Minkowski reduced bases for Fokker groups of certain vals
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For some purposes, eg [[Fokker blocks]], it is convenient to have a good basis for the wedgies of rank two temperaments supported by a given val. Below are listed some Minkowski reduced bases relative to [[generator complexity]] as a metric, with TE complexity used to break any ties. Given a p-limit val V, there is a corresponding p-limit JI subgroup of elements q such that <V|q> = 0; if V is an edo val, this is the comma group corresponding to V. We can expand this to a larger group by adding 2 as a generator; now this group G is a subgroup of the full JI group of finite index. There is then a corresponding group of vals, and an isomorphic group of bivals obtained by wedging these vals with V. This is the Fokker group Fokk(V) of bivals associated to V. An algorithm to find it is to take a basis for G in monzo form, take the transpose of the inverse of the corresponding matrix, clear denominators in the resulting vals, wedge each val with V and reduce to wedgies, and that is a basis for Fokk(V). =5-limit= <12 19 28|: <<1 4 4||, <<3 0 -7|| <15 24 35|: <<3 0 -7||, <<3 5 1|| <17 27 39|: <<2 1 -3||, <<1 9 12|| <17 27 40|: <<1 4 4||, <<4 -1 -11|| <19 30 44|: <<1 4 4||, <<5 1 -10|| <22 35 51|: <<3 5 1||, <<2 -4 -11|| <31 49 72|: <<1 4 4||, <<8 1 -17|| <34 54 79|: <<2 -4 -11||, <<6 5 -6|| <41 65 95|: <<5 1 -10||, <<4 9 5|| <46 73 107|: <<2 -4 -11||, <<7 9 -2|| <53 84 123|: <<6 5 -6||, <<1 -8 -15|| =7-limit= ==Temperaments== ammonite7: <<9 15 19 3 5 2|| armodue7: <<1 -3 5 -7 5 20|| augene7: <<3 0 -6 -7 -18 -14|| august7: <<3 0 6 -7 1 14|| baba7: <<2 -2 1 -8 -4 8|| beatles7: <<2 -9 -4 -19 -12 16|| beep7: <<2 3 1 0 -4 -6|| bipelog7: <<2 -6 -6 -14 -15 3|| blacksmith7: <<0 5 0 8 0 -14|| catalan7: <<6 5 -12 -6 -36 -42|| charon7: <<2 4 4 2 1 -2|| decimal7: <<4 2 2 -6 -8 -1|| dichotic7: <<2 1 -4 -3 -12 -12|| dicot7: <<2 1 3 -3 -1 4|| diminished7: <<4 4 4 -3 -5 -2|| dominant7: <<1 4 -2 4 -6 -16|| father7: <<1 -1 3 -4 2 10|| fifive7: <<10 14 14 -1 -6 -7|| flattone7: <<1 4 -9 4 -17 -32|| garibaldi7: <<1 -8 -14 -15 -25 -10|| godzilla7: <<2 8 1 8 -4 -20|| hystrix7: <<3 5 1 1 -7 -12|| immunity7: <<2 13 1 16 -4 -34|| inflated7: <<3 0 9 -7 6 21|| injera7: <<2 8 8 8 7 -4|| jamesbond7: <<0 0 7 0 11 16|| keemun7: <<6 5 3 -6 -12 -7|| lemba7: <<6 -2 -2 -17 -20 1|| magic7: <<5 1 12 -10 5 25|| meantone7: <<1 4 10 4 13 12|| mother7: <<1 -1 -2 -4 -6 -2|| mothra7: <<3 12 -1 12 -10 -36|| nautilus7: <<6 10 3 2 -12 -21|| negri7: <<4 -3 2 -14 -8 13|| orwell7: <<7 -3 8 -21 -7 27|| pajara7: <<2 -4 -4 -11 -12 2|| passion7: <<5 -4 -10 -18 -30 -12|| pelogic7: <<1 -3 -4 -7 -9 -1|| plutus7: <<1 4 5 4 5 0|| porcupine7: <<3 5 -6 1 -18 -28|| progress7: <<3 -5 -6 -15 -18 0|| progression7: <<5 3 7 -7 -3 