Marvel temperaments: Difference between revisions
- CTE & POTE tunings |
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Temperaments considered in families and clans are: | Temperaments considered in families and clans are: | ||
* ''[[Pelogic]]'' → [[Mavila family #Pelogic|Mavila family]] | * ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]] | ||
* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] | * [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]] | ||
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] | * [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]] | ||
* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] | * [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]] | ||
* ''[[Sharpie]]'' → [[Dicot family #Sharpie|Dicot family]] | * ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]] | ||
* ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] | * ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]] | ||
* ''[[August]]'' → [[Augmented family #August|Augmented family]] | * ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]] | ||
* ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+ | * ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]] | ||
* [[Negri]] → [[ | * [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]] | ||
* [[Magic]] → [[Magic family #Magic|Magic family]] | * [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]] | ||
* ''[[Passive]]'' → [[Passion family #Passive|Passion family]] | * [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]] | ||
* ''[[ | * ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]] | ||
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]] | |||
* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] | * ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]] | ||
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] | * [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]] | ||
* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] | * [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]] | ||
* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] | * ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]] | ||
* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] | * [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]] | ||
* ''[[Snipes]]'' → [[Wesley family #Snipes|Wesley family]] | * ''[[Snipes]]'' (+6125/5832) → [[Wesley family #Snipes|Wesley family]] | ||
* ''[[ | * ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]] | ||
* ''[[Escapist]]'' → [[Escapade family #Escapist|Escapade family]] | * ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]] | ||
* ''[[ | * ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]] | ||
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]] | |||
* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] | * [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]] | ||
* | * ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]] | ||
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]] | |||
* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] | * ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]] | ||
* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] | * [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]] | ||
* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+ | |||
Considered below are | 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. | ||
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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}}.) | ||
== | == Tritonic == | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].'' | |||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments # | |||
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val. | |||
[[ | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 209: | Line 67: | ||
Comma list: 121/120, 225/224, 441/440 | Comma list: 121/120, 225/224, 441/440 | ||
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 224: | Line 82: | ||
Comma list: 105/104, 121/120, 196/195, 275/273 | Comma list: 105/104, 121/120, 196/195, 275/273 | ||
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 239: | 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|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 254: | 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|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 269: | 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|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 284: | Line 142: | ||
Comma list: 225/224, 385/384, 27783/27500 | Comma list: 225/224, 385/384, 27783/27500 | ||
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 295: | Line 153: | ||
== Septimin == | == Septimin == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].'' | ||
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: 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 319: | Line 179: | ||
Comma list: 225/224, 245/242, 385/384 | Comma list: 225/224, 245/242, 385/384 | ||
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 325: | 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 334: | Line 194: | ||
Comma list: 105/104, 144/143, 196/195, 245/242 | Comma list: 105/104, 144/143, 196/195, 245/242 | ||
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 340: | 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 | ||
== Merman == | == Merman == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].'' | ||
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot. | |||
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 369: | Line 233: | ||
Comma list: 225/224, 441/440, 1344/1331 | Comma list: 225/224, 441/440, 1344/1331 | ||
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 379: | Line 243: | ||
Badness (Sintel): 1.20 | Badness (Sintel): 1.20 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 144/143, 225/224, 364/363, 441/440 | Comma list: 144/143, 225/224, 364/363, 441/440 | ||
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 393: | Line 257: | ||
Badness (Sintel): 1.14 | Badness (Sintel): 1.14 | ||
=== Mermaid === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 225/224, 385/384, 532400/531441 | |||
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }} | |||
Badness (Sintel): 1.46 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 225/224, 325/324, 385/384, 10648/10647 | |||
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }} | |||
Badness (Sintel): 1.47 | |||
== Slender == | == Slender == | ||
Slender | Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot. | ||
