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~~.

The '''optimal patent val''' for a [[regular temperament]] is the unique [[patent val]] that [[support]]s the temperament with the lowest [[error]].

Given any temperament, which is characterized by the [[comma]]s it [[tempering out|tempers out]], there is a finite list of [[patent val]]s that temper out all the commas of the temperament in the same [[subgroup]]. 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 one which has the lowest [[TE error]]; this is the (TE) optimal patent val for the temperament. Note that other definitions of error lead to different results.

On this wiki, the optimal patent val for each temperament is given as the last patent val in the [[optimal ET sequence]], or stated explicitly in case it is not a member of the sequence.

== Instructions ==

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[[Meantone family|Dominant]]: [[12edo|12et]] 5&7 20.690

[[~~Jubilismic clan~~|Diminished]]: [[12edo|12et]] 4&12 22.401

[[Diminished family #Septimal diminished|Diminished]]: [[12edo|12et]] 4&12 22.401

[[Augmented family|August]]: [[12edo|12et]] 9&12 26.459

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[[Meantone family#Godzilla-Semafour|Semafour]]: [[5edo|5et]] 5&14c 28.510

[[Meantone family#~~Maqamic~~|~~Maqamic~~]]: [[7edo|7et]] 7&10c 40.240

[[Meantone family#Neutrominant|Neutrominant]]: [[7edo|7et]] 7&10c 40.240

[[Meantone family#Dominant-Arnold|Arnold]]: [[7edo|7et]] 5&7 26.141

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[[Dicot family#Decimal|Decibel]]: [[10edo|10et]] 4&6 32.385

[[~~Jubilismic clan~~|Diminished]]: [[12edo|12et]] 4&12 22.132

[[Diminished family #Septimal diminished|Diminished]]: [[12edo|12et]] 4&12 22.132

[[Diaschismic family#Pajara-Pajaric|Pajaric]]: [[12edo|12et]] 2&10 23.798

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[[Mint temperaments#Ripple|Ripple]]: [[12edo|12et]] 1ce&12 38.811

[~~[Jubilismic clan~~#~~Diminished-~~Hemidim|Hemidim]]: [[12edo|12et]] 12&20b 54.965

[Diminished family #Hemidim|Hemidim]]: [[12edo|12et]] 12&20b 54.965

[[Mint temperaments#Smate|Smate]]: [[14edo|14et]] 14&17c 42.518

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[[Augmented family#Hemiaug|Hemiaug]]: [[24edo|24et]] 24&27e 38.232

[[~~Dimipent~~ family#Hemidim|Hemidim]]: [[24edo|24et]] 4e&24 56.576

[[Diminished family #Hemidim|Hemidim]]: [[24edo|24et]] 4e&24 56.576

[[~~Dimipent~~ family#~~Semidim~~|~~Semidim~~]]: [[24edo|24et]] 24&32c 30.597

[[Diminished family #Octonion|Octonion]]: [[24edo|24et]] 24&32c 30.597

[[Meantone family#Injera-11-limit|Injera]]: [[26edo|26et]] 12&26 23.124

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[[Sensipent family#Sensi-Sensa|Sensa]]: [[27edo|27et]] 19e&27 36.835

[[~~Jubilismic clan~~#~~Diminished-~~Demolished|Demolished]]: [[28edo|28et]] 12&28 26.574

[[Diminished family #Demolished|Demolished]]: [[28edo|28et]] 12&28 26.574

[[Porcupine family|Nautilus]]: [[29edo|29et]] 15&29 26.023

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[[Meantone family#Liese-Elisa|Elisa]]: [[36edo|36e]] 19e&36e 41.592

[[~~Pythagorean~~ family#~~Catler temperament-Catcall~~|~~Catcall~~]]: [[36edo|36et]] 12&24 34.478

[[Compton family#Catnip|Catnip]]: [[36edo|36et]] 12&24 34.478

[[Augmented family#Niner|Niner]]: [[36edo|36et]] 9&36 34.861

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Name: ET w/ optimal patent val: Val name: 1000*badness

[[~~Jubilismic clan~~#~~Diminished-13-limit~~|Diminished]]: [[4edo|4et]] 4&8d 19.509

[[Diminished family #Septimal diminished|Diminished]]: [[4edo|4et]] 4&8d 19.509

[[Meantone family#~~Maqamic~~-13-limit|~~Maqamic~~]]: [[7edo|7et]] 7&10c 27.214

[[Meantone family#Neutrominant-13-limit|Neutrominant]]: [[7edo|7et]] 7&10c 27.214

[[Tetracot family#Modus|Modus]]: [[7edo|7et]] 7&27e 23.806

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[[Archytas clan#Quasisuper-Quasisupra-13-limit|Quasisupra]]: [[22edo|22et]] 17c&22 30.219

[[~~Pythagorean~~ family#~~Catler temperament~~|~~Catcall~~]]: [[24edo|24et]] 12f&24 28.363

[[Compton family#Catnip|Catnip]]: [[24edo|24et]] 12f&24 28.363

[[Augmented family|Triforce]]: [[24edo|24et]] 9&15 20.248

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[[Augmented family#Hemiaug-13-limit|Hemiaug]]: [[24edo|24et]] 3de&24 30.159

