View source for Minkowski reduced bases for Fokker groups of certain vals

{{Todo|inline=1|rework|text=Try explaining without wedgies. }}

It is sometimes convenient (e.g. when working with Fokker blocks) 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 val V, there is a corresponding group of bivals, Fokk(V), consisting of all W = V∧A for vals A in the group to which V belongs. Since V∧A + V∧B = V∧(A + B), Fokk(V) consists of simple bivals ("blades"); meaning they each reduce to a wedgie and correspond, except for the zero element, to an [[Abstract_regular_temperament|abstract regular temperament]]. Fokk(V) can be given a basis consisting of wedgies, and these wedgies can be made to be those of low complexity, which the Minkowski reduction accomplishes. If low complexity is not a consideration, the Hermite normal form reduction of [V∧e2, V∧e3, ... V∧ep], (where ek is the valuation val corresponding to the prime k, meaning ek(k)=1, ek(l)=0 for all other primes) gives a canonically defined basis. 

=5-limit=

&lt;5 8 12|: &lt;&lt;1 -1 -4||, &lt;&lt;2 3 0||

&lt;7 11 16|: &lt;&lt;2 1 -3||, &lt;&lt;1 4 4||

&lt;8 13 19|: &lt;&lt;1 -1 -4||, &lt;&lt;3 5 1||

&lt;9 14 21|: &lt;&lt;2 3 0||, &lt;&lt;3 0 -7||

&lt;10 16 23|: &lt;&lt;2 1 -3||, &lt;&lt;0 5 8||

&lt;12 19 28|: &lt;&lt;1 4 4||, &lt;&lt;3 0 -7||

&lt;14 22 33|: &lt;&lt;2 3 0||, &lt;&lt;4 -1 -11||

&lt;15 24 35|: &lt;&lt;3 0 -7||, &lt;&lt;3 5 1||

&lt;16 25 37|: &lt;&lt;1 -3 -7||, &lt;&lt;4 4 -3||

&lt;17 27 39|: &lt;&lt;2 1 -3||, &lt;&lt;1 9 12||

&lt;17 27 40|: &lt;&lt;1 4 4||, &lt;&lt;4 -1 -11||

&lt;18 29 42}: &lt;&lt;3 0 -7||, &lt;&lt;2 6 5||

&lt;19 30 44|: &lt;&lt;1 4 4||, &lt;&lt;5 1 -10||

&lt;22 35 51|: &lt;&lt;3 5 1||, &lt;&lt;2 -4 -11||

&lt;31 49 72|: &lt;&lt;1 4 4||, &lt;&lt;8 1 -17||

&lt;34 54 79|: &lt;&lt;2 -4 -11||, &lt;&lt;6 5 -6||

&lt;41 65 95|: &lt;&lt;5 1 -10||, &lt;&lt;4 9 5||

&lt;46 73 107|: &lt;&lt;2 -4 -11||, &lt;&lt;7 9 -2||

&lt;53 84 123|: &lt;&lt;6 5 -6||, &lt;&lt;1 -8 -15||

=7-limit=

==Temperaments==

ammonite7: &lt;&lt;9 15 19 3 5 2||

armodue7: &lt;&lt;1 -3 5 -7 5 20||

augene7: &lt;&lt;3 0 -6 -7 -18 -14||

august7: &lt;&lt;3 0 6 -7 1 14||

baba7: &lt;&lt;2 -2 1 -8 -4 8||

beatles7: &lt;&lt;2 -9 -4 -19 -12 16||

beep7: &lt;&lt;2 3 1 0 -4 -6||

bipelog7: &lt;&lt;2 -6 -6 -14 -15 3||

blacksmith7: &lt;&lt;0 5 0 8 0 -14||

catalan7: &lt;&lt;6 5 -12 -6 -36 -42||

charon7: &lt;&lt;2 4 4 2 1 -2||

