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Mercator family: Difference between revisions

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[[Category:Fractional-octave temperaments]]
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]


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]]. 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.  
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[[Comma list]]: {{monzo| -84 53 }}
[[Comma list]]: {{monzo| -84 53 }}


[[Mapping]]: [{{val| 53 84 123 }}, {{val| 0 0 1 }}]
[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]


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


[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.264
[[Optimal tuning]]s:
* [[CTE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.3137¢
* [[CWE]]: ~531441/524288 = 22.6415¢ (1 ⧵ 53), ~5/4 = 386.2804¢


{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}
{{Optimal ET sequence|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }}


[[Badness]]: 0.284323
[[Badness]] (Sintel): 6.670


== Schismerc ==
== Schismerc ==
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[[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


[[Optimal tuning]] ([[POTE]]): ~225/224 = 5.3666
[[Optimal tuning]]s:
* [[CTE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.1741¢
* [[CWE]]: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 231.6299¢


{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}


[[Badness]]: 0.087022
[[Badness]] (Sintel): 2.202


=== Cartography ===
=== Cartography ===
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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 }}]


Mapping generators: ~81/80, ~7/1
Optimal tunings:
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5178¢


Optimal tuning (POTE): ~225/224 = 6.1204
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371d, 583cde }}


{{Optimal ET sequence|legend=1| 53, 106d, 159, 212, 371d, 583cde }}
Badness (Sintel): 1.800
 
Badness: 0.054452


==== 13-limit ====
==== 13-limit ====
Line 75: Line 78:
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 }}


Mapping generators: ~81/80, ~7/1
Optimal tunings:
 
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.4299¢
Optimal tuning (POTE): ~225/224 = 6.1430
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢


{{Optimal ET sequence|legend=1| 53, 106d, 159, 212, 371df, 583cdeff }}
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


Badness: 0.029980
Badness (Sintel): 1.239


=== 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
Subgroup: 2.3.5.7.11
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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 }}]


Mapping generators: ~81/80, ~7/1
Optimal tunings:
 
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5956¢
Optimal tuning (POTE): ~385/384 = 4.1494
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5697¢


{{Optimal ET sequence|legend=1| 53, 159e, 212e, 265, 318, 583c }}
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}


Badness: 0.115066
Badness (Sintel): 3.804


==== 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 minisma 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
Subgroup: 2.3.5.7.11.13
Line 109: Line 112:
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 }}


Mapping generators: ~81/80, ~7/1
Optimal tuning (POTE): ~385/384 = 3.9850


Optimal tuning (POTE): ~385/384 = 3.9850
Optimal tunings:
* CTE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4057¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.4008¢


{{Optimal ET sequence|legend=1| 53, 159ef, 212ef, 265, 318, 583cf }}
{{Optimal ET sequence|legend=0| 53, 159ef, 212ef, 265, 318, 583cf }}


Badness: 0.061158
Badness (Sintel): 2.527


=== Boiler ===
=== Boiler ===
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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 }}]


Mapping generators: ~2835/2816, ~7
: mapping generators: ~2835/2816, ~7


Optimal tuning (POTE): ~225/224 = 6.3976 or ~441/440 = 4.9232
Optimal tunings:
* CTE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 230.6341¢
* CWE: ~2835/2816 = 11.3208¢ (1 ⧵ 106), ~8/7 = 231.1634¢


{{Optimal ET sequence|legend=1| 106, 212 }}
{{Optimal ET sequence|legend=0| 106, 212 }}


Badness: 0.109648
Badness (Sintel): 3.625


== Joliet ==
== Joliet ==
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[[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: ~50/49, ~11


[[Optimal tuning]] ([[POTE]]): ~176/175 = 8.8283
[[Optimal tuning]]s:
* [[CTE]]: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
* [[CWE]]: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 552.0415¢


{{Optimal ET sequence|legend=1| 53, 106, 159d }}
{{Optimal ET sequence|legend=1| 53, 106, 159d }}


[[Badness]]: 0.063254
[[Badness]] (Sintel): 2.091


=== 13-limit ===
=== 13-limit ===
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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 }}]


Mapping generators: ~81/80, ~11/1
Optimal tunings:
 
* CTE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.3179¢
Optimal tuning (POTE): ~176/175 = 8.1254
* CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.4859¢


{{Optimal ET sequence|legend=1| 53, 106, 159d }}
{{Optimal ET sequence|legend=0| 53, 106, 159d }}


Badness: 0.036988
Badness (Sintel): 1.528


== Iodine ==
== Iodine ==
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[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]
[[Mapping]]: [{{val| 53 84 2 -53 }}, {{val| 0 0 3 5 }}]


Mapping generators: ~3125/3087, 6075/3584
: mapping generators: ~3125/3087, ~6075/3584


[[Optimal tuning]] ([[CTE]]): ~6075/3584 = 913.7347
[[Optimal tuning]]s:
* [[CTE]]: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7347¢
* [[CWE]]: ~3125/3087 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7301¢


{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}


[[Badness]]: 0.477
[[Badness]] (Sintel): 12.075


=== 11-limit ===
=== 11-limit ===
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Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]
Mapping: [{{val| 53 84 2 -53 143 }}, {{val| 0 0 3 5 1 }}]


Optimal tuning (CTE): ~6075/3584 = 913.7322
Optimal tunings:
* CTE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7322¢
* CWE: ~1815/1792 = 22.6415¢ (1 ⧵ 53), ~6075/3584 = 913.7345¢


{{Optimal ET sequence|legend=1| 159, 424cd, 583, 742, 2385d, 3127d }}
{{Optimal ET sequence|legend=0| 159, 424cd, 583, 742, 2385d, 3127d }}


Badness: 0.0875
Badness (Sintel): 2.893


=== 13-limit ===
=== 13-limit ===
Line 205: Line 218:
Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]
Mapping: [{{val| 53 84 2 -53 143 -46 }}, {{val| 0 0 3 5 1 6 }}]


Optimal tuning (CTE): ~441/260 = 913.7115
Optimal tunings:
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7115¢
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7126¢


{{Optimal ET sequence|legend=1| 159, 424cdff, 583f, 742, 1643 }}
{{Optimal ET sequence|legend=0| 159, 424cdff, 583f, 742, 1643 }}


Badness: 0.0476
Badness (Sintel): 1.967


=== 17-limit ===
=== 17-limit ===
Line 218: Line 233:
Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]
Mapping: [{{val| 53 84 2 -53 143 -46 257 }}, {{val| 0 0 3 5 1 6 -1 }}]


Optimal tuning (CTE): ~441/260 = 913.7131
Optimal tunings:
* CTE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7131¢
* CWE: ~78/77 = 22.6415¢ (1 ⧵ 53), ~441/260 = 913.7208¢


{{Optimal ET sequence|legend=1| 159, 583f, 742 }}
{{Optimal ET sequence|legend=0| 159, 583f, 742 }}


Badness: 0.0328
Badness (Sintel): 1.568


{{Navbox fractional-octave|53}}
{{Navbox fractional-octave|53}}


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Mercator family]] <!-- main article -->
[[Category:Mercator family]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mercator_family"