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Mercator family: Difference between revisions

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{{Technical data page}}
<div class="toccolours" style="float: right">
<center>'''[[Fractional-octave temperaments]]'''</center>
----
<small>← [[{{Ordinal|{{#expr:53-1}}}}-octave temperaments]]</small> 53rd-octave temperaments <small>[[{{Ordinal|{{#expr:53+1}}}}-octave temperaments]] →</small>
</div>
[[Category:53edo]]
[[Category:Fractional-octave temperaments]]
[[Category:Temperament collections]]
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.  
Discussed elsewhere are:
* ''[[Aemilic]]'' (+250047/250000) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]


== Mercator ==
== Mercator ==
Line 6: Line 21:
[[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


{{Multival|legend=1| 0 53 84 }}
[[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 tuning]] ([[POTE]]): ~5/4 = 386.264
{{Optimal ET sequence|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]] (Sintel): 6.670
 
[[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.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 27: Line 42:
[[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


{{Multival|legend=1| 0 0 53 0 84 123 }}
[[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 tuning]] ([[POTE]]): ~225/224 = 5.3666
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}


{{Val list|legend=1| 53, 159, 212, 689c, 901cc }}
[[Badness]] (Sintel): 2.202
 
[[Badness]]: 0.087022


=== Cartography ===
=== Cartography ===
Line 46: Line 61:
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 GPV sequence: {{Val list| 53, 106d, 159, 212, 371d, 583cde }}
Badness (Sintel): 1.800
 
Badness: 0.054452


==== 13-limit ====
==== 13-limit ====
Line 63: 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¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 232.5397¢


Optimal tuning (POTE): ~225/224 = 6.1430
{{Optimal ET sequence|legend=0| 53, 106d, 159, 212, 371df, 583cdeff }}


Optimal GPV sequence: {{Val list| 53, 106d, 159, 212, 371df, 583cdeff }}
Badness (Sintel): 1.239
 
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]]. 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
Line 80: Line 95:
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¢
* CWE: ~81/80 = 22.6415¢ (1 ⧵ 53), ~8/7 = 230.5697¢


Optimal tuning (POTE): ~385/384 = 4.1494
{{Optimal ET sequence|legend=0| 53, 159e, 212e, 265, 318, 583c }}


Optimal GPV sequence: {{Val list| 53, 159e, 212e, 265, 318, 583c }}
Badness (Sintel): 3.804
 
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 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 97: 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 GPV sequence: {{Val list| 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 ===
Line 114: Line 131:
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 GPV sequence: {{Val list| 106, 212 }}
{{Optimal ET sequence|legend=0| 106, 212 }}


Badness: 0.109648
Badness (Sintel): 3.625


== Joliet ==
== Joliet ==
Line 131: Line 150:
[[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¢


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


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


=== 13-limit ===
=== 13-limit ===
Line 146: Line 167:
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¢
* CWE: ~50/49 = 22.6415¢ (1 ⧵ 53), ~11/8 = 551.4859¢


Optimal tuning (POTE): ~176/175 = 8.1254
{{Optimal ET sequence|legend=0| 53, 106, 159d }}


Optimal GPV sequence: {{Val list| 53, 106, 159d }}
Badness (Sintel): 1.528
 
Badness: 0.036988


== Iodine ==
== Iodine ==
Proposed by Eliora, the name of ''iodine'' is taken from the convention of naming some fractional-octave temperaments after elements, in this case the 53rd chemical element. It can be expressed as the 159 & 742 temperament.  
Proposed by Eliora, the name of ''iodine'' is taken from the convention of naming some fractional-octave temperaments after elements, in this case the 53rd chemical element. It can be expressed as the 159 & 742 temperament. 2 periods + 3 less than 600 cent generators correspond to [[8/5]]. 5 less than 600 cent generators (minus 1 octave) correspond to [[8/7]].  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 163: Line 184:
[[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¢


{{Val list|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 ===
Line 180: Line 203:
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 GPV sequence: {{Val list| 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 193: 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 GPV sequence: {{Val list| 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 206: 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=0| 159, 583f, 742 }}


Optimal GPV sequence: {{Val list| 159, 583f, 742 }}
Badness (Sintel): 1.568


Badness: 0.0328
{{Navbox fractional-octave|53}}


[[Category:Temperament families]]
[[Category:Temperament families]]
[[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"