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 419: | Line 315: | ||
Comma list: 225/224, 385/384, 1331/1323 | Comma list: 225/224, 385/384, 1331/1323 | ||
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 434: | Line 330: | ||
Comma list: 225/224, 275/273, 385/384, 1331/1323 | Comma list: 225/224, 275/273, 385/384, 1331/1323 | ||
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 445: | Line 341: | ||
== Triton == | == Triton == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].'' | ||
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 469: | Line 367: | ||
Comma list: 45/44, 56/55, 1029/1000 | Comma list: 45/44, 56/55, 1029/1000 | ||
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 502: | Line 400: | ||
Comma list: 225/224, 441/440, 4000/3993 | Comma list: 225/224, 441/440, 4000/3993 | ||
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 517: | Line 415: | ||
Comma list: 169/168, 225/224, 364/363, 441/440 | Comma list: 169/168, 225/224, 364/363, 441/440 | ||
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 532: | 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|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 547: | 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|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 556: | 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 580: | Line 641: | ||
Comma list: 225/224, 385/384, 12005/11979 | Comma list: 225/224, 385/384, 12005/11979 | ||
{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 595: | Line 656: | ||
Comma list: 169/168, 225/224, 364/363, 1716/1715 | Comma list: 169/168, 225/224, 364/363, 1716/1715 | ||
{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 606: | Line 667: | ||
== Gracecordial == | == Gracecordial == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].'' | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 618: | 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. | * [[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 630: | Line 691: | ||
Comma list: 225/224, 385/384, 236328125/234365481 | Comma list: 225/224, 385/384, 236328125/234365481 | ||
{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 645: | Line 706: | ||
Comma list: 225/224, 325/324, 385/384, 831875/830466 | Comma list: 225/224, 325/324, 385/384, 831875/830466 | ||
{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 660: | 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|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 675: | 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|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 690: | 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|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 705: | 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|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 720: | 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|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 735: | Line 796: | ||
Comma list: 225/224, 441/440, 109375/107811 | Comma list: 225/224, 441/440, 109375/107811 | ||
{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 750: | Line 811: | ||
Comma list: 225/224, 364/363, 441/440, 6125/6084 | Comma list: 225/224, 364/363, 441/440, 6125/6084 | ||
{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 765: | 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|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 780: | 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|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 813: | Line 874: | ||
Comma list: 225/224, 385/384, 12250/11979 | Comma list: 225/224, 385/384, 12250/11979 | ||
{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 824: | Line 885: | ||
== Misneb == | == Misneb == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].'' | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 848: | Line 909: | ||
Comma list: 99/98, 176/175, 1310720/1294139 | Comma list: 99/98, 176/175, 1310720/1294139 | ||
{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 863: | Line 924: | ||
Comma list: 99/98, 176/175, 640/637, 847/845 | Comma list: 99/98, 176/175, 640/637, 847/845 | ||
{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 878: | Line 939: | ||
Comma list: 225/224, 385/384, 66550/64827 | Comma list: 225/224, 385/384, 66550/64827 | ||
{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 889: | Line 950: | ||
== Untriton == | == Untriton == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].'' | ||
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 913: | Line 976: | ||
Comma list: 121/120, 225/224, 22000/21609 | Comma list: 121/120, 225/224, 22000/21609 | ||
{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 928: | Line 991: | ||
Comma list: 121/120, 225/224, 275/273, 1040/1029 | Comma list: 121/120, 225/224, 275/273, 1040/1029 | ||
{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 939: | Line 1,002: | ||
== Naiadical == | == Naiadical == | ||
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,002: | Line 1,067: | ||
== Quintannic == | == Quintannic == | ||
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,024: | Line 1,091: | ||
Comma list: 225/224, 441/440, 43923/43750 | Comma list: 225/224, 441/440, 43923/43750 | ||
{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 1,039: | Line 1,106: | ||
Comma list: 225/224, 441/440, 1001/1000, 1188/1183 | Comma list: 225/224, 441/440, 1001/1000, 1188/1183 | ||
{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 1,054: | 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|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 1,063: | 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 == | ||
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].'' | |||
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,089: | Line 1,196: | ||
Comma list: 99/98, 176/175, 65536/65219 | Comma list: 99/98, 176/175, 65536/65219 | ||
{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 1,101: | Line 1,208: | ||
== Tertiosec == | == Tertiosec == | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].'' | ||
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,124: | Line 1,233: | ||
Comma list: 225/224, 3840/3773, 12005/11979 | Comma list: 225/224, 3840/3773, 12005/11979 | ||
{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 1,134: | Line 1,243: | ||
Badness (Sintel): 5.74 | Badness (Sintel): 5.74 | ||
== References == | |||
[[Category:Marvel temperaments| ]] <!-- main article --> | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||