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 val V, there is a corresponding group if 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 simple bivals ("blades"); meaning they each reduce to a wedgie and correspond, except for the zero element, to a 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.

=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

Original HTML content:

<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 val V, there is a corresponding group if 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 simple bivals (&quot;blades&quot;); meaning they each reduce to a wedgie and correspond, except for the zero element, to a 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x5-limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit</h1>
<br />
&lt;12 19 28|: &lt;&lt;1 4 4||, &lt;&lt;3 0 -7||<br />
&lt;15 24 35|: &lt;&lt;3 0 -7||, &lt;&lt;3 5 1||<br />
&lt;17 27 39|: &lt;&lt;2 1 -3||, &lt;&lt;1 9 12||<br />
&lt;17 27 40|: &lt;&lt;1 4 4||, &lt;&lt;4 -1 -11||<br />
&lt;19 30 44|: &lt;&lt;1 4 4||, &lt;&lt;5 1 -10||<br />
&lt;22 35 51|: &lt;&lt;3 5 1||, &lt;&lt;2 -4 -11||<br />
&lt;31 49 72|: &lt;&lt;1 4 4||, &lt;&lt;8 1 -17||<br />
&lt;34 54 79|: &lt;&lt;2 -4 -11||, &lt;&lt;6 5 -6||<br />
&lt;41 65 95|: &lt;&lt;5 1 -10||, &lt;&lt;4 9 5||<br />
&lt;46 73 107|: &lt;&lt;2 -4 -11||, &lt;&lt;7 9 -2||<br />
&lt;53 84 123|: &lt;&lt;6 5 -6||, &lt;&lt;1 -8 -15||<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x7-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit</h1>
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x7-limit-Temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Temperaments</h2>
<br />
ammonite7: &lt;&lt;9 15 19 3 5 2||<br />
armodue7: &lt;&lt;1 -3 5 -7 5 20||<br />
augene7: &lt;&lt;3 0 -6 -7 -18 -14||<br />
august7: &lt;&lt;3 0 6 -7 1 14||<br />
baba7: &lt;&lt;2 -2 1 -8 -4 8||<br />
beatles7: &lt;&lt;2 -9 -4 -19 -12 16||<br />
beep7: &lt;&lt;2 3 1 0 -4 -6||<br />
bipelog7: &lt;&lt;2 -6 -6 -14 -15 3||<br />
blacksmith7: &lt;&lt;0 5 0 8 0 -14||<br />
catalan7: &lt;&lt;6 5 -12 -6 -36 -42||<br />
charon7: &lt;&lt;2 4 4 2 1 -2||<br />
decimal7: &lt;&lt;4 2 2 -6 -8 -1||<br />
dichotic7: &lt;&lt;2 1 -4 -3 -12 -12||<br />
dicot7: &lt;&lt;2 1 3 -3 -1 4||<br />
diminished7: &lt;&lt;4 4 4 -3 -5 -2||<br />
dominant7: &lt;&lt;1 4 -2 4 -6 -16||<br />
father7: &lt;&lt;1 -1 3 -4 2 10||<br />
fifive7: &lt;&lt;10 14 14 -1 -6 -7||<br />
flattone7: &lt;&lt;1 4 -9 4 -17 -32||<br />
garibaldi7: &lt;&lt;1 -8 -14 -15 -25 -10||<br />
godzilla7: &lt;&lt;2 8 1 8 -4 -20||<br />
hystrix7: &lt;&lt;3 5 1 1 -7 -12||<br />
immunity7: &lt;&lt;2 13 1 16 -4 -34||<br />
inflated7: &lt;&lt;3 0 9 -7 6 21||<br />
injera7: &lt;&lt;2 8 8 8 7 -4||<br />
jamesbond7: &lt;&lt;0 0 7 0 11 16||<br />
keemun7: &lt;&lt;6 5 3 -6 -12 -7||<br />
lemba7: &lt;&lt;6 -2 -2 -17 -20 1||<br />
magic7: &lt;&lt;5 1 12 -10 5 25||<br />
meantone7: &lt;&lt;1 4 10 4 13 12||<br />
mother7: &lt;&lt;1 -1 -2 -4 -6 -2||<br />
mothra7: &lt;&lt;3 12 -1 12 -10 -36||<br />
nautilus7: &lt;&lt;6 10 3 2 -12 -21||<br />
negri7: &lt;&lt;4 -3 2 -14 -8 13||<br />
orwell7: &lt;&lt;7 -3 8 -21 -7 27||<br />
pajara7: &lt;&lt;2 -4 -4 -11 -12 2||<br />
passion7: &lt;&lt;5 -4 -10 -18 -30 -12||<br />
