Optimal patent val: Difference between revisions
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Given any collection of ''p''-limit commas, there is a finite list of ''p''-limit [[patent val]]s tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[TE error]]; this is the (TE) '''optimal patent val''' for the temperament defined by the commas. Note that other definitions of error, such as maximum ''p''-limit error, or maximum ''q''-limit error where q is the largest odd number less than the prime above p, lead to different results. | Given any collection of ''p''-limit commas, there is a finite list of ''p''-limit [[patent val]]s tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[TE error]]; this is the (TE) '''optimal patent val''' for the temperament defined by the commas. Note that other definitions of error, such as maximum ''p''-limit error, or maximum ''q''-limit error where q is the largest odd number less than the prime above p, lead to different results. | ||
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7629394531250/7625597484987: [[3501edo|3501et]] 17.191 | 7629394531250/7625597484987: [[3501edo|3501et]] 17.191 | ||
===7-limit rank two=== | === 7-limit rank two === | ||
Name: ET w/ optimal patent val: Val name: 1000*badness | Name: ET w/ optimal patent val: Val name: 1000*badness | ||
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[[Archytas clan|Mother]]: [[5edo|5et]] 2&3 24.152 | [[Archytas clan|Mother]]: [[5edo|5et]] 2&3 24.152 | ||
[[Sharptone]]: [[5edo|5et]] 5&7d 24.848 | [[Meantone family|Sharptone]]: [[5edo|5et]] 5&7d 24.848 | ||
[[Father family#Baba|Baba]]: [[5edo|5et]] 1&5 44.321 | [[Father family#Baba|Baba]]: [[5edo|5et]] 1&5 44.321 | ||
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[[Meantone family|Squares]]: [[31edo|31et]] 31&45 45.993 | [[Meantone family|Squares]]: [[31edo|31et]] 31&45 45.993 | ||
[[Immunity family# | [[Immunity family#Septimal immunity|Immunity]]: [[34edo|34et]] 5&29 77.631 | ||
[[Meantone family|Injera]]: [[38edo|38et]] 12&26 31.130 | [[Meantone family|Injera]]: [[38edo|38et]] 12&26 31.130 | ||
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[[Marvel temperaments|Wizard]]: [[72edo|72et]] 22&50 40.846 | [[Marvel temperaments|Wizard]]: [[72edo|72et]] 22&50 40.846 | ||
[[Unicorn family | [[Unicorn family|Unicorn]] [[77edo|77et]] 19&58 40.913 | ||
[[Diaschismic family|Bidia]]: [[80edo|80et]] 12&56 56.474 | [[Diaschismic family|Bidia]]: [[80edo|80et]] 12&56 56.474 | ||
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[[Semicomma family|Orwell]]: [[137edo|137et]] 9&22 20.375 | [[Semicomma family|Orwell]]: [[137edo|137et]] 9&22 20.375 | ||
[[Breedsmic temperaments|Tertiaseptal]]: [[171edo|171et]] 31&109 12.995 | [[Breedsmic temperaments#Tertiaseptal|Tertiaseptal]]: [[171edo|171et]] 31&109 12.995 | ||
[[Breedsmic temperaments#Septidiasemi|Septidiasemi]]: [[171edo|171et]] 10&151 44.115 | [[Breedsmic temperaments#Septidiasemi|Septidiasemi]]: [[171edo|171et]] 10&151 44.115 | ||
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[[Schismatic family|Bischismic]]: [[378edo|378et]] 12&118 54.744 | [[Schismatic family|Bischismic]]: [[378edo|378et]] 12&118 54.744 | ||