8|| quartonic7: <<11 18 5 3 -23 -39|| rodan7: <<3 17 -1 20 -10 -50|| schism7: <<1 -8 -2 -15 -6 18|| sensi7: <<7 9 13 -2 1 5|| sharp7: <<2 1 6 -3 4 11|| sidi7: <<4 2 9 -6 3 15|| superkleismic7: <<9 10 -3 -5 -30 -35|| superpyth7: <<1 9 -2 12 -6 -30|| ternary7: <<0 0 3 0 5 7|| valentine7: <<9 5 -3 -13 -30 -21|| walid7: <<2 -2 -2 -8 -9 1|| wollemia7: <<4 9 19 5 19 19|| würschmidt7: <<8 1 18 -17 6 39|| ==Bases== <5 8 12 14|: beep7, mother7, father7 <6 10 14 17|: ternary7, charon7, baba7 <7 11 16 20|: dicot7, plutus7, hystrix7 <8 13 19 23|: father7, walid7, hystrix7 <9 14 21 25|: beep7, pelogic7, august7 <10 16 23 28|: sharp7, blacksmith7, decimal7 <12 19 28 34|: august7, dominant7, pajara7 <14 22 32 39|: jamesbond7, decimal7, godzilla7 <15 24 35 42|: blacksmith7, inflated7, keemun7 <16 25 37 45|: diminished7, armodue7, bipelog7 <17 27 39 48|: dichotic7, sidi7, schism7 <17 27 40 48|: dominant7, progression7, progress7 <19 30 44 53: godzilla7, meantone7, keemun7 <22 35 51 62|: pajara7, magic7, porcupine7 <26 41 60 73|: injera7, lemba7, flattone7 <27 43 63 76|: augene7, superpyth7, sensi7 <29 46 67 81|: negri7, nautilus7, garibaldi7 <31 49 72 87|: meantone7, mothra7, orwell7 <34 54 79 95|: keemun7, immunity7, wollemia7 <34 54 79 96|: pajara7, fifive7, würschmidt7 <37 59 86 104|: porcupine7, beatles7, ammonite7 <41 65 95 115|: magic7, garibaldi7, superkleismic7 <46 73 107 129|: sensi7, valentine7, rodan7 <49 78 114 138}: superpyth7, passion7, catalan7 <53 84 123 149|: garibaldi7, orwell7, quartonic7 =11-limit= ==Temperaments== august11: <<3 0 6 6 -7 1 -1 14 14 -4|| diminished11: <<4 4 4 0 -3 -5 -14 -2 -14 -14|| domineering11: <<1 4 -2 6 4 -6 6 -16 0 24|| godzilla11: <<2 8 1 12 8 -4 12 -20 0 30|| keemun11: <<6 5 3 -2 -6 -12 -24 -7 -22 -16|| meanenneadecal11: <<1 4 10 6 4 13 6 12 0 -18|| negri11: <<4 -3 2 5 -14 -8 -6 13 22 7|| pajaric11: <<2 -4 -4 0 -11 -12 -7 2 14 14|| ==Bases== <12 19 28 34 42|: august11, domineering11, diminished11, pajaric11 <19 30 44 53 66|: godzilla11, meanenneadecal11, negri11, keemun11
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<html><head><title>Minkowski reduced bases for Fokker groups of certain vals</title></head><body>For some purposes, eg <a class="wiki_link" href="/Fokker%20blocks">Fokker blocks</a>, it is convenient to have a good basis for the wedgies of rank two temperaments supported by a given val. Below are listed some Minkowski reduced bases relative to <a class="wiki_link" href="/generator%20complexity">generator complexity</a> as a metric, with TE complexity used to break any ties.