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Mercator family: Difference between revisions

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The '''mercator family''' of [[regular temperament|temperaments]] [[tempering out|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''' of [[regular temperament|temperaments]] [[tempering out|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 16: Line 13:
[[Comma list]]: {{monzo| -84 53 }}
[[Comma list]]: {{monzo| -84 53 }}


[[Mapping]]: [{{val| 53 84 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 53 84 0 | 0 0 1 }}
: mapping generators: ~531441/524288, ~5
: mapping generators: ~531441/524288, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.3137{{c}}
* [[WE]]: ~531441/524288 = 22.6419{{c}}, ~5/4 = 386.2707{{c}} (~32805/32768 = 1.3581{{c}})
* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}}
: [[error map]]: {{val| +0.022 -0.034 -0.000 }}
* [[CWE]]: ~531441/524288 = 22.6415{{c}}, ~5/4 = 386.2804{{c}} (~32805/32768 = 1.3747{{c}})
: error map: {{val| 0.000 -0.068 -0.033 }}


{{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]] (Sintel): 6.670
[[Badness]] (Sintel): 6.67


== 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 11-limit extensions are cartography, pentacontatritonic, boiler, and auric.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 38: Line 37:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~81/80 = 22.6415{{c}}, ~8/7 = 231.1741{{c}}
* [[WE]]: ~81/80 = 22.6456{{c}}, ~7/4 = 968.3928{{c}} (~225/224 = 5.3675{{c}})
* [[CWE]]: ~81/80 = 22.641{{c}}, ~8/7 = 231.6299{{c}}
: [[error map]]: {{val| +0.216 +0.275 -0.906 -0.001 }}
* [[CWE]]: ~81/80 = 22.6415{{c}}, ~7/4 = 968.3701{{c}} (~225/224 = 5.2148{{c}})
: error map: {{val| 0.000 -0.068 -1.408 -0.456 }}


{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc }}
{{Optimal ET sequence|legend=1| 53, 159, 212, 689c, 901cc, 1113ccd }}


[[Badness]] (Sintel): 2.202
[[Badness]] (Sintel): 2.20


=== 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.
 
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
Subgroup: 2.3.5.7.11
Line 52: Line 55:
Comma list: 385/384, 6250/6237, 19712/19683
Comma list: 385/384, 6250/6237, 19712/19683


Mapping: {{mapping| 53 84 123 0 332 | 0 0 0 1 -1 }}
{{Mapping|legend=0| 53 84 123 0 332 | 0 0 0 1 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.641{{c}}, ~8/7 = 232.4299{{c}}
* WE: ~81/80 = 22.6478{{c}}, ~7/4 = 967.7312{{c}} (~225/224 = 6.1221{{c}})
* CWE: ~81/80 = 22.641{{c}}, ~8/7 = 232.5178{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 967.4822{{c}} (~225/224 = 6.1027{{c}})


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


Badness (Sintel): 1.800
Badness (Sintel): 1.80


==== 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.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 625/624, 19712/19683
Comma list: 325/324, 385/384, 625/624, 19712/19683


Mapping: {{mapping| 53 84 123 0 332 196 | 0 0 0 1 -1 0 }}
{{Mapping|legend=0| 53 84 123 0 332 196 | 0 0 0 1 -1 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.4299{{c}}
* WE: ~81/80 = 22.6491{{c}}, ~7/4 = 967.7656{{c}} (~225/224 = 6.1450{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 232.5397{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 967.4603{{c}} (~225/224 = 6.1246{{c}})


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


Badness (Sintel): 1.239
Badness (Sintel): 1.24


=== 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 [[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.
 
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
Subgroup: 2.3.5.7.11
Line 86: Line 89:
Comma list: 540/539, 15625/15552, 32805/32768
Comma list: 540/539, 15625/15552, 32805/32768


Mapping: {{mapping| 53 84 123 0 481 | 0 0 0 1 -2 }}
{{Mapping|legend=0| 53 84 123 0 481 | 0 0 0 1 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5956{{c}}
* WE: ~81/80 = 22.6427{{c}}, ~7/4 = 969.4874{{c}} (~385/384 = 4.1496{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.5697{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 969.4303{{c}} (~385/384 = 4.1546{{c}})


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


Badness (Sintel): 3.804
Badness (Sintel): 3.80


==== 13-limit ====
==== 13-limit ====
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


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


Mapping: {{mapping| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}
{{Mapping|legend=0| 53 84 123 0 481 345 | 0 0 0 1 -2 1 }}
 
Optimal tuning (POTE): ~385/384 = 3.9850


Optimal tunings:  
Optimal tunings:  
* CTE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4057{{c}}
* WE: ~81/80 = 22.6416{{c}}, ~7/4 = 969.6046{{c}} (~385/384 = 3.9850{{c}})
* CWE: ~81/80 = 22.6415{{c}}, ~8/7 = 230.4008{{c}}
* CWE: ~81/80 = 22.6415{{c}}, ~7/4 = 969.5992{{c}} (~385/384 = 3.9858{{c}})


