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.  
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 }}


Comma list: {{monzo| -84 53 }}
[[Mapping]]: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
 
[[POTE generator]]: ~5/4 = 386.264
 
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: 0.2843
[[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


POTE generator: ~225/224 = 5.3666
[[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: 0.0870
[[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
POTE generator: ~225/224 = 6.1204


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|legend=1| 53, 106d, 159, 212, 371d, 583cde }}
POTE generator: ~225/224 = 6.1204
 
Vals: {{Val list| 53, 106d, 159, 212, 371d, 583cde }}


Badness: 0.0545
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
POTE generator: ~225/224 = 6.1430


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|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}
POTE generator: ~225/224 = 6.1430
 
Vals: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }}


Badness: 0.0300
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
POTE generator: ~385/384 = 4.1494


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|legend=1| 53, 159e, 212e, 265, 318, 583c }}
POTE generator: ~385/384 = 4.1494
 
Vals: {{Val list| 53, 159e, 212e, 265, 318, 583c }}


Badness: 0.1151
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
POTE generator: ~385/384 = 3.9850


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|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}
POTE generator: ~385/384 = 3.9850
 
Vals: {{Val list| 53, 159ef, 212ef, 265, 318, 583cf }}


Badness: 0.0612
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. 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.
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
POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232


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


{{Val list|legend=1| 106, 212 }}
POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232


Badness: 0.1096
Vals: {{Val list| 106, 212 }}


== Joliet  ==
Badness: 0.109648
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
== 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.


POTE generator: ~176/175 = 8.8283
[[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: 0.0633
[[Badness]]: 0.063254
 
=== 13-limit  ===


=== 13-limit ===
Comma list: 169/168, 225/224, 325/324, 640/637
Comma list: 169/168, 225/224, 325/324, 640/637
POTE generator: ~176/175 = 8.1254


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|legend=1| 53, 106, 159d }}
POTE generator: ~176/175 = 8.1254
 
Vals: {{Val list| 53, 106, 159d }}


Badness: 0.0370
Badness: 0.036988


[[Category:Theory]]
[[Category:Theory]]

Revision as of 10:51, 5 June 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⟩

Mapping: [⟨53 84 123], ⟨0 0 1]]

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

Wedgie: ⟨⟨ 0 53 84 ]]

POTE generator: ~5/4 = 386.264

Template:Val list

Badness: 0.284323

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.

Comma list: 15625/15552, 32805/32768

Mapping: [⟨53 84 123 0], ⟨0 0 0 1]]

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

Wedgie: ⟨⟨ 0 0 53 0 84 123 ]]

POTE generator: ~225/224 = 5.3666

Template:Val list

Badness: 0.087022

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.

Comma list: 385/384, 6250/6237, 19712/19683

Mapping: [⟨53 84 123 0 332], ⟨0 0 0 1 -1]]

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

POTE generator: ~225/224 = 6.1204

Vals: Template:Val list

Badness: 0.054452

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.

Commas: 325/324, 385/384, 625/624, 19712/19683

Mapping: [⟨53 84 123 0 332 196], ⟨0 0 0 1 -1 0]

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

POTE generator: ~225/224 = 6.1430

Vals: Template:Val list

Badness: 0.029980

Pentacontatritonic

First proposed by 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

Mapping: [⟨53 84 123 0 481], ⟨0 0 0 1 -2]]

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

POTE generator: ~385/384 = 4.1494

Vals: Template:Val list

Badness: 0.115066

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.

Comma list: 540/539, 729/728, 4096/4095, 13750/13689

Mapping: [⟨53 84 123 0 481 345], ⟨0 0 0 1 -2 1]

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

POTE generator: ~385/384 = 3.9850

Vals: Template:Val list

Badness: 0.061158

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. 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

Mapping: [⟨106 168 246 0 69], ⟨0 0 0 1 1]]

Mapping generators: ~2835/2816, ~7

POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232

Vals: Template:Val list

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: [⟨53 84 123 149 0], ⟨0 0 0 0 1]]

Mapping generators: ~81/80, ~11/1

POTE generator: ~176/175 = 8.8283

Template:Val list

Badness: 0.063254

13-limit

Comma list: 169/168, 225/224, 325/324, 640/637

Mapping: [⟨53 84 123 149 0 196], ⟨0 0 0 0 1 0]]

Mapping generators: ~81/80, ~11/1

POTE generator: ~176/175 = 8.1254

Vals: Template:Val list

Badness: 0.036988

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