<br /> <br /> Given a p-limit val V, there is a corresponding p-limit JI subgroup of elements q such that <V|q> = 0; if V is an edo val, this is the comma group corresponding to V. We can expand this to a larger group by adding 2 as a generator; now this group G is a subgroup of the full JI group of finite index. There is then a corresponding group of vals, and an isomorphic group of bivals obtained by wedging these vals with V. This is the Fokker group Fokk(V) of bivals associated to V. An algorithm to find it is to take a basis for G in monzo form, take the transpose of the inverse of the corresponding matrix, clear denominators in the resulting vals, wedge each val with V and reduce to wedgies, and that is a basis for Fokk(V).<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x5-limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit</h1> <br /> <12 19 28|: <<1 4 4||, <<3 0 -7||<br /> <15 24 35|: <<3 0 -7||, <<3 5 1||<br /> <17 27 39|: <<2 1 -3||, <<1 9 12||<br /> <17 27 40|: <<1 4 4||, <<4 -1 -11||<br /> <19 30 44|: <<1 4 4||, <<5 1 -10||<br /> <22 35 51|: <<3 5 1||, <<2 -4 -11||<br /> <31 49 72|: <<1 4 4||, <<8 1 -17||<br /> <34 54 79|: <<2 -4 -11||, <<6 5 -6||<br /> <41 65 95|: <<5 1 -10||, <<4 9 5||<br /> <46 73 107|: <<2 -4 -11||, <<7 9 -2||<br /> <53 84 123|: <<6 5 -6||, <<1 -8 -15||<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="x7-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit</h1> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x7-limit-Temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Temperaments</h2> <br /> ammonite7: <<9 15 19 3 5 2||<br /> armodue7: <<1 -3 5 -7 5 20||<br /> augene7: <<3 0 -6 -7 -18 -14||<br /> august7: <<3 0 6 -7 1 14||<br /> baba7: <<2 -2 1 -8 -4 8||<br /> beatles7: <<2 -9 -4 -19 -12 16||<br /> beep7: <<2 3 1 0 -4 -6||<br /> bipelog7: <<2 -6 -6 -14 -15 3||<br /> blacksmith7: <<0 5 0 8 0 -14||<br /> catalan7: <<6 5 -12 -6 -36 -42||<br /> charon7: <<2 4 4 2 1 -2||<br /> decimal7: <<4 2 2 -6 -8 -1||<br /> dichotic7: <<2 1 -4 -3 -12 -12||<br /> dicot7: <<2 1 3 -3 -1 4||<br /> diminished7: <<4 4 4 -3 -5 -2||<br /> dominant7: <<1 4 -2 4 -6 -16||<br /> father7: <<1 -1 3 -4 2 10||<br /> fifive7: <<10 14 14 -1 -6 -7||<br /> flattone7: <<1 4 -9 4 -17 -32||<br /> garibaldi7: <<1 -8 -14 -15 -25 -10||<br /> godzilla7: <<2 8 1 8 -4 -20||<br /> hystrix7: <<3 5 1 1 -7 -12||<br /> immunity7: <<2 13 1 16 -4 -34||<br /> inflated7: <<3 0 9 -7 6 21||<br /> injera7: <<2 8 8 8 7 -4||<br /> jamesbond7: <<0 0 7 0 11 16||<br /> keemun7: <<6 5 3 -6 -12 -7||<br /> lemba7: <<6 -2 -2 -17 -20 1||<br /> magic7: <<5 1 12 -10 5 25||<br /> meantone7: <<1 4 10 4 13 12||<br /> mother7: <<1 -1 -2 -4 -6 -2||<br /> mothra7: <<3 12 -1 12 -10 -36||<br /> nautilus7: <<6 10 3 2 -12 -21||<br /> negri7: <<4 -3 2 -14 -8 13||<br /> orwell7: <<7 -3 8 -21 -7 27||<br /> pajara7: <<2 -4 -4 -11 -12 2||<br /> passion7: <<5 -4 -10 -18 -30 -12||<br /> pelogic7: <<1 -3 -4 -7 -9 -1||<br /> plutus7: <<1 4 5 4 5 0||<br /> porcupine7: <<3 5 -6 1 -18 -28||<br /> progress7: <<3 -5 -6 -15 -18 0||<br /> progression7: <<5 3 7 -7 -3 8||<br /> quartonic7: <<11 18 5 3 -23 -39||<br /> rodan7: <<3 17 -1 20 -10 -50||<br /> schism7: <<1 -8 -2 -15 -6 18||<br /> sensi7: <<7 9 13 -2 1 5||<br /> sharp7: <<2 1 6 -3 4 11||<br /> sidi7: <<4 2 9 -6 3 15||<br /> superkleismic7: <<9 10 -3 -5 -30 -35||<br /> superpyth7: <<1 9 -2 12 -6 -30||<br /> ternary7: <<0 0 3 0 5 7||<br /> valentine7: <<9 5 -3 -13 -30 -21||<br /> walid7: <<2 -2 -2 -8 -9 1||<br /> wollemia7: <<4 9 19 5 19 19||<br /> würschmidt7: <<8 1 18 -17 6 39||<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x7-limit-Bases"></a><!-- ws:end:WikiTextHeadingRule:6 -->Bases</h2> <br /> <5 8 12 14|: beep7, mother7, father7<br /> <6 10 14 17|: ternary7, charon7, baba7<br /> <7 11 16 20|: dicot7, plutus7, hystrix7<br /> <8 13 19 23|: father7, walid7, hystrix7<br /> <9 14 21 25|: beep7, pelogic7, august7<br /> <10 16 23 28|: sharp7, blacksmith7, decimal7<br /> <12 19 28 34|: august7, dominant7, pajara7<br /> <14 22 32 39|: jamesbond7, decimal7, godzilla7<br /> <15 24 35 42|: blacksmith7, inflated7, keemun7<br /> <16 25 37 45|: diminished7, armodue7, bipelog7<br /> <17 27 39 48|: dichotic7, sidi7, schism7<br /> <17 27 40 48|: dominant7, progression7, progress7<br /> <19 30 44 53: godzilla7, meantone7, keemun7<br /> <22 35 51 62|: pajara7, magic7, porcupine7<br /> <26 41 60 73|: injera7, lemba7, flattone7<br /> <27 43 63 76|: augene7, superpyth7, sensi7<br /> <29 46 67 81|: negri7, nautilus7, garibaldi7<br /> <31 49 72 87|: meantone7, mothra7, orwell7<br /> <34 54 79 95|: keemun7, immunity7, wollemia7<br /> <34 54 79 96|: pajara7, fifive7, würschmidt7<br /> <37 59 86 104|: porcupine7, beatles7, ammonite7<br /> <41 65 95 115|: magic7, garibaldi7, superkleismic7<br /> <46 73 107 129|: sensi7, valentine7, rodan7<br /> <49 78 114 138}: superpyth7, passion7, catalan7<br /> <53 84 123 149|: garibaldi7, orwell7, quartonic7<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="x11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h1> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x11-limit-Temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->Temperaments</h2> <br /> august11: <<3 0 6 6 -7 1 -1 14 14 -4||<br /> diminished11: <<4 4 4 0 -3 -5 -14 -2 -14 -14||<br /> domineering11: <<1 4 -2 6 4 -6 6 -16 0 24||<br /> godzilla11: <<2 8 1 12 8 -4 12 -20 0 30||<br /> keemun11: <<6 5 3 -2 -6 -12 -24 -7 -22 -16||<br /> meanenneadecal11: <<1 4 10 6 4 13 6 12 0 -18||<br /> negri11: <<4 -3 2 5 -14 -8 -6 13 22 7||<br /> pajaric11: <<2 -4 -4 0 -11 -12 -7 2 14 14||<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x11-limit-Bases"></a><!-- ws:end:WikiTextHeadingRule:12 -->Bases</h2> <br /> <12 19 28 34 42|: august11, domineering11, diminished11, pajaric11<br /> <19 30 44 53 66|: godzilla11, meanenneadecal11, negri11, keemun11</body></html>