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


Badness (Sintel): 2.527
Badness (Sintel): 2.53


=== 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 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.
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
Subgroup: 2.3.5.7.11
Line 122: Line 121:
Comma list: 9801/9800, 15625/15552, 32805/32768
Comma list: 9801/9800, 15625/15552, 32805/32768


Mapping: {{mapping| 106 168 246 0 69 | 0 0 0 1 1 }}
{{Mapping|legend=0| 106 168 246 0 69 | 0 0 0 1 1 }}
: mapping generators: ~2835/2816, ~7
: mapping generators: ~2835/2816, ~7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 230.6341{{c}}
* WE: ~2835/2816 = 11.3229{{c}}, ~7/4 = 968.8469{{c}} (~225/224 = 4.9241{{c}})
* CWE: ~2835/2816 = 11.3208{{c}}, ~8/7 = 231.1634{{c}}
* CWE: ~2835/2816 = 11.3208{{c}}, ~7/4 = 968.8633{{c}} (~225/224 = 4.7483{{c}})
 
{{Optimal ET sequence|legend=0| 106, 212, 530c, 742ce }}
 
Badness (Sintel): 3.62
 
=== Auric ===
Auric, named after [[Aura]], is defined by setting the 2.3.5.11 mapping to that of 159edo and the generator for the prime 7 being independent.
 
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 15625/15552, 32805/32768
 
{{Mapping|legend=0| 159 252 369 0 550 | 0 0 0 1 0 }}
: mappping generators: ~243/242, ~7
 
Optimal tunings:
* WE: ~243/242 = 7.5484{{c}}, ~7/4 = 968.4469{{c}} (~3024/3025 = 2.2569{{c}})
* CWE: ~243/242 = 7.5472{{c}}, ~7/4 = 968.4005{{c}} (~3024/3025 = 2.3627{{c}})


{{Optimal ET sequence|legend=0| 106, 212 }}
{{Optimal ET sequence|legend=0| 159, 318, 477c }}


Badness (Sintel): 3.625
Badness (Sintel): 5.23


== 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
[[Subgroup]]: 2.3.5.7.11
Line 141: Line 158:


{{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 = 22.641{{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 ===
Line 156: Line 175:
Comma list: 169/168, 225/224, 325/324, 640/637
Comma list: 169/168, 225/224, 325/324, 640/637


Mapping: {{mapping| 53 84 123 149 0 196 | 0 0 0 0 1 0 }}
{{Mapping|legend=0| 53 84 123 149 0 196 | 0 0 0 0 1 0 }}


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
Line 177: Line 198:


[[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


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


Mapping: {{mapping| 53 84 2 -53 143 | 0 0 3 5 1 }}
{{Mapping|legend=0| 53 84 2 -53 143 | 0 0 3 5 1 }}


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 ===
Line 206: Line 227:
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488
Comma list: 6656/6655, 34398/34375, 43904/43875, 59535/59488


Mapping: {{mapping| 53 84 2 -53 143 -46 | 0 0 3 5 1 6 }}
{{Mapping|legend=0| 53 84 2 -53 143 -46 | 0 0 3 5 1 6 }}


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 ===
Line 221: Line 242:
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872
Comma list: 1701/1700, 6656/6655, 8624/8619, 12376/12375, 14875/14872


Mapping: {{mapping| 53 84 2 -53 143 -46 257 | 0 0 3 5 1 6 -1 }}
{{Mapping|legend=0| 53 84 2 -53 143 -46 257 | 0 0 3 5 1 6 -1 }}


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'' 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}}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| -84 53 }}
 
{{Mapping|legend=1| 159 252 0 -292 | 0 0 1 2 }}
: mapping generators: ~225/224, ~5
 
[[Optimal tuning]]s
* [[WE]]: ~225/224 = 7.5473{{c}}, ~5/4 = 386.2754{{c}} (~32805/32768 = 1.3629{{c}})
: [[error map]]: {{val| +0.021 -0.035 +0.004 -0.003 }}
* [[CWE]]: ~225/224 = 7.5472{{c}}, ~5/4 = 386.2798{{c}} (~32805/32768 = 1.3741{{c}})
: error map: {{val| 0.000 -0.068 -0.034 -0.040 }}
 
{{Optimal ET sequence|legend=1| 159, …, 795, 954, 1749, 6201, 7950, 9699bd, 11448bd }}
 
[[Badness]] (Sintel): 4.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 102487/102400, {{monzo| 34 19 8 4 -1 }}
 
{{Mapping|legend=0| 159 252 0 -292 550 | 0 0 1 2 0 }}
 
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 }}
 
Badness (Sintel): 2.79
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 102487/102400, 123201/123200, 4100625/4100096
 
{{Mapping|legend=0| 159 252 0 -292 550 -150 | 0 0 1 2 0 2 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 159, …, 795, 954, 2703e, 3657eef }}
 
Badness (Sintel): 1.42
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2431/2430, 3025/3024, 57375/57344, 102487/102400, 123201/123200
 
{{Mapping|legend=0| 159 252 0 -292 550 -150 1019 | 0 0 1 2 0 2 -1 }}
 
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 }}
 
Badness (Sintel): 1.64


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