[[Nessafof]]: [[381edo|381et]] 15&69 45.478 | [[Porwell temperaments|Nessafof]]: [[381edo|381et]] 15&69 45.478 | ||
[[Hemifamity temperaments#Septiquarter|Septiquarter]]: [[391edo|391et]] 5&94 53.760 | [[Hemifamity temperaments#Septiquarter|Septiquarter]]: [[391edo|391et]] 5&94 53.760 | ||
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[[Hemimean clan#Sengagen|Sengagen]]: [[446edo|446et]] 49&50 57.978 | [[Hemimean clan#Sengagen|Sengagen]]: [[446edo|446et]] 49&50 57.978 | ||
[[Chromat]]: [[456edo|456et]] 60&99 57.499 | [[Landscape microtemperaments|Chromat]]: [[456edo|456et]] 60&99 57.499 | ||
[[Kleismic family|Countercata]]: [[473edo|473et]] 34&53 52.128 | [[Kleismic family|Countercata]]: [[473edo|473et]] 34&53 52.128 | ||
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[[Breedsmic temperaments#Emmthird|Emmthird]] [[1255edo|1255et]] 58&113 16.736 | [[Breedsmic temperaments#Emmthird|Emmthird]] [[1255edo|1255et]] 58&113 16.736 | ||
[[Landscape microtemperaments#Septichrome| | [[Landscape microtemperaments#Septichrome|Septichrome]] [[1308edo|1308et]] 60&111 16.814 | ||
[[Schismatic family|Sesquiquartififths]]: [[1498edo|1498et]] 41&89 11.244 | [[Schismatic family|Sesquiquartififths]]: [[1498edo|1498et]] 41&89 11.244 | ||
[[Brahmagupta]]: [[1547edo|1547et]] 7&217 29.122 | [[Ragismic microtemperaments|Brahmagupta]]: [[1547edo|1547et]] 7&217 29.122 | ||
[[Breedsmic temperaments#Maviloid|Maviloid]]: [[1614edo|1614et]] 76&99 57.632 | [[Breedsmic temperaments#Maviloid|Maviloid]]: [[1614edo|1614et]] 76&99 57.632 | ||
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[[Marvel family #Potassium|Potassium]]: [[19edo|19et]] 2&9&10 46.3665 | [[Marvel family #Potassium|Potassium]]: [[19edo|19et]] 2&9&10 46.3665 | ||
[[Jubilismic family #Festival|Festival]]: [[26edo|26et]] 2&10&26 68.8510 | |||
[[Mint family #Nickel|Nickel]]: [[28edo|28et]] 7&9&12 46.3463 | [[Mint family #Nickel|Nickel]]: [[28edo|28et]] 7&9&12 46.3463 | ||
[[Starling family #Thrasher|Thrasher]]: [[34edo|34et]] 3&4&12 48.0188 | [[Starling family #Thrasher|Thrasher]]: [[34edo|34et]] 3&4&12 48.0188 | ||
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[[Horwell family #Zelda|Zelda]]: [[118edo|118et]]: 9&22&87 64.1882 | [[Horwell family #Zelda|Zelda]]: [[118edo|118et]]: 9&22&87 64.1882 | ||
[[Rastmic temperaments #Parahemif|Parahemif]]: 17&24&58 134.547 | [[Rastmic temperaments #Parahemif|Parahemif]]: [[123edo|123et]] 17&24&58 134.547 | ||
[[Valinorismic temperaments #Varda|Varda]]: [[126edo|126et]] 12&22&46 72.8125 | [[Valinorismic temperaments #Varda|Varda]]: [[126edo|126et]] 12&22&46 72.8125 | ||
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[[Starling temperaments|Lupercalia]]: [[31edo|31et]] 15&31 21.328 | [[Starling temperaments|Lupercalia]]: [[31edo|31et]] 15&31 21.328 | ||
[[Starling temperaments#Casablanca | [[Starling temperaments#Casablanca|Murakuc]]: [[31edo|31et]] 31&73f 41.394 | ||
[[Starling temperaments#Nusecond | [[Starling temperaments#Nusecond|Nusecond]]: [[31edo|31et]] 31&70f 23.323 | ||
[[Starling temperaments#Cypress-13-limit|Cypress]]: [[31edo|31et]] 11cdeef&31 37.849 | [[Starling temperaments#Cypress-13-limit|Cypress]]: [[31edo|31et]] 11cdeef&31 37.849 | ||
