Marvel temperaments: Difference between revisions
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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]] | * [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]] | ||
* | * ''[[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]] | |||
* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] | |||
Considered below are wizard, tritonic, septimin, slender, triton | Considered below are wizard, 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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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #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 | 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 | [[Subgroup]]: 2.3.5.7 | ||
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* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}} | * [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}} | ||
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }} | : error map: {{val| 0.000 -1.169 -3.111 -0.849 }} | ||
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }} | {{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }} | ||
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* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}} | * WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}} | ||
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}} | * CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}} | ||
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }} | {{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }} | ||
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* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}} | * WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}} | ||
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}} | * CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}} | ||
{{Optimal ET sequence|legend=0| 22, 50, 72 }} | {{Optimal ET sequence|legend=0| 22, 50, 72 }} | ||
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* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}} | * WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}} | ||
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}} | * CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}} | ||
{{Optimal ET sequence|legend=0| 22, 50, 72 }} | {{Optimal ET sequence|legend=0| 22, 50, 72 }} | ||
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* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}} | * WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}} | ||
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}} | * CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}} | ||
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }} | {{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }} | ||
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* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}} | * WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}} | ||
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}} | * CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}} | ||
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }} | {{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }} | ||
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* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}} | * WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}} | ||
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}} | * CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}} | ||
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }} | {{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }} | ||
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* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}} | * WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}} | ||
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}} | * CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}} | ||
{{Optimal ET sequence|legend=0| 72, 94, 166g }} | {{Optimal ET sequence|legend=0| 72, 94, 166g }} | ||
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* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}} | * WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}} | ||
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}} | * CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}} | ||
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }} | {{Optimal ET sequence|legend=0| 22, 50e, 72ee }} | ||
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== Tritonic == | == Tritonic == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].'' | ||
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 | ||
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}} | ||
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }} | : error map: {{val| 0.000 -3.566 -2.769 -4.959 }} | ||
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }} | {{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }} | ||
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* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}} | * WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}} | ||
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }} | {{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }} | ||
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* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}} | * WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}} | ||
{{Optimal ET sequence|legend=0| 29, 31, 60e }} | {{Optimal ET sequence|legend=0| 29, 31, 60e }} | ||
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* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}} | * WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}} | ||
{{Optimal ET sequence|legend=0| 29g, 31, 60e }} | {{Optimal ET sequence|legend=0| 29g, 31, 60e }} | ||
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* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}} | * WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}} | ||
{{Optimal ET sequence|legend=0| 29g, 31, 60e }} | {{Optimal ET sequence|legend=0| 29g, 31, 60e }} | ||
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* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}} | * WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}} | ||
{{Optimal ET sequence|legend=0| 29g, 31, 60e }} | {{Optimal ET sequence|legend=0| 29g, 31, 60e }} | ||
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* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}} | * WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }} | {{Optimal ET sequence|legend=0| 31, 91, 122, 153d }} | ||
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== 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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* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}} | ||
: 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| 41, 91, 132d }} | ||
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* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}} | * WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}} | ||
* 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| 41, 91, 223cdef }} | ||