pelogic7: &lt;&lt;1 -3 -4 -7 -9 -1||<br />
plutus7: &lt;&lt;1 4 5 4 5 0||<br />
porcupine7: &lt;&lt;3 5 -6 1 -18 -28||<br />
progress7: &lt;&lt;3 -5 -6 -15 -18 0||<br />
progression7: &lt;&lt;5 3 7 -7 -3 8||<br />
quartonic7: &lt;&lt;11 18 5 3 -23 -39||<br />
rodan7: &lt;&lt;3 17 -1 20 -10 -50||<br />
schism7: &lt;&lt;1 -8 -2 -15 -6 18||<br />
sensi7: &lt;&lt;7 9 13 -2 1 5||<br />
sharp7: &lt;&lt;2 1 6 -3 4 11||<br />
sidi7: &lt;&lt;4 2 9 -6 3 15||<br />
superkleismic7: &lt;&lt;9 10 -3 -5 -30 -35||<br />
superpyth7: &lt;&lt;1 9 -2 12 -6 -30||<br />
ternary7: &lt;&lt;0 0 3 0 5 7||<br />
valentine7: &lt;&lt;9 5 -3 -13 -30 -21||<br />
walid7: &lt;&lt;2 -2 -2 -8 -9 1||<br />
wollemia7: &lt;&lt;4 9 19 5 19 19||<br />
würschmidt7: &lt;&lt;8 1 18 -17 6 39||<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x7-limit-Bases"></a><!-- ws:end:WikiTextHeadingRule:6 -->Bases</h2>
<br />
&lt;5 8 12 14|: beep7, mother7, father7<br />
&lt;6 10 14 17|: ternary7, charon7, baba7<br />
&lt;7 11 16 20|: dicot7, plutus7, hystrix7<br />
&lt;8 13 19 23|: father7, walid7, hystrix7<br />
&lt;9 14 21 25|: beep7, pelogic7, august7<br />
&lt;10 16 23 28|: sharp7, blacksmith7, decimal7<br />
&lt;12 19 28 34|: august7, dominant7, pajara7<br />
&lt;14 22 32 39|: jamesbond7, decimal7, godzilla7<br />
&lt;15 24 35 42|: blacksmith7, inflated7, keemun7<br />
&lt;16 25 37 45|: diminished7, armodue7, bipelog7<br />
&lt;17 27 39 48|: dichotic7, sidi7, schism7<br />
&lt;17 27 40 48|: dominant7, progression7, progress7<br />
&lt;19 30 44 53: godzilla7, meantone7, keemun7<br />
&lt;22 35 51 62|: pajara7, magic7, porcupine7<br />
&lt;26 41 60 73|: injera7, lemba7, flattone7<br />
&lt;27 43 63 76|: augene7, superpyth7, sensi7<br />
&lt;29 46 67 81|: negri7, nautilus7, garibaldi7<br />
&lt;31 49 72 87|: meantone7, mothra7, orwell7<br />
&lt;34 54 79 95|: keemun7, immunity7, wollemia7<br />
&lt;34 54 79 96|: pajara7, fifive7, würschmidt7<br />
&lt;37 59 86 104|: porcupine7, beatles7, ammonite7<br />
&lt;41 65 95 115|: magic7, garibaldi7, superkleismic7<br />
&lt;46 73 107 129|: sensi7, valentine7, rodan7<br />
&lt;49 78 114 138}: superpyth7, passion7, catalan7<br />
&lt;53 84 123 149|: garibaldi7, orwell7, quartonic7<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="x11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h1>
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x11-limit-Temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->Temperaments</h2>
<br />
august11: &lt;&lt;3 0 6 6 -7 1 -1 14 14 -4||<br />
diminished11: &lt;&lt;4 4 4 0 -3 -5 -14 -2 -14 -14||<br />
domineering11: &lt;&lt;1 4 -2 6 4 -6 6 -16 0 24||<br />
godzilla11: &lt;&lt;2 8 1 12 8 -4 12 -20 0 30||<br />
keemun11: &lt;&lt;6 5 3 -2 -6 -12 -24 -7 -22 -16||<br />
meanenneadecal11: &lt;&lt;1 4 10 6 4 13 6 12 0 -18||<br />
negri11: &lt;&lt;4 -3 2 5 -14 -8 -6 13 22 7||<br />
pajaric11: &lt;&lt;2 -4 -4 0 -11 -12 -7 2 14 14||<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x11-limit-Bases"></a><!-- ws:end:WikiTextHeadingRule:12 -->Bases</h2>
<br />
&lt;12 19 28 34 42|: august11, domineering11, diminished11, pajaric11<br />
&lt;19 30 44 53 66|: godzilla11, meanenneadecal11, negri11, keemun11</body></html>