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 }} | ||
| Line 15: | Line 17: | ||
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 }} | ||
| Line 32: | Line 34: | ||
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 }} | ||
| Line 49: | Line 49: | ||
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 }} | ||
| Line 66: | Line 64: | ||
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 }} | ||
| Line 83: | Line 79: | ||
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 | ||
Revision as of 07:28, 11 March 2021
The Mercator family tempers out Mercator's comma, [-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: [-84 53⟩
POTE generator: ~5/4 = 386.264
Mapping: [⟨53 84 123], ⟨0 0 1]]
Mapping generators: ~81/80, ~5
⟨⟨0 53 84]]
Badness: 0.2843
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. 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
Mapping: [⟨53 84 123 0], ⟨0 0 0 1]]
Mapping generators: ~81/80, ~7
⟨⟨0 0 53 0 84 123]]
Badness: 0.0870
Cartography
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
Mapping: [⟨53 84 123 0 332 196], ⟨0 0 0 1 -1 0]]
Mapping generators: ~81/80, ~7
Badness: 0.0545
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.
Commas: 325/324, 385/384, 625/624, 19712/19683
POTE generator: ~225/224 = 6.1430
Mapping: [⟨53 84 123 0 332 196], ⟨0 0 0 1 -1 0]
Mapping generators: ~81/80, ~7
Badness: 0.0300
Pentacontatritonic
First proposed by 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
Mapping: [⟨53 84 123 0 481], ⟨0 0 0 1 -2]]
Mapping generators: ~81/80, ~7
Badness: 0.1151
13-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
Mapping: [⟨53 84 123 0 481 345], ⟨0 0 0 1 -2 1]
Mapping generators: ~81/80, ~7
Badness: 0.0612