crepuscular7: &lt;&lt;10 14 14 -1 -6 -7||

decimal7: &lt;&lt;4 2 2 -6 -8 -1||

dichotic7: &lt;&lt;2 1 -4 -3 -12 -12||

dicot7: &lt;&lt;2 1 3 -3 -1 4||

diminished7: &lt;&lt;4 4 4 -3 -5 -2||

dominant7: &lt;&lt;1 4 -2 4 -6 -16||

father7: &lt;&lt;1 -1 3 -4 2 10||

flattone7: &lt;&lt;1 4 -9 4 -17 -32||

garibaldi7: &lt;&lt;1 -8 -14 -15 -25 -10||

godzilla7: &lt;&lt;2 8 1 8 -4 -20||

hystrix7: &lt;&lt;3 5 1 1 -7 -12||

immunity7: &lt;&lt;2 13 1 16 -4 -34||

inflated7: &lt;&lt;3 0 9 -7 6 21||

injera7: &lt;&lt;2 8 8 8 7 -4||

jamesbond7: &lt;&lt;0 0 7 0 11 16||

keemun7: &lt;&lt;6 5 3 -6 -12 -7||

lemba7: &lt;&lt;6 -2 -2 -17 -20 1||

magic7: &lt;&lt;5 1 12 -10 5 25||

meantone7: &lt;&lt;1 4 10 4 13 12||

mother7: &lt;&lt;1 -1 -2 -4 -6 -2||

mothra7: &lt;&lt;3 12 -1 12 -10 -36||

nautilus7: &lt;&lt;6 10 3 2 -12 -21||

negri7: &lt;&lt;4 -3 2 -14 -8 13||

orwell7: &lt;&lt;7 -3 8 -21 -7 27||

pajara7: &lt;&lt;2 -4 -4 -11 -12 2||

passion7: &lt;&lt;5 -4 -10 -18 -30 -12||

pelogic7: &lt;&lt;1 -3 -4 -7 -9 -1||

plutus7: &lt;&lt;1 4 5 4 5 0||

porcupine7: &lt;&lt;3 5 -6 1 -18 -28||

progress7: &lt;&lt;3 -5 -6 -15 -18 0||

progression7: &lt;&lt;5 3 7 -7 -3 8||

quartonic7: &lt;&lt;11 18 5 3 -23 -39||

rodan7: &lt;&lt;3 17 -1 20 -10 -50||

schism7: &lt;&lt;1 -8 -2 -15 -6 18||

sensi7: &lt;&lt;7 9 13 -2 1 5||

sharp7: &lt;&lt;2 1 6 -3 4 11||

sidi7: &lt;&lt;4 2 9 -6 3 15||

superkleismic7: &lt;&lt;9 10 -3 -5 -30 -35||

superpyth7: &lt;&lt;1 9 -2 12 -6 -30||

ternary7: &lt;&lt;0 0 3 0 5 7||

valentine7: &lt;&lt;9 5 -3 -13 -30 -21||

walid7: &lt;&lt;2 -2 -2 -8 -9 1||

wollemia7: &lt;&lt;4 9 19 5 19 19||

würschmidt7: &lt;&lt;8 1 18 -17 6 39||

==Bases==

&lt;5 8 12 14|: beep7, mother7, father7

&lt;6 10 14 17|: ternary7, charon7, baba7

&lt;7 11 16 20|: dicot7, plutus7, hystrix7

&lt;8 13 19 23|: father7, walid7, hystrix7

&lt;9 14 21 25|: beep7, pelogic7, august7

&lt;10 16 23 28|: sharp7, blacksmith7, decimal7

&lt;12 19 28 34|: august7, dominant7, pajara7

&lt;14 22 32 39|: jamesbond7, decimal7, godzilla7

&lt;15 24 35 42|: blacksmith7, inflated7, keemun7

&lt;16 25 37 45|: diminished7, armodue7, bipelog7

&lt;17 27 39 48|: dichotic7, sidi7, schism7

&lt;17 27 40 48|: dominant7, progression7, progress7

&lt;19 30 44 53: godzilla7, meantone7, keemun7

&lt;22 35 51 62|: pajara7, magic7, porcupine7

&lt;26 41 60 73|: injera7, lemba7, flattone7

&lt;27 43 63 76|: augene7, superpyth7, sensi7

