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Marvel temperaments: Difference between revisions

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* ''[[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, decic, 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 39: 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 210: 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 225: 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 240: 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 255: 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 270: 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 285: 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 313: 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 322: 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 328: 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 337: 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 343: 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 376: 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 391: 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 406: 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 421: 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 458: 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 473: 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 510: 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 543: 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 558: 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 573: 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 588: 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 627: Line 484:
Comma list: 225/224, 385/384, 3087/3025
Comma list: 225/224, 385/384, 3087/3025


Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}


Optimal tunings:  
Optimal tunings:  
Line 642: Line 499:
Comma list: 105/104, 144/143, 196/195, 2200/2197
Comma list: 105/104, 144/143, 196/195, 2200/2197


Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Optimal tunings:  
Optimal tunings:  
Line 657: Line 514:
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
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 }}
{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 674: Line 531:
Comma list: 225/224, 1617/1600, 2401/2376
Comma list: 225/224, 1617/1600, 2401/2376


Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 689: Line 546:
Comma list: 105/104, 196/195, 1001/1000, 1188/1183
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Optimal tunings:  
Optimal tunings:  
Line 704: Line 561:
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
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 }}
{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 721: Line 578:
Comma list: 225/224, 441/440, 5929/5832
Comma list: 225/224, 441/440, 5929/5832


Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Optimal tunings:  
Optimal tunings:  
Line 736: Line 593:
Comma list: 105/104, 196/195, 275/273, 5929/5832
Comma list: 105/104, 196/195, 275/273, 5929/5832


Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Optimal tunings:  
Optimal tunings:  
Line 751: Line 608:
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
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 }}
{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 784: 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 799: 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 822: 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 834: 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 849: 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 864: 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 879: 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 894: 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 909: 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 924: 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 939: 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 954: 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 969: 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 984: 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 1,017: 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 1,052: 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 1,067: 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 1,082: 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 1,119: 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 1,134: 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 1,234: 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 1,249: 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 1,264: 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 1,273: 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 1,301: 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 1,338: 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 1,350: 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"