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* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}} | * WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}} | ||
* 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| 41, 91 }} | ||
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== 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 | ||
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}} | ||
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }} | : error map: {{val| 0.000 -1.104 -2.423 +0.657 }} | ||
{{Optimal ET sequence|legend=1| 41, 84, 125 }} | {{Optimal ET sequence|legend=1| 41, 84, 125 }} | ||
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* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}} | * WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}} | ||
{{Optimal ET sequence|legend=0| 41, 84, 125e }} | {{Optimal ET sequence|legend=0| 41, 84, 125e }} | ||
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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 | ||
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* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}} | * WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}} | ||
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }} | {{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }} | ||
Badness (Sintel): 1.14 | Badness (Sintel): 1.14 | ||
=== Mermaid === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 225/224, 385/384, 532400/531441 | |||
Mapping: {{mapping| 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: {{mapping| 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 | ||
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}} | ||
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }} | : error map: {{val| 0.000 -1.257 -2.235 +0.727 }} | ||
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }} | {{Optimal ET sequence|legend=1| 31, 94, 125, 406c }} | ||
| Line 458: | Line 462: | ||
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}} | * WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }} | {{Optimal ET sequence|legend=0| 31, 63, 94, 125 }} | ||
| Line 474: | Line 477: | ||
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}} | * WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 63, 94 }} | {{Optimal ET sequence|legend=0| 31, 63, 94 }} | ||
| Line 481: | Line 483: | ||
== 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 | ||
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}} | ||
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }} | : error map: {{val| 0.000 -9.007 -3.192 -16.687 }} | ||
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }} | {{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }} | ||
| Line 511: | Line 514: | ||
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}} | * WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}} | ||
{{Optimal ET sequence|legend=0| 2, 17d, 19 }} | {{Optimal ET sequence|legend=0| 2, 17d, 19 }} | ||
| Line 530: | Line 532: | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}} | ||
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }} | : error map: {{val| 0.000 -2.139 -2.398 -1.362 }} | ||
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }} | {{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }} | ||
| Line 546: | Line 547: | ||
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}} | * WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}} | ||
{{Optimal ET sequence|legend=0| 29, 43, 72 }} | {{Optimal ET sequence|legend=0| 29, 43, 72 }} | ||
| Line 562: | Line 562: | ||
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}} | * WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}} | ||
{{Optimal ET sequence|legend=0| 29, 43, 72 }} | {{Optimal ET sequence|legend=0| 29, 43, 72 }} | ||
| Line 578: | Line 577: | ||
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}} | * WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}} | ||
{{Optimal ET sequence|legend=0| 29g, 43, 72 }} | {{Optimal ET sequence|legend=0| 29g, 43, 72 }} | ||
| Line 594: | Line 592: | ||
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}} | * WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}} | ||
{{Optimal ET sequence|legend=0| 29g, 43, 72 }} | {{Optimal ET sequence|legend=0| 29g, 43, 72 }} | ||
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 == | ||
| Line 613: | Line 773: | ||
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}}) | * [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}}) | ||
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }} | : error map: {{val| 0.000 -1.361 -3.277 -1.565 }} | ||
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }} | {{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }} | ||
| Line 629: | Line 788: | ||
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}}) | * WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}}) | ||
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}}) | * CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}}) | ||
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }} | {{Optimal ET sequence|legend=0| 9, 54, 63, 72 }} | ||
| Line 645: | Line 803: | ||
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}}) | * WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}}) | ||
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}}) | * CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}}) | ||
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }} | {{Optimal ET sequence|legend=0| 9, 54, 63, 72 }} | ||
| Line 652: | Line 809: | ||
== 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 666: | Line 823: | ||
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}} | * [[CWE]]: ~2 = 1200.3333{{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 }} | ||
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }} | {{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }} | ||
| Line 682: | Line 838: | ||
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}} | * WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }} | {{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }} | ||
| Line 698: | Line 853: | ||
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}} | * WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }} | {{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }} | ||
| Line 714: | Line 868: | ||
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}} | * WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }} | {{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }} | ||
| Line 730: | Line 883: | ||
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}} | * WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }} | {{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }} | ||
| Line 746: | Line 898: | ||
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}} | * WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }} | {{Optimal ET sequence|legend=0| 12e, 113, 238cfi }} | ||
| Line 762: | Line 913: | ||
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}} | * WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }} | {{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }} | ||