&lt;29 46 67 81|: negri7, nautilus7, garibaldi7

&lt;31 49 72 87|: meantone7, mothra7, orwell7

&lt;34 54 79 95|: keemun7, immunity7, wollemia7

&lt;34 54 79 96|: pajara7, crepuscular7, würschmidt7

&lt;37 59 86 104|: porcupine7, beatles7, ammonite7

&lt;41 65 95 115|: magic7, garibaldi7, superkleismic7

&lt;46 73 107 129|: sensi7, valentine7, rodan7

&lt;49 78 114 138|: superpyth7, passion7, catalan7

&lt;53 84 123 149|: garibaldi7, orwell7, quartonic7

=11-limit=

==Temperaments==

august11: &lt;&lt;3 0 6 6 -7 1 -1 14 14 -4||

cassandra11: &lt;&lt;1 -8 -14 -18 -15 -25 -32 -10 -14 -2||

cataclysmic11: &lt;&lt;6 5 22 32 -6 18 30 37 57 14||

catcall11: &lt;&lt;0 0 12 12 0 19 19 28 28 -8||

compton11: &lt;&lt;0 12 24 36 19 38 57 22 42 18||

decibel11: &lt;&lt;4 2 2 0 -6 -8 -14 -1 -7 -7||

diaschismic11: &lt;&lt;2 -4 -16 -24 -11 -31 -45 -26 -42 -12||

dicot11: &lt;&lt;2 1 3 5 -3 -1 1 4 8 4||

diminished11: &lt;&lt;4 4 4 0 -3 -5 -14 -2 -14 -14||

domineering11: &lt;&lt;1 4 -2 6 4 -6 6 -16 0 24||

doublewide11: &lt;&lt;8 6 6 -4 -9 -13 -34 -3 -30 -32||

ferrier11: &lt;&lt;0 5 0 10 8 0 16 -14 6 28||

echidna11: &lt;&lt;6 -12 10 -14 -33 -1 -43 57 9 -74||

eudicot11: &lt;&lt;2 1 3 -2 -3 -1 -10 4 -8 -16||

ferrum11:  &lt;&lt;0 5 0 5 8 0 8 -14 -6 14||

flattone11: &lt;&lt;1 4 -9 6 4 -17 6 -32 0 48|| 

godzilla11: &lt;&lt;2 8 1 12 8 -4 12 -20 0 30||

hedgehog11: &lt;&lt;6 10 10 8 2 -1 -8 -5 -16 -12||

hemififths11: &lt;&lt;2 25 13 5 35 15 1 -40 -75 -31||

hemikleismic11: &lt;&lt;12 10 -9 11 -12 -48 -24 -49 -9 62||

hemithirds11: &lt;&lt;15 -2 -5 22 -38 -50 -17 -6 58 79||

hystrix11: &lt;&lt;3 5 1 4 1 -7 -4 -12 -8 8||

inflated11: &lt;&lt;3 0 9 9 -7 6 4 21 21 -6|| 

injera11: &lt;&lt;2 8 8 12 8 7 12 -4 0 6||

keemun11: &lt;&lt;6 5 3 -2 -6 -12 -24 -7 -22 -16||

lemba11: &lt;&lt;6 -2 -2 10 -17 -20 -5 1 30 35||

magic11: &lt;&lt;5 1 12 -8 -10 5 -30 25 -22 -64||

maja11: &lt;&lt;17 23 27 20 -3 -5 -27 -2 -33 -37||

meanenneadecal11: &lt;&lt;1 4 10 6 4 13 6 12 0 -18||

meansept7: &lt;&lt;1 4 5 6 4 5 6 0 0 0||

meantone11: &lt;&lt;1 4 10 18 4 13 25 12 28 16||

miracle11: &lt;&lt;6 -7 -2 15 -25 -20 3 15 59 49||

mohamaq11: &lt;&lt;2 8 13 5 8 15 1 8 -16 -31||

myna11: &lt;&lt;10 9 7 25 -9 -17 5 -9 27 46||

mystery11: &lt;&lt;0 29 29 29 46 46 46 -14 -33 -19||

nautilus11: &lt;&lt;6 10 3 8 2 -12 -8 -21 -16 12||

negri11: &lt;&lt;4 -3 2 5 -14 -8 -6 13 22 7||

negroni11: &lt;&lt;4 -3 2 15 -14 -8 10 13 45 35||