[[Starling temperaments# | [[Starling temperaments#Maneh|Maneh]]: [[31edo|31et]] 4ef&31 29.868 | ||
[[31st-octave temperaments #Prajapati|Prajapati]]: [[31edo|31et]] 31&93f 37.885 | [[31st-octave temperaments #Prajapati|Prajapati]]: [[31edo|31et]] 31&93f 37.885 | ||
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[[Ragismic microtemperaments#Ennealimmal-Ennealiminal-13-limit|Ennealiminal]]: [[72edo|72et]] 27&72 30.325 | [[Ragismic microtemperaments#Ennealimmal-Ennealiminal-13-limit|Ennealiminal]]: [[72edo|72et]] 27&72 30.325 | ||
[[Starling temperaments#Casablanca | [[Starling temperaments#Casablanca|Marrakesh]]: [[73edo|73et]] 31&73 40.774 | ||
[[Meantone family#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone|Grosstone]]: [[74edo|74et]] 12&31 25.899 | [[Meantone family#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone|Grosstone]]: [[74edo|74et]] 12&31 25.899 | ||
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[[Diaschismic family|Bidia]]: [[80edo|80et]] 12&68 41.137 | [[Diaschismic family|Bidia]]: [[80edo|80et]] 12&68 41.137 | ||
[[Ragismic microtemperaments#Parakleismic | [[Ragismic microtemperaments#Parakleismic|Paradigmic]]: [[80edo|80et]] 19&80 35.781 | ||
[[Orwellismic temperaments#Quartonic-13-limit|Quartonic]]: [[80edo|80et]] 53&80 23.875 | [[Orwellismic temperaments#Quartonic-13-limit|Quartonic]]: [[80edo|80et]] 53&80 23.875 | ||
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[[Schismatic family#Sanjaab-13-limit|Sanjaab]]: [[94edo|94et]] 29&94 33.849 | [[Schismatic family#Sanjaab-13-limit|Sanjaab]]: [[94edo|94et]] 29&94 33.849 | ||
[[Hemifamity temperaments#Supers|Supers]]: [[94edo|94et]] 58&94 21.644 | |||
[[Marvel temperaments#Submajor-Interpental-13-limit|Interpental]]: [[96edo|96et]] 43&53 29.680 | [[Marvel temperaments#Submajor-Interpental-13-limit|Interpental]]: [[96edo|96et]] 43&53 29.680 | ||
[[Meantone family#Mothra|Mosura]]: [[98edo|98et]] 31&67 36.857 | [[Meantone family#Mothra|Mosura]]: [[98edo|98et]] 31&67 36.857 | ||
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[[Breedsmic temperaments|Harry]]: [[130edo|130et]] 58&72 13.046 | [[Breedsmic temperaments|Harry]]: [[130edo|130et]] 58&72 13.046 | ||
[[ | [[26th-octave temperaments#Bosonic|Bosonic]]: [[130edo|130et]] 26&130 32.946 | ||
[[Sensipent family#Bison-13-limit|Bison]]: [[130edo|130et]] 46&84 23.504 | [[Sensipent family#Bison-13-limit|Bison]]: [[130edo|130et]] 46&84 23.504 | ||
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[[Horwell temperaments#Emkay|Emkay]]: [[535edo|535et]] 87&137 17.853 | [[Horwell temperaments#Emkay|Emkay]]: [[535edo|535et]] 87&137 17.853 | ||
[[ | [[26th-octave temperaments#Fermionic|Fermionic]]: [[546edo|546et]] 130&286 43.581 | ||
[[Schismatic family#Pogo-13-limit|Pogo]]: [[578edo|578et]] 36&94 17.514 | [[Schismatic family#Pogo-13-limit|Pogo]]: [[578edo|578et]] 36&94 17.514 | ||
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[[Breed family #Freya|Freya]]: [[2615edo|2615et]] 10&31&229 85.515 | [[Breed family #Freya|Freya]]: [[2615edo|2615et]] 10&31&229 85.515 | ||
= See also = | == See also == | ||
* [[Associated temperament]] | * [[Associated temperament]] | ||