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|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.  


== Mercator ==
== Mercator ==
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -84 53 }}
[[Comma list]]: {{monzo| -84 53 }}


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{{Multival|legend=1| 0 53 84 }}
{{Multival|legend=1| 0 53 84 }}


[[POTE generator]]: ~5/4 = 386.264
[[Optimal tuning]] ([[POTE]]): ~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 }}
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== 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.
Subgroup: 2.3.5..7


[[Comma list]]: 15625/15552, 32805/32768
[[Comma list]]: 15625/15552, 32805/32768
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{{Multival|legend=1| 0 0 53 0 84 123 }}
{{Multival|legend=1| 0 0 53 0 84 123 }}


[[POTE generator]]: ~225/224 = 5.3666
[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666


{{Val list|legend=1| 53, 159, 212, 689c, 901cc }}
{{Val list|legend=1| 53, 159, 212, 689c, 901cc }}
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=== Cartography ===
=== Cartography ===
Cartography is a strong extension to Schismerc that 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 is a strong extension to Schismerc that 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.
Subgroup: 2.3.5.7.11


Comma list: 385/384, 6250/6237, 19712/19683
Comma list: 385/384, 6250/6237, 19712/19683
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


POTE generator: ~225/224 = 6.1204
Optimal tuning (POTE): ~225/224 = 6.1204


Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371d, 583cde }}
Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371d, 583cde }}
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==== 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
Subgroup: 2.3.5.7.11.13
 
Comma list: 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 }}
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


POTE generator: ~225/224 = 6.1430
Optimal tuning (POTE): ~225/224 = 6.1430


Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }}
Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }}
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=== 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]].  Like Cartography, Pentacontatritonic is a strong extension to Schismerc.
First proposed by [[User:Xenllium|Xenllium]], this temperament nails down both the 7-limit and the 11-limit by tempering out the [[swetisma]].  Like Cartography, Pentacontatritonic is a strong extension to Schismerc.
Subgroup: 2.3.5.7.11


Comma list: 540/539, 15625/15552, 32805/32768
Comma list: 540/539, 15625/15552, 32805/32768
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


POTE generator: ~385/384 = 4.1494
Optimal tuning (POTE): ~385/384 = 4.1494


Optimal GPV sequence: {{Val list| 53, 159e, 212e, 265, 318, 583c }}
Optimal GPV sequence: {{Val list| 53, 159e, 212e, 265, 318, 583c }}
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==== 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.
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 4096/4095, 13750/13689
Comma list: 540/539, 729/728, 4096/4095, 13750/13689
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Mapping generators: ~81/80, ~7/1
Mapping generators: ~81/80, ~7/1


POTE generator: ~385/384 = 3.9850
Optimal tuning (POTE): ~385/384 = 3.9850


Optimal GPV sequence: {{Val list| 53, 159ef, 212ef, 265, 318, 583cf }}
Optimal GPV sequence: {{Val list| 53, 159ef, 212ef, 265, 318, 583cf }}
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=== 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 is not only a weak extension, but lacks a good 13-limit extension of its own. 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 is not only a weak extension, but lacks a clear 13-limit extension of its own. 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.
 
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 15625/15552, 32805/32768
Comma list: 9801/9800, 15625/15552, 32805/32768
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Mapping generators: ~2835/2816, ~7
Mapping generators: ~2835/2816, ~7


POTE generator: ~225/224 = 6.3976 or ~441/440 = 4.9232
Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232


Optimal GPV sequence: {{Val list| 106, 212 }}
Optimal GPV sequence: {{Val list| 106, 212 }}
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== Joliet ==
== 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.
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.
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 1728/1715, 3125/3087
[[Comma list]]: 225/224, 1728/1715, 3125/3087
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Mapping generators: ~81/80, ~11/1
Mapping generators: ~81/80, ~11/1


[[POTE generator]]: ~176/175 = 8.8283
[[Optimal tuning]] ([[POTE]]): ~176/175 = 8.8283


{{Val list|legend=1| 53, 106, 159d }}
{{Val list|legend=1| 53, 106, 159d }}
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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 325/324, 640/637
Comma list: 169/168, 225/224, 325/324, 640/637


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


POTE generator: ~176/175 = 8.1254
Optimal tuning (POTE): ~176/175 = 8.1254


Optimal GPV sequence: {{Val list| 53, 106, 159d }}
Optimal GPV sequence: {{Val list| 53, 106, 159d }}

Revision as of 14:32, 29 August 2022

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

Subgroup: 2.3.5

Comma list: [-84 53⟩

Mapping: [⟨53 84 123], ⟨0 0 1]]

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

Wedgie: ⟨⟨ 0 53 84 ]]

Optimal tuning (POTE): ~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.

Subgroup: 2.3.5..7

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 ]]

Optimal tuning (POTE): ~225/224 = 5.3666

Template:Val list

Badness: 0.087022

Cartography

Cartography is a strong extension to Schismerc that 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.

Subgroup: 2.3.5.7.11

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

Optimal tuning (POTE): ~225/224 = 6.1204

Optimal GPV sequence: 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.

Subgroup: 2.3.5.7.11.13

Comma list: 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

Optimal tuning (POTE): ~225/224 = 6.1430

Optimal GPV sequence: 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. Like Cartography, Pentacontatritonic is a strong extension to Schismerc.

Subgroup: 2.3.5.7.11

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

Optimal tuning (POTE): ~385/384 = 4.1494

Optimal GPV sequence: 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.

Subgroup: 2.3.5.7.11.13

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

Optimal tuning (POTE): ~385/384 = 3.9850

Optimal GPV sequence: 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 is not only a weak extension, but lacks a clear 13-limit extension of its own. 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.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 15625/15552, 32805/32768

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

Mapping generators: ~2835/2816, ~7

Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232

Optimal GPV sequence: 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.

Subgroup: 2.3.5.7.11

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

Optimal tuning (POTE): ~176/175 = 8.8283

Template:Val list

Badness: 0.063254

13-limit

Subgroup: 2.3.5.7.11.13

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

Optimal tuning (POTE): ~176/175 = 8.1254

Optimal GPV sequence: Template:Val list

Badness: 0.036988

Iodine

Iodine is named after the 53rd chemical element and can be expressed as the 159 & 742 temperament.

Subgroup: 2.3.5.7

Comma list: [-19 14 -5 3⟩, [8 3 -20 12⟩

Mapping: [⟨53 84 2 -53], ⟨0 0 3 5]]

Mapping generators: ~3125/3087, 6075/3584

Optimal tuning (CTE): ~6075/3584 = 913.7347

Template:Val list

Badness: 0.477

11-limit

Subgroup: 2.3.5.7.11

Comma list: 160083/160000, 820125/819896, 4302592/4296875

Mapping: [⟨53 84 2 -53 143], ⟨0 0 3 5 1]]

Optimal tuning (CTE): ~6075/3584 = 913.7322

Optimal GPV sequence: Template:Val list

Badness: 0.0875

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488

Mapping: [⟨53 84 2 -53 143 -46], ⟨0 0 3 5 1 6]]

Optimal tuning (CTE): ~441/260 = 913.7115

Optimal GPV sequence: Template:Val list

Badness: 0.0476

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872

Mapping: [⟨53 84 2 -53 143 -46 257], ⟨0 0 3 5 1 6 -1]]

Optimal tuning (CTE): ~441/260 = 913.7131

Optimal GPV sequence: Template:Val list

Badness: 0.0328

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