octokaidecal11: &lt;&lt;2 6 6 0 5 4 -7 -3 -21 -21||

opossum11: &lt;&lt;3 5 9 4 1 6 -4 7 -8 -20||

orwell11: &lt;&lt;7 -3 8 2 -21 -7 -21 27 15 -22||

pajaric11: &lt;&lt;2 -4 -4 0 -11 -12 -7 2 14 14||

pajara11: &lt;&lt;2 -4 -4 -12 -11 -12 -26 2 -14 -20||

pelogic11: &lt;&lt;1 -3 -4 -1 -7 -9 -5 -1 8 11||

pento11: &lt;&lt;2 3 1 7 0 -4 4 -6 6 16||

pentoid11: &lt;&lt;2 3 1 -2 0 -4 -10 -6 -15 -9||

porcupine11: &lt;&lt;3 5 -6 4 1 -18 -4 -28 -8 32||

porky11: &lt;&lt;3 5 16 4 1 17 -4 23 -8 -44||

progression11: &lt;&lt;5 3 7 4 -7 -3 -11 8 -1 -13||

rodan11: &lt;&lt;3 17 -1 -13 20 -10 -31 -50 -89 -33||

sensa11: &lt;&lt;9 -7 26 -10 -32 16 -47 80 1 -118||

sensor11: &lt;&lt;7 9 13 -15 -2 1 -48 5 -66 -87||

sharp11: &lt;&lt;2 1 6 5 -3 4 1 11 8 -7||

shrutar11: &lt;&lt;4 -8 14 -2 -22 11 -17 55 23 -54||

squares11: &lt;&lt;4 16 9 10 16 3 2 -24 -32 -3||

superkleismic11: &lt;&lt;9 10 -3 2 -5 -30 -28 -35 -30 16||

telepathy11: &lt;&lt;5 1 12 14 -10 5 5 25 29 -2||

triforce11: &lt;&lt;6 0 3 3 -14 -12 -16 7 7 -2||

unidec11: &lt;&lt;12 22 -4 -6 7 -40 -51 -71 -90 -3||

valentine11: &lt;&lt;9 5 -3 7 -13 -30 -20 -21 -1 30|| 

varan11: &lt;&lt;2 8 1 17 8 -4 20 -20 12 44||

wizard11: &lt;&lt;12 -2 20 -6 -31 -2 -51 52 -7 -86||

==Bases==

&lt;7 11 16 20 24|: dicot11, meansept7, eudicot11, hystrix11

&lt;9 14 21 25 31|: pentoid11, pento11, pelogic11, progression11

&lt;10 16 23 28 35|: sharp11, ferrum11, decibel11, octokaidecal11

&lt;12 19 28 34 42|: august11, domineering11, diminished11, pajaric11

&lt;15 24 35 42 52|: ferrier11, opossum11, inflated11, triforce11

&lt;19 30 44 53 66|: godzilla11, meanenneadecal11, negri11, keemun11

&lt;22 35 51 62 76|: telepathy11, porcupine11, hedgehog11, pajara11

&lt;24 38 56 67 83|: triforce11, catcall11, mohamaq11, varan11

&lt;26 41 60 73 90|: injera11, lemba11, flattone11, doublewide11

&lt;29 46 67 81 100|: nautilus11, negroni11, porky11, cassandra11

&lt;31 49 72 87 107|: meantone11, orwell11, valentine11, squares11

&lt;41 65 95 115 142|: cassandra11, magic11, superkleismic11, miracle11

&lt;46 73 107 129 159|: valentine11, diaschismic11, shrutar11, sensor11

&lt;53 84 123 149 183|: orwell11, cataclysmic11, maja11, hemikleismic11

&lt;58 92 135 163 201|: myna11, diaschismic11, echidna11, hemififths11

&lt;72 114 167 202 249|: miracle11, wizard11, compton11, unidec11

&lt;87 138 202 244 301: rodan11, hemithirds11, sensa11, mystery11

[[Category:Fokker block]][[Category:Math]]