[[~~Dimipent~~ family#Hemidim|Hemidim]]: [[24edo|24et]] 4ef&24 39.030

[[Diminished family #Hemidim|Hemidim]]: [[24edo|24et]] 4ef&24 39.030

[[~~Dimipent~~ family#~~Semidim~~|~~Semidim~~]]: [[24edo|24et]] 24&32c 30.597

[[Diminished family #Octonion|Octonion]]: [[24edo|24et]] 24&32c 30.597

[[Meantone family#Mohamaq-13-limit|Mohamaq]]: [[24edo|24et]] 17c&24 28.738

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[[Porwell temperaments#Septisuperfourth|Septisuperfourth]]: [[282edo|282et]] 130&282 22.887

[[~~Sensamagic clan~~|~~Sensa~~]]: [[283edo|283et]] 22&65 31.366

[[Escapade family#Escaped|Escaped]]: [[283edo|283et]] 22&65 31.366

[[Schismatic family|~~Sextilififths~~]]: [[289edo|289et]] 29&130 25.276

[[Schismatic family#Sextilifourths|Sextilifourths]]: [[289edo|289et]] 29&130 25.276

[[Ragismic microtemperaments#Quincy|Quincy]]: [[289edo|289et]] 72&145 23.862

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[[Würschmidt family#Hemiwürschmidt-11-limit-13-limit|Hemiwürschmidt]]: [[291edo|291et]] 31&130 23.074

