TM-reduced basis

Hi,

Could someone point me to a good definition of a TM-reduced basis?
I can see that it is arrived at by manipulating rows of a matrix,
(adding and subtracting). Is it always two or more fractions that
have the smallest numerator/denominator combo?

Thanx

Paul

>Could someone point me to a good definition of a TM-reduced basis?
>I can see that it is arrived at by manipulating rows of a matrix,
>(adding and subtracting). Is it always two or more fractions that
>have the smallest numerator/denominator combo?

Hi Paul,

Here are some bits from the aether...

>>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of
>>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q
>>and r/s independent is Minkowski reduced iff the only ratios t/u
>>in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers
>>of p/q.
>
>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 that's
>simpler than the more complex of the pair is by itself.

//

>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.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Could someone point me to a good definition of a TM-reduced basis?
> >I can see that it is arrived at by manipulating rows of a matrix,
> >(adding and subtracting). Is it always two or more fractions that
> >have the smallest numerator/denominator combo?
>
> Hi Paul,
>
> Here are some bits from the aether...
>
> >>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of
> >>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q
> >>and r/s independent is Minkowski reduced iff the only ratios t/u
> >>in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers
> >>of p/q.
> >
> >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 that's
> >simpler than the more complex of the pair is by itself.

Really. Not even in between the two unison vectors?
> //
>
> >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.
>
> -Carl

I'll mull this over. Thanks

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >Could someone point me to a good definition of a TM-reduced
basis?
> > >I can see that it is arrived at by manipulating rows of a matrix,
> > >(adding and subtracting). Is it always two or more fractions that
> > >have the smallest numerator/denominator combo?
> >
> > Hi Paul,
> >
> > Here are some bits from the aether...
> >
> > >>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of
> > >>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q
> > >>and r/s independent is Minkowski reduced iff the only ratios t/u
> > >>in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are
powers
> > >>of p/q.
> > >
> > >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 that's
> > >simpler than the more complex of the pair is by itself.
>
> Really. Not even in between the two unison vectors?

Don't know what you mean by "between" but, a simple example is 81:80
and 128:125 (defining 5-limit 12-equal). You can't get a simpler
comma by multiplying and dividing powers of these.