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Marvel temperaments: Difference between revisions

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* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
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* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]


Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].  
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].  


Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
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Badness (Sintel): 0.895
Badness (Sintel): 0.895
== Decic ==
{{Main| Decic }}
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 225/224, 16807/16384
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
[[Badness]] (Sintel): 2.26
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 3087/3025
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
Badness (Sintel): 2.11
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 196/195, 2200/2197
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
Optimal tunings:
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
Badness (Sintel): 1.52
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
Optimal tunings:
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})
{{Optimal ET sequence|legend=0| 10, 40, 50 }}
Badness (Sintel): 1.28
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.
Subgroup: 2.3.5.7.11
Comma list: 225/224, 1617/1600, 2401/2376
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
Optimal tunings:
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}
Badness (Sintel): 1.98
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 1001/1000, 1188/1183
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
Badness (Sintel): 1.57
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
Optimal tunings:
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}
Badness (Sintel): 1.33
=== Prodecic ===
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 5929/5832
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
Optimal tunings:
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Badness (Sintel): 2.20
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 275/273, 5929/5832
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
Optimal tunings:
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Badness (Sintel): 1.73
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
Optimal tunings:
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
Badness (Sintel): 1.41


== Enneaportent ==
== Enneaportent ==
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"