| Line 778: | Line 928: | ||
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}} | * WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }} | {{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }} | ||
| Line 794: | Line 943: | ||
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}} | * WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}} | ||
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }} | {{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }} | ||
| Line 810: | Line 958: | ||
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}} | * WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }} | {{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }} | ||
| Line 826: | Line 973: | ||
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}} | * WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }} | {{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }} | ||
| Line 842: | Line 988: | ||
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}} | * WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }} | {{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }} | ||
| Line 861: | Line 1,006: | ||
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}} | * [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}} | ||
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }} | : error map: {{val| 0.000 -0.936 -5.382 -4.924 }} | ||
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }} | {{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }} | ||
| Line 877: | Line 1,021: | ||
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}} | * WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}} | ||
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }} | {{Optimal ET sequence|legend=0| 13d, 28d, 41 }} | ||
| Line 884: | Line 1,027: | ||
== 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 898: | Line 1,041: | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}} | ||
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }} | : error map: {{val| 0.000 -0.506 -0.999 +4.701 }} | ||
{{Optimal ET sequence|legend=1| 21, 32, 53 }} | {{Optimal ET sequence|legend=1| 21, 32, 53 }} | ||
| Line 914: | Line 1,056: | ||
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}} | * WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}} | ||
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }} | {{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }} | ||
| Line 930: | Line 1,071: | ||
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}} | * WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}} | ||
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }} | {{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }} | ||
| Line 946: | Line 1,086: | ||
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}} | * WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}} | ||
{{Optimal ET sequence|legend=0| 21e, 32, 53 }} | {{Optimal ET sequence|legend=0| 21e, 32, 53 }} | ||
| Line 953: | Line 1,092: | ||
== 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 967: | Line 1,108: | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}} | ||
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }} | : error map: {{val| 0.000 +0.298 -2.181 +3.946 }} | ||
{{Optimal ET sequence|legend=1| 51, 53 }} | {{Optimal ET sequence|legend=1| 51, 53 }} | ||
| Line 983: | Line 1,123: | ||
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}} | * WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}} | ||
{{Optimal ET sequence|legend=0| 51, 53 }} | {{Optimal ET sequence|legend=0| 51, 53 }} | ||
| Line 999: | Line 1,138: | ||
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}} | * WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}} | ||
{{Optimal ET sequence|legend=0| 51f, 53 }} | {{Optimal ET sequence|legend=0| 51f, 53 }} | ||
| Line 1,006: | Line 1,144: | ||
== 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 1,069: | Line 1,209: | ||
== 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 1,081: | Line 1,223: | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}} | * [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}} | ||
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }} | : error map: {{val| 0.000 -2.863 -2.136 -2.287 }} | ||
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }} | {{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }} | ||
| Line 1,097: | Line 1,238: | ||
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}} | * WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}} | ||
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }} | {{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }} | ||
| Line 1,113: | Line 1,253: | ||
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}} | * WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}} | ||
{{Optimal ET sequence|legend=0| 43, 60e, 103 }} | {{Optimal ET sequence|legend=0| 43, 60e, 103 }} | ||
| Line 1,129: | Line 1,268: | ||
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}} | * WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}} | ||
{{Optimal ET sequence|legend=0| 43, 60e, 103 }} | {{Optimal ET sequence|legend=0| 43, 60e, 103 }} | ||
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: {{mapping| 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 1,150: | Line 1,328: | ||
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}} | * [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}} | ||
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }} | : error map: {{val| 0.000 -0.844 +0.492 +7.007 }} | ||
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }} | {{Optimal ET sequence|legend=1| 22, 74, 96, 118d }} | ||
| Line 1,166: | Line 1,343: | ||
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}} | * WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}} | ||
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}} | * CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}} | ||
{{Optimal ET sequence|legend=0| 22, 74, 96 }} | {{Optimal ET sequence|legend=0| 22, 74, 96 }} | ||
| Line 1,174: | Line 1,350: | ||
== 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 1,187: | Line 1,365: | ||
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}} | * [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}} | ||
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }} | : error map: {{val| 0.000 -0.284 -0.276 +6.592 }} | ||
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }} | {{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }} | ||
| Line 1,203: | Line 1,380: | ||
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}} | * WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}} | ||
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}} | * CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}} | ||
{{Optimal ET sequence|legend=0| 21, 54, 75e }} | {{Optimal ET sequence|legend=0| 21, 54, 75e }} | ||
Badness (Sintel): 5.74 | Badness (Sintel): 5.74 | ||
== References == | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Marvel temperaments| ]] <!-- main article --> | [[Category:Marvel temperaments| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||