> i asked Gene for a good definition for TM reduction a
> long time ago ... and Gene, if you gave it to me and i
> lost it in the shuffle, i apologize. can you send it again?
First we need to define Tenney height: if p/q is a positive rational
number in reduced form, then the Tenney height is TH(p/q) = p q.
Now suppose {q1, ..., qn} are n multiplicatively linearly independent
positive rational numbers. Linear independence can be equated, for
instance, with the condition that rank of the matrix whose rows are
the monzos for qi is n. Then {q1, ..., qn} is a basis for a lattice
L, consisting of every positive rational number of the form q1^e1 ...
qn^en where the ei are integers and where the log of the Tenney
height defines a norm. Let t1>1 be the shortest (in terms of Tenney
height) rational number in L greater than 1. Define ti>1 inductively
as the shortest number in L independent of {t1, ... t_{i-1}} and such
that {t1, ..., ti} can be extended to be a basis for L. In this way
we obtain {t1, ..., tn}, the TM reduced basis of L.
hey Carl (and Gene, even moreso), ... i asked Gene for a good definition for TM reduction a long time ago ... and Gene, if you gave it to me and i lost it in...
... First we need to define Tenney height: if p/q is a positive rational number in reduced form, then the Tenney height is TH(p/q) = p q. Now suppose {q1, ...,...
what's the point of defining tenney height as p*q if you're only going to use the log anyway, and tenney harmonic distance is already log(p*q)? ... rational ...
... Aside from the fact that I don't know what elements are in a set notated like {(p/q)^i (r/s)^i}, and I can't fathom the function of t/u in this definition,...
... the last i should be a j. that's the set of ratios that can be expressed as (p/q)^i *times* (r/s)^j. ... the definition should have read, "the only numbers...
... Now it makes sense... ... So IOW, if you have a pair of unison vectors for a PB, you shouldn't be able to stack them both in some way to get an interval...
... Whoops, finger failure. ... I figured it was a multiply, but didn't realize this was what it meant. ... Uh... ... Ok. So how can we get this into monz'...
... yes, a reduced basis will have good straightness, because the set of basis vectors is, in some sense, as short as possible. and, as we discussed before,...
... vector ... are ... space ... i'm confused about that, because wouldn't b_2, b_2 + b_1, b_2 + 2*b_1, b_2 - b+1, etc., all have the same length in the...
hi paul and Carl, ... i understand this, in a nutshell, to mean that the reduction process places the bounding vectors of the periodicity-block as close as...
... These are what I called "chord blocks", which are 7-limit scales analogous to Fokker blocks. This works because 7-limit tetrads, uniquely among prime...
... What are cm1, c1, and s1? ... I remember this stuff. But I don't remember the bit about the 7-limit being unique in this. What is it that makes, say, the...
... s1 is the scale. If I recall correctly, cm1 is the comma basis, and c1 is something obtained from cm1 and used to calculate s1. ... 5-limit triads can be...
... it ... One step takes a C minor chord to a C major chord. Where does the next step go? It doesn't go to a chord at all--we don't have a group, since we...
... Doesn't go to a chord? Aren't you connecting the centers of the triangles? Then I get Cm->CM->C#m. If you connect the roots, or any of the vertices, I...
... I'm connecting the centers of the triangles with a line whenever there is a common line between two of the triangles. This gives hexagons, where you have a...
... So CM and C#m aren't connected then? Why wouldn't you connect them? ... So does connecting the centers of all the triangles. ... If you continue in the...
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