val list|legend=1| 53, 159, 212, 689c, 901cc }} | {{Val list|legend=1| 53, 159, 212, 689c, 901cc }} | ||
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Badness: 0.0870 | Badness: 0.0870 | ||
==Cartography== | == Cartography == | ||
Cartography nails down the 7-limit by adding the [[symbiotic comma]] to Schismerc's list of tempered commas. | Cartography nails down the 7-limit by adding the [[symbiotic comma]] to Schismerc's list of tempered commas. | ||
Comma list: 385/384, 6250/6237, 19712/19683 | |||
POTE generator: ~225/224 = 6.1430 | POTE generator: ~225/224 = 6.1430 | ||
Mapping: [ | Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }}] | ||
Mapping generators: ~81/80, ~7 | Mapping generators: ~81/80, ~7 | ||
{{Val list|legend=1| 53, 106d, 159, 212, 371d, 583cde }} | {{Val list|legend=1| 53, 106d, 159, 212, 371d, 583cde }} | ||
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Badness: 0.0545 | Badness: 0.0545 | ||
===13-limit=== | === 13-limit === | ||
13-limit Cartography adds the island comma to the list of tempered commas, and while this extension is connected to the 5-limit, it is independent of the 11-limit and 7-limit, so it can just as easily be added by itself to make a no-sevens no-elevens version of Cartography. | 13-limit Cartography adds the island comma to the list of tempered commas, and while this extension is connected to the 5-limit, it is independent of the 11-limit and 7-limit, so it can just as easily be added by itself to make a no-sevens no-elevens version of Cartography. | ||
Commas: | Commas: 325/324, 385/384, 625/624, 19712/19683 | ||
POTE generator: ~225/224 = 6.1430 | POTE generator: ~225/224 = 6.1430 | ||
Mapping: [ | Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }} | ||
Mapping generators: ~81/80, ~7 | Mapping generators: ~81/80, ~7 | ||
{{Val list|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }} | {{Val list|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }} | ||
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Badness: 0.0300 | Badness: 0.0300 | ||
==Pentacontatritonic== | == Pentacontatritonic == | ||
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 | 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 | |||
POTE generator: ~385/384 = 4.1494 | POTE generator: ~385/384 = 4.1494 | ||
Mapping: [ | Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}] | ||
Mapping generators: ~81/80, ~7 | Mapping generators: ~81/80, ~7 | ||
{{Val list|legend=1| 53, 159e, 212e, 265, 318, 583c }} | {{Val list|legend=1| 53, 159e, 212e, 265, 318, 583c }} | ||
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Badness: 0.1151 | Badness: 0.1151 | ||
===13-limit=== | === 13-limit === | ||
13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out- this extension is connected to the 7-limit. | 13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out- this extension is connected to the 7-limit. | ||
Comma list: 540/539, 729/728, 4096/4095, 13750/13689 | |||
POTE generator: ~385/384 = 3.9850 | POTE generator: ~385/384 = 3.9850 | ||
Mapping: [ | Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }} | ||
Mapping generators: ~81/80, ~7 | |||
{{Val list|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }} | {{Val list|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }} | ||
Badness: 0.0612 | Badness: 0.0612 | ||