[[~~Porwell temperaments~~|Hemischis]]: [[313edo|313et]] 53&130 20.816

[[Schismatic family#Hemischis|Hemischis]]: [[313edo|313et]] 53&130 20.816

[[Werckismic temperaments#Octowerck|Octowerck]]: [[320edo|320et]] 72&248 27.632

[[~~Pental~~ family|~~Decal~~]]: [[320edo|320et]] 130&190 23.948

[[Quintile family#Decile|Decile]]: [[320edo|320et]] 130&190 23.948

[[Kleismic family#Novemkleismic|Novemkleismic]]: [[333edo|333et]] 72&261 39.072

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{{todo

| improve layout

~~| add introduction~~

~~| introduce lemma~~

| increase applicability

| text = add structure by switching to tables

@@ Line 1: / Line 1: @@
-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.
+The '''optimal patent val''' for a [[regular temperament]] is the unique [[patent val]] that [[support]]s the temperament with the lowest [[error]].
+Given any temperament, which is characterized by the [[comma]]s it [[tempering out|tempers out]], there is a finite list of [[patent val]]s that temper out all the commas of the temperament in the same [[subgroup]]. 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 one which has the lowest [[TE error]]; this is the (TE) optimal patent val for the temperament. Note that other definitions of error lead to different results.
+On this wiki, the optimal patent val for each temperament is given as the last patent val in the [[optimal ET sequence]], or stated explicitly in case it is not a member of the sequence.
 == Instructions ==
@@ Line 99: / Line 103: @@
 [[Meantone family|Dominant]]: [[12edo|12et]] 5&7 20.690
-[[Jubilismic clan|Diminished]]: [[12edo|12et]] 4&12 22.401
+[[Diminished family #Septimal diminished|Diminished]]: [[12edo|12et]] 4&12 22.401
 [[Augmented family|August]]: [[12edo|12et]] 9&12 26.459
@@ Line 437: / Line 441: @@
 [[Meantone family#Godzilla-Semafour|Semafour]]: [[5edo|5et]] 5&14c 28.510
-[[Meantone family#Maqamic|Maqamic]]: [[7edo|7et]] 7&10c 40.240
+[[Meantone family#Neutrominant|Neutrominant]]: [[7edo|7et]] 7&10c 40.240
 [[Meantone family#Dominant-Arnold|Arnold]]: [[7edo|7et]] 5&7 26.141
@@ Line 469: / Line 473: @@
 [[Dicot family#Decimal|Decibel]]: [[10edo|10et]] 4&6 32.385
-[[Jubilismic clan|Diminished]]: [[12edo|12et]] 4&12 22.132
+[[Diminished family #Septimal diminished|Diminished]]: [[12edo|12et]] 4&12 22.132
 [[Diaschismic family#Pajara-Pajaric|Pajaric]]: [[12edo|12et]] 2&10 23.798
@@ Line 489: / Line 493: @@
 [[Mint temperaments#Ripple|Ripple]]: [[12edo|12et]] 1ce&12 38.811
-[[Jubilismic clan#Diminished-Hemidim|Hemidim]]: [[12edo|12et]] 12&20b 54.965
+[Diminished family #Hemidim|Hemidim]]: [[12edo|12et]] 12&20b 54.965
 [[Mint temperaments#Smate|Smate]]: [[14edo|14et]] 14&17c 42.518
@@ Line 577: / Line 581: @@
 [[Augmented family#Hemiaug|Hemiaug]]: [[24edo|24et]] 24&27e 38.232
-[[Dimipent family#Hemidim|Hemidim]]: [[24edo|24et]] 4e&24 56.576
+[[Diminished family #Hemidim|Hemidim]]: [[24edo|24et]] 4e&24 56.576
-[[Dimipent family#Semidim|Semidim]]: [[24edo|24et]] 24&32c 30.597
+[[Diminished family #Octonion|Octonion]]: [[24edo|24et]] 24&32c 30.597
 [[Meantone family#Injera-11-limit|Injera]]: [[26edo|26et]] 12&26 23.124
@@ Line 597: / Line 601: @@
 [[Sensipent family#Sensi-Sensa|Sensa]]: [[27edo|27et]] 19e&27 36.835
-[[Jubilismic clan#Diminished-Demolished|Demolished]]: [[28edo|28et]] 12&28 26.574
+[[Diminished family #Demolished|Demolished]]: [[28edo|28et]] 12&28 26.574
 [[Porcupine family|Nautilus]]: [[29edo|29et]] 15&29 26.023
@@ Line 649: / Line 653: @@
 [[Meantone family#Liese-Elisa|Elisa]]: [[36edo|36e]] 19e&36e 41.592
-[[Pythagorean family#Catler temperament-Catcall|Catcall]]: [[36edo|36et]] 12&24 34.478
+[[Compton family#Catnip|Catnip]]: [[36edo|36et]] 12&24 34.478
 [[Augmented family#Niner|Niner]]: [[36edo|36et]] 9&36 34.861
@@ Line 1,223: / Line 1,227: @@
 Name: ET w/ optimal patent val: Val name: 1000*badness
-[[Jubilismic clan#Diminished-13-limit|Diminished]]: [[4edo|4et]] 4&8d 19.509
+[[Diminished family #Septimal diminished|Diminished]]: [[4edo|4et]] 4&8d 19.509
-[[Meantone family#Maqamic-13-limit|Maqamic]]: [[7edo|7et]] 7&10c 27.214
+[[Meantone family#Neutrominant-13-limit|Neutrominant]]: [[7edo|7et]] 7&10c 27.214
 [[Tetracot family#Modus|Modus]]: [[7edo|7et]] 7&27e 23.806
@@ Line 1,353: / Line 1,357: @@
 [[Archytas clan#Quasisuper-Quasisupra-13-limit|Quasisupra]]: [[22edo|22et]] 17c&22 30.219
-[[Pythagorean family#Catler temperament|Catcall]]: [[24edo|24et]] 12f&24 28.363
+[[Compton family#Catnip|Catnip]]: [[24edo|24et]] 12f&24 28.363
 [[Augmented family|Triforce]]: [[24edo|24et]] 9&15 20.248
@@ Line 1,359: / Line 1,363: @@
 [[Augmented family#Hemiaug-13-limit|Hemiaug]]: [[24edo|24et]] 3de&24 30.159
-[[Dimipent family#Hemidim|Hemidim]]: [[24edo|24et]] 4ef&24 39.030
+[[Diminished family #Hemidim|Hemidim]]: [[24edo|24et]] 4ef&24 39.030
-[[Dimipent family#Semidim|Semidim]]: [[24edo|24et]] 24&32c 30.597
+[[Diminished family #Octonion|Octonion]]: [[24edo|24et]] 24&32c 30.597
 [[Meantone family#Mohamaq-13-limit|Mohamaq]]: [[24edo|24et]] 17c&24 28.738
@@ Line 1,803: / Line 1,807: @@
 [[Porwell temperaments#Septisuperfourth|Septisuperfourth]]: [[282edo|282et]] 130&282 22.887
-[[Sensamagic clan|Sensa]]: [[283edo|283et]] 22&65 31.366
+[[Escapade family#Escaped|Escaped]]: [[283edo|283et]] 22&65 31.366
-[[Schismatic family|Sextilififths]]: [[289edo|289et]] 29&130 25.276
+[[Schismatic family#Sextilifourths|Sextilifourths]]: [[289edo|289et]] 29&130 25.276
 [[Ragismic microtemperaments#Quincy|Quincy]]: [[289edo|289et]] 72&145 23.862
@@ Line 1,811: / Line 1,815: @@
 [[Würschmidt family#Hemiwürschmidt-11-limit-13-limit|Hemiwürschmidt]]: [[291edo|291et]] 31&130 23.074
-[[Porwell temperaments|Hemischis]]: [[313edo|313et]] 53&130 20.816
+[[Schismatic family#Hemischis|Hemischis]]: [[313edo|313et]] 53&130 20.816
 [[Werckismic temperaments#Octowerck|Octowerck]]: [[320edo|320et]] 72&248 27.632
-[[Pental family|Decal]]: [[320edo|320et]] 130&190 23.948
+[[Quintile family#Decile|Decile]]: [[320edo|320et]] 130&190 23.948
 [[Kleismic family#Novemkleismic|Novemkleismic]]: [[333edo|333et]] 72&261 39.072
@@ Line 2,023: / Line 2,027: @@
 {{todo
 | improve layout
-| add introduction
-| introduce lemma
 | increase applicability
 | text = add structure by switching to tables