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Marvel temperaments: Difference between revisions

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* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
Line 25: Line 24:
* ''[[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]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton 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 tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, 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.
Line 41: Line 39:


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 225/224, 118098/117649
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
[[Badness]] (Sintel): 1.03
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 4000/3993
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
Badness (Sintel): 0.613
==== Lizard ====
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 351/350, 364/363, 385/384
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
Badness (Sintel): 0.900
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
Badness (Sintel): 0.741
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
Badness (Sintel): 0.955
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 1573/1568
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
Badness (Sintel): 0.837
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
Badness (Sintel): 0.694
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
Badness (Sintel): 0.901
=== Mage ===
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 1331/1296
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
Badness (Sintel): 1.91


== Tritonic ==
== Tritonic ==
Line 212: Line 67:
Comma list: 121/120, 225/224, 441/440
Comma list: 121/120, 225/224, 441/440


Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 227: Line 82:
Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Optimal tunings:  
Optimal tunings:  
Line 242: Line 97:
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 257: Line 112:
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 272: Line 127:
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Optimal tunings:  
Optimal tunings:  
Line 287: Line 142:
Comma list: 225/224, 385/384, 27783/27500
Comma list: 225/224, 385/384, 27783/27500


Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Optimal tunings:  
Optimal tunings:  
Line 315: Line 170:
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


{{Optimal ET sequence|legend=1| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


[[Badness]] (Sintel): 1.38
[[Badness]] (Sintel): 1.38
Line 324: Line 179:
Comma list: 225/224, 245/242, 385/384
Comma list: 225/224, 245/242, 385/384


Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Optimal tunings:  
Optimal tunings:  
Line 330: Line 185:
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Badness (Sintel): 1.04
Badness (Sintel): 1.04
Line 339: Line 194:
Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Optimal tunings:  
Optimal tunings:  
Line 345: Line 200:
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


{{Optimal ET sequence|legend=0| 41, 91 }}
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


Badness (Sintel): 0.955
Badness (Sintel): 0.955
Line 378: Line 233:
Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 441/440, 1344/1331


Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 393: Line 248:
Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 144/143, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Optimal tunings:  
Optimal tunings:  
Line 408: Line 263:
Comma list: 225/224, 385/384, 532400/531441
Comma list: 225/224, 385/384, 532400/531441


Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Optimal tunings:  
Optimal tunings:  
Line 423: Line 278:
Comma list: 225/224, 325/324, 385/384, 10648/10647
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Optimal tunings:  
Optimal tunings:  
Line 460: Line 315:
Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 385/384, 1331/1323


Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 475: Line 330:
Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Optimal tunings:  
Optimal tunings:  
Line 512: Line 367:
Comma list: 45/44, 56/55, 1029/1000
Comma list: 45/44, 56/55, 1029/1000


Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 545: Line 400:
Comma list: 225/224, 441/440, 4000/3993
Comma list: 225/224, 441/440, 4000/3993


Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}


Optimal tunings:  
Optimal tunings:  
Line 560: Line 415:
Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 169/168, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Optimal tunings:  
Optimal tunings:  
Line 575: Line 430:
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Optimal tunings:  
Optimal tunings:  
Line 590: Line 445:
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Optimal tunings:  
Optimal tunings:  
Line 599: Line 454:


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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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 ==
Line 623: Line 641:
Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 385/384, 12005/11979


Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}


Optimal tunings:  
Optimal tunings:  
Line 638: Line 656:
Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Optimal tunings:  
Optimal tunings:  
Line 661: Line 679:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


Line 673: Line 691:
Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Optimal tunings:  
Optimal tunings:  
Line 688: Line 706:
Comma list: 225/224, 325/324, 385/384, 831875/830466
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Optimal tunings:  
Optimal tunings:  
Line 703: Line 721:
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Optimal tunings:  
Optimal tunings:  
Line 718: Line 736:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 733: Line 751:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Optimal tunings:  
Optimal tunings:  
Line 748: Line 766:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Optimal tunings:  
Optimal tunings:  
Line 763: Line 781:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Optimal tunings:  
Optimal tunings:  
Line 778: Line 796:
Comma list: 225/224, 441/440, 109375/107811
Comma list: 225/224, 441/440, 109375/107811


Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Optimal tunings:  
Optimal tunings:  
Line 793: Line 811:
Comma list: 225/224, 364/363, 441/440, 6125/6084
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Optimal tunings:  
Optimal tunings:  
Line 808: Line 826:
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Optimal tunings:  
Optimal tunings:  
Line 823: Line 841:
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 856: Line 874:
Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 385/384, 12250/11979


Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 891: Line 909:
Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Optimal tunings:  
Optimal tunings:  
Line 906: Line 924:
Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Optimal tunings:  
Optimal tunings:  
Line 921: Line 939:
Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 385/384, 66550/64827


Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Optimal tunings:  
Optimal tunings:  
Line 958: Line 976:
Comma list: 121/120, 225/224, 22000/21609
Comma list: 121/120, 225/224, 22000/21609


Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 973: Line 991:
Comma list: 121/120, 225/224, 275/273, 1040/1029
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Optimal tunings:  
Optimal tunings:  
Line 1,073: Line 1,091:
Comma list: 225/224, 441/440, 43923/43750
Comma list: 225/224, 441/440, 43923/43750


Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Optimal tunings:  
Optimal tunings:  
Line 1,088: Line 1,106:
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Optimal tunings:  
Optimal tunings:  
Line 1,103: Line 1,121:
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 1,112: Line 1,130:


Badness (Sintel): 1.17
Badness (Sintel): 1.17
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 225/224, 122880/117649
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
[[Badness]] (Sintel): 4.37
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 352/343
{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}
{{Optimal ET sequence|legend=0| 22, 55d }}
Badness (Sintel): 2.46


== Gwazy ==
== Gwazy ==
Line 1,140: Line 1,196:
Comma list: 99/98, 176/175, 65536/65219
Comma list: 99/98, 176/175, 65536/65219


Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 1,177: Line 1,233:
Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 1,189: Line 1,245:
== References ==
== References ==


[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"