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Mercator family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
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{{Mapping|legend=1| 53 84 123 149 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 53 84 123 149 0 | 0 0 0 0 1 }}
: mapping generators: ~50/49, ~11
: mapping generators: ~81/80, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}
* [[WE]]: ~81/80 = 22.6365{{c}}, ~11/8 = 551.1028{{c}} (~176/175 = 8.8263{{c}})
* [[CWE]]: ~50/49 = 22.6415{{c}}, ~11/8 = 552.0415{{c}}
: [[error map]]: {{val| -0.264 -0.487 -2.022 +4.015 -0.009 }}
* [[CWE]]: ~81/80 = 22.6415{{c}}, ~11/8 = 552.0415{{c}} (~176/175 = 8.6452{{c}})
: error map: {{val| 0.000 -0.068 -1.408 +4.759 +0.724 }}


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


[[Badness]] (Sintel): 2.091
[[Badness]] (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~50/49 = 22.6415{{c}}, ~11/8 = 551.3179{{c}}
* WE: ~81/80 = 22.6403{{c}}, ~11/8 = 551.4931{{c}} (~176/175 = 8.1250{{c}})
* CWE: ~50/49 = 22.6415{{c}}, ~11/8 = 551.4859{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~11/8 = 551.4859{{c}} (~176/175 = 8.0897{{c}})


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


Badness (Sintel): 1.528
Badness (Sintel): 1.53


== 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. 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]].  
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. 3 generators plus 2 periods reach the [[5/1|5th harmonic]]. 5 generators reach the [[14/1|14th harmonic]].
 
In the 11-limit, the [[~]][[11/8]] can serve as a generator, being 16 periods less than the canonical generator.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7347{{c}}
* [[WE]]: ~3125/3087 = 22.6417{{c}}, ~6075/3584 = 913.7361{{c}} (~225/224 = 8.0675{{c}})
* [[CWE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7301{{c}}
: [[error map]]: {{val| +0.011 -0.051 +0.178 -0.156 }}
* [[CWE]]: ~3125/3087 = 22.6415{{c}}, ~6075/3584 = 913.7301{{c}} (~225/224 = 8.0697{{c}})
: error map: {{val| 0.000 -0.068 +0.159 -0.176 }}


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


[[Badness]] (Sintel): 12.075
[[Badness]] (Sintel): 12.1


=== 11-limit ===
=== 11-limit ===
24 periods plus the reduced generator correspond to [[11/8]].
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Optimal tunings:  
Optimal tunings:  
* CTE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7322{{c}}
* WE: ~1815/1792 = 22.6414{{c}}, ~6075/3584 = 913.7318{{c}} (~225/224 = 8.0750{{c}})
* CWE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7345{{c}}
* CWE: ~1815/1792 = 22.6415{{c}}, ~6075/3584 = 913.7345{{c}} (~225/224 = 8.0742{{c}})


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


Badness (Sintel): 2.893
Badness (Sintel): 2.89


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7115{{c}}
* WE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7113{{c}} (~225/224 = 8.0526{{c}})
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7126{{c}}
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7126{{c}} (~225/224 = 8.0522{{c}})


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


Badness (Sintel): 1.967
Badness (Sintel): 1.97


=== 17-limit ===
=== 17-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7131{{c}}
* WE: ~78/77 = 22.6412{{c}}, ~441/260 = 913.7109{{c}} (~225/224 = 8.0623{{c}})
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7208{{c}}
* CWE: ~78/77 = 22.6415{{c}}, ~441/260 = 913.7208{{c}} (~225/224 = 8.0605{{c}})


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


Badness (Sintel): 1.568
Badness (Sintel): 1.57


== Aemilic ==
== Aemilic ==
''Aemilic'' is described as the 159 & 954 temperament in the 17-limit and is named after the minor planet {{w|159 Aemilia}}.
''Aemilic'' tempers out the [[landscape comma]] alongisde the mercator comma. It may be described as the 159 & 954 temperament. It was named after the minor planet {{w|159 Aemilia}}.  
 
In the 7-limit, aemilic tempers out the [[landscape comma]] alongisde the [[Mercator comma]], and also {{monzo| -76 41 12 -6 }}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~225/224, ~5
: mapping generators: ~225/224, ~5


[[Optimal tuning]] ([[CTE]]): ~225/224 = 7.547{{c}}, ~5/4 = 386.303{{c}}
[[Optimal tuning]]s
* [[WE]]: ~225/224 = 7.5473{{c}}, ~5/4 = 386.2754{{c}} (~32805/32768 = 1.3629{{c}})
* [[CWE]]: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2798{{c}} (~32805/32768 = 1.3741{{c}})


[[Support]]ing [[ET]]s: {{EDOs| 159, 795, 954, 1749, 2544, 2703 }}
{{Optimal ET sequence|legend=1| 159, …, 795, 954, 1749, 6201, 7950, 9699bd, 11448bd }}
 
Badness (Sintel): 4.86


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 159 252 0 -292 550 | 0 0 1 2 0 }}
Mapping: {{mapping| 159 252 0 -292 550 | 0 0 1 2 0 }}


Optimal tuning (CTE): ~225/224 = 7.547{{c}}, ~5/4 = 386.303{{c}}
Optimal tunings:
* WE: ~225/224 = 7.5475{{c}}, ~5/4 = 386.2382{{c}} (~32805/32768 = 1.3168{{c}})
* CWE: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2421{{c}} (~32805/32768 = 1.3365{{c}})
 
{{Optimal ET sequence|legend=0| 159, …, 795, 954, 1749e, 2703e, 3657ee }}


Supporting ETs: {{EDOs| 159, 795, 954 }}
Badness (Sintel): 2.79


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}
Mapping: {{mapping| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}


Optimal tuning (CTE): ~225/224 = 7.547{{c}}, ~5/4 = 386.303{{c}}
Optimal tunings:
* WE: ~225/224 = 7.5475{{c}}, ~5/4 = 386.2350{{c}} (~32805/32768 = 1.3136{{c}})
* CWE: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2395{{c}} (~32805/32768 = 1.3338{{c}})


Supporting ETs: {{EDOs| 159, 795, 954 }}
{{Optimal ET sequence|legend=0| 159, …, 795, 954, 2703e, 3657eef }}
 
Badness (Sintel): 1.42


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}
Mapping: {{mapping| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}


Optimal tuning (CTE): ~225/224 = 7.547{{c}}, ~5/4 = 386.263{{c}}
Optimal tunings:
* WE: ~225/224 = 7.5476{{c}}, ~5/4 = 386.1835{{c}} (~32805/32768 = 1.2536{{c}})
* CWE: ~225/224 = 7.5472{{c}}, ~5/4 = 386.1765{{c}} (~32805/32768 = 1.2708{{c}})
 
{{Optimal ET sequence|legend=0| 159, 795g, 954 }}


Supporting ETs: {{EDOs| 159, 795g, 954 }}
Badness (Sintel): 1.64


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