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Date: Mon, 19 Nov 2001.
Subject: Re: Minkowski reduced lattices
Done. The thinking goes as follows. The "length" of a unison vector n/d,
n~=d, in the Tenney lattice with taxicab metric, or van Prooijen lattice
with triangular-taxicab metric, is proportional to log(n) + log(d)
(hence approx. proportional to log(d)), and also to the "number" (in
some weighted sense) of consonant intervals making up that unison
vector. Thus, in order to temper this unison vector out (assuming that
other UVs being tempered out, if any, are orthogonal to this one), one
must temper each consonant interval involved by an "average" amount
proportional to w/log(d), where w is the musical width of the unison
vector.
w = log(n/d)
w ~= n/d-1
w ~= (n-d)/d
Hence the amount of tempering implied by the unison vector is approx.
proportional to...
(n-d)/(d*log(d))
Yes?
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