Minkowski reduced lattices

The classic Minkowski reduced lattice definition in the case of a two-
dimensional lattice reduces to saying that one basis element is as
short as possible, and the other is the shortest such that together
with the first it generates the lattice. In small dimensions like
this, it is possible to reduce lattices without worrying about
exponential time. The definition of Minkowski reduced can also be
applied using the Tenney metric.

--- In tuning-math@y..., genewardsmith@j... wrote:
> The classic Minkowski reduced lattice definition in the case of a
two-
> dimensional lattice reduces to saying that one basis element is as
> short as possible, and the other is the shortest such that together
> with the first it generates the lattice. In small dimensions like
> this, it is possible to reduce lattices without worrying about
> exponential time. The definition of Minkowski reduced can also be
> applied using the Tenney metric.

Minkowski reduced basis . . . it it unique?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Minkowski reduced basis . . . it it unique?

Yes, that's basically why I brought it up. I think usually the LLL-
reduced basis in the 7-limit will also be Minkowski reduced, but if
it isn't it would be easy enough to fix.

Here's some questions for you: how did you pick the commas you wanted
to start out from, both this time and the last time? What would be
good cut-off criteria for exluding a basis--for instance, numerator
times denominator > 2000, let us say, and small basis element not
less than a third of the large one?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Minkowski reduced basis . . . it it unique?
>
> Yes, that's basically why I brought it up. I think usually the LLL-
> reduced basis in the 7-limit will also be Minkowski reduced

On the Tenney lattice with a taxicab metric?

>, but if
> it isn't it would be easy enough to fix.
>
> Here's some questions for you: how did you pick the commas you
wanted
> to start out from, both this time and the last time? What would be
> good cut-off criteria for exluding a basis--for instance, numerator
> times denominator > 2000, let us say, and small basis element not
> less than a third of the large one?

I'll have to think hard about this. For now, does the conjecture I've
posted about the relationship between the numbers in a comma's ratio,
and the amount of tempering the comma implied for the constituent
consonant intervals, make any sense to you? This keyboard won't allow
me to type this right now, but you can search for "conjecture" . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > Yes, that's basically why I brought it up. I think usually the
LLL-
> > reduced basis in the 7-limit will also be Minkowski reduced

> On the Tenney lattice with a taxicab metric?

I think so--we are talking about only two vectors, after all, and a
Euclidean distance adjusted to be as much like Tenney as possible.

I'll get back to you if I can find this conjecture you speak of.

--- In tuning-math@y..., genewardsmith@j... wrote:

> I'll get back to you if I can find this conjecture you speak of.

I think you either need to find a better keyboard, or give me an
article number.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > I'll get back to you if I can find this conjecture you speak of.
>
> I think you either need to find a better keyboard

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?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Yes?

Sounds reasonable. Is this a conjecture, and observation or a
heuristic argument? If it's a conjecture it needs to be stated more
precisely.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Yes?
>
> Sounds reasonable. Is this a conjecture, and observation or a
> heuristic argument? If it's a conjecture it needs to be stated more
> precisely.

Heuristic argument. If it can be made more precise I'd be most
grateful to whoever does so.