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 | 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|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 | == Mercator == | ||
[[Comma list]]: {{monzo| -84 53 }} | |||
[[Mapping]]: [{{val| 53 84 123 }}, {{val| 0 0 1 }}] | |||
[[ | |||
Mapping generators: ~531441/524288, ~5/1 | Mapping generators: ~531441/524288, ~5/1 | ||
{{Multival|legend=1| 0 53 84 }} | {{Multival|legend=1| 0 53 84 }} | ||
[[POTE generator]]: ~5/4 = 386.264 | |||
{{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 }} | ||
Badness: 0. | [[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. | ||
Comma list: 15625/15552, 32805/32768 | [[Comma list]]: 15625/15552, 32805/32768 | ||
[[Mapping]]: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}] | |||
Mapping: [{{val| 53 84 123 0 }}, {{val| 0 0 0 1 }}] | |||
Mapping generators: ~81/80, ~7/1 | Mapping generators: ~81/80, ~7/1 | ||
{{Multival|legend=1| 0 0 53 0 84 123 }} | {{Multival|legend=1| 0 0 53 0 84 123 }} | ||
[[POTE generator]]: ~225/224 = 5.3666 | |||
{{Val list|legend=1| 53, 159, 212, 689c, 901cc }} | {{Val list|legend=1| 53, 159, 212, 689c, 901cc }} | ||
Badness: 0. | [[Badness]]: 0.087022 | ||
=== Cartography | === Cartography === | ||
Cartography nails down both the 7-limit and the 11-limit by adding the [[symbiotic comma]] to Schismerc's list of tempered commas. The name for this temperament comes from how good the mappings are, and also from the idea of "Mercator" being a dual reference to both Nicolas Mercator and Gerardus Mercator. | Cartography nails down both the 7-limit and the 11-limit by adding the [[symbiotic comma]] to Schismerc's list of tempered commas. The name for this temperament comes from how good the mappings are, and also from the idea of "Mercator" being a dual reference to both Nicolas Mercator and Gerardus Mercator. | ||
Comma list: 385/384, 6250/6237, 19712/19683 | Comma list: 385/384, 6250/6237, 19712/19683 | ||
Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}] | Mapping: [{{val| 53 84 123 0 332 }}, {{val| 0 0 0 1 -1 }}] | ||
| Line 45: | Line 42: | ||
Mapping generators: ~81/80, ~7/1 | Mapping generators: ~81/80, ~7/1 | ||
{{Val list | POTE generator: ~225/224 = 6.1204 | ||
Vals: {{Val list| 53, 106d, 159, 212, 371d, 583cde }} | |||
Badness: 0. | Badness: 0.054452 | ||
==== 13-limit | ==== 13-limit ==== | ||
13-limit Cartography adds the [[island comma]] to the list of tempered commas- a development which fits well with the ideas of mapmaking and geography. The harmonic 13 in this extension is part of the period and independent of the generator for harmonics 7 and 11. | 13-limit Cartography adds the [[island comma]] to the list of tempered commas- a development which fits well with the ideas of mapmaking and geography. The harmonic 13 in this extension is part of the period and independent of the generator for harmonics 7 and 11. | ||
Commas: 325/324, 385/384, 625/624, 19712/19683 | Commas: 325/324, 385/384, 625/624, 19712/19683 | ||
Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }} | Mapping: [{{val| 53 84 123 0 332 196 }}, {{val| 0 0 0 1 -1 0 }} | ||
| Line 60: | Line 57: | ||
Mapping generators: ~81/80, ~7/1 | Mapping generators: ~81/80, ~7/1 | ||
{{Val list | POTE generator: ~225/224 = 6.1430 | ||
Vals: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }} | |||
Badness: 0. | Badness: 0.029980 | ||
=== Pentacontatritonic | === Pentacontatritonic === | ||
First proposed by [[User:Xenllium|Xenllium]], this temperament nails down both the 7-limit and the 11-limit by tempering out the [[swetisma]]. | First proposed by [[User:Xenllium|Xenllium]], this temperament nails down both the 7-limit and the 11-limit by tempering out the [[swetisma]]. | ||
Comma list: 540/539, 15625/15552, 32805/32768 | Comma list: 540/539, 15625/15552, 32805/32768 | ||
Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}] | Mapping: [{{val| 53 84 123 0 481 }}, {{val| 0 0 0 1 -2 }}] | ||
| Line 75: | Line 72: | ||
Mapping generators: ~81/80, ~7/1 | Mapping generators: ~81/80, ~7/1 | ||
{{Val list | POTE generator: ~385/384 = 4.1494 | ||
Vals: {{Val list| 53, 159e, 212e, 265, 318, 583c }} | |||
Badness: 0. | Badness: 0.115066 | ||
==== 13-limit | ==== 13-limit ==== | ||
13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out – in this extension the harmonic 13 is connected to the generator. | 13-limit Pentacontatritonic adds the schismina to the list of commas being tempered out – in this extension the harmonic 13 is connected to the generator. | ||
Comma list: 540/539, 729/728, 4096/4095, 13750/13689 | Comma list: 540/539, 729/728, 4096/4095, 13750/13689 | ||
Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }} | Mapping: [{{val| 53 84 123 0 481 345 }}, {{val| 0 0 0 1 -2 1 }} | ||
| Line 90: | Line 87: | ||
Mapping generators: ~81/80, ~7/1 | Mapping generators: ~81/80, ~7/1 | ||
{{Val list | POTE generator: ~385/384 = 3.9850 | ||
Vals: {{Val list| 53, 159ef, 212ef, 265, 318, 583cf }} | |||
Badness: 0. | Badness: 0.061158 | ||
=== Boiler | === Boiler === | ||
Boiler nails down both the 7-limit and the 11-limit by adding the [[kalisma]] to Schismerc's list of tempered commas, though unlike with the other extensions of Schismerc, this temperament lacks a good 13-limit extension. | Boiler nails down both the 7-limit and the 11-limit by adding the [[kalisma]] to Schismerc's list of tempered commas, though unlike with the other extensions of Schismerc, this temperament lacks a good 13-limit extension. The name for this temperament is a reference to how 212 degrees Fahrenheit is the boiling point of water, as well as to a number of mechanical devices that boil water for various purposes. | ||
Comma list: 9801/9800, 15625/15552, 32805/32768 | Comma list: 9801/9800, 15625/15552, 32805/32768 | ||
Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}] | Mapping: [{{val| 106 168 246 0 69 }}, {{val| 0 0 0 1 1 }}] | ||
| Line 105: | Line 102: | ||
Mapping generators: ~2835/2816, ~7 | Mapping generators: ~2835/2816, ~7 | ||
POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232 | |||
Vals: {{Val list| 106, 212 }} | |||
Badness: 0.109648 | |||
== Joliet == | |||
Joliet can be characterized as the 53 & 106 temperament, having 7-limit representation akin to 53EDO with the addition of harmonic 11 represented by an independent generator. The name for this temperament is a reference to 106 being the maximum number of characters in the Joliet extension to the ISO 9660 file system. | |||
[[Comma list]]: 225/224, 1728/1715, 3125/3087 | |||
Mapping: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}] | [[Mapping]]: [{{val| 53 84 123 149 0 }}, {{val| 0 0 0 0 1 }}] | ||
Mapping generators: ~81/80, ~11/1 | Mapping generators: ~81/80, ~11/1 | ||
[[POTE generator]]: ~176/175 = 8.8283 | |||
{{Val list|legend=1| 53, 106, 159d }} | {{Val list|legend=1| 53, 106, 159d }} | ||
Badness: 0. | [[Badness]]: 0.063254 | ||
=== 13-limit === | |||
Comma list: 169/168, 225/224, 325/324, 640/637 | Comma list: 169/168, 225/224, 325/324, 640/637 | ||
Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}] | Mapping: [{{val| 53 84 123 149 0 196 }}, {{val| 0 0 0 0 1 0 }}] | ||
| Line 134: | Line 130: | ||
Mapping generators: ~81/80, ~11/1 | Mapping generators: ~81/80, ~11/1 | ||
{{Val list | POTE generator: ~176/175 = 8.1254 | ||
Vals: {{Val list| 53, 106, 159d }} | |||
Badness: 0. | Badness: 0.036988 | ||
[[Category:Theory]] | [[Category:Theory]] | ||