LLL basis for ETs

It would be great to see the LLL bases for all the "good" ETs in each
limit. This should use the log(p) weighting, and, if possible, the
taxicab metric. Also, I'd like to understand the parameter you
mentioned and the optimization criterion too.

Gene, it looks, from your post on the tuning list, that a basis need
not be complete for LLL to be performed on it. So can you perform LLL
on the list of, say, 7-limit linear temperaments represented by any
pair of commatic unison vectors from the list:

49:50
63:64
80:81
125:126
225:224
245:243
1024:1029
2400:2401
4374:4375
and (optionally, if the task is not to onerous)
1715:1728
3125:3136
4374:4375

I'd expect we'd end up with a relatively manageable list of distinct
temperaments, for which the LLL basis will, if nothing else, serve as
a label for each equivalence class of bases.

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

> I'd expect we'd end up with a relatively manageable list of
distinct
> temperaments, for which the LLL basis will, if nothing else, serve
as
> a label for each equivalence class of bases.

That's a thought; I had in mind a similar project, which was to go
through pairs of ets in the 11-limit, and LLL reduce to linear
temperaments. Sometimes there is more that one good way to do this; I
was thinking at least of posting the example of 12 and 34.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I'd expect we'd end up with a relatively manageable list of
> distinct
> > temperaments, for which the LLL basis will, if nothing else,
serve
> as
> > a label for each equivalence class of bases.
>
> That's a thought;

Is it? I think something of the sort, with pretty color graphics and
lots of musical explanations, would make a great paper for XH18
(coming out in Februrary). Would you like to co-author such a paper
with me, or perhaps write one paper upon which I'd draw in one of my
own . . . ?

> I had in mind a similar project, which was to go
> through pairs of ets in the 11-limit, and LLL reduce to linear
> temperaments. Sometimes there is more that one good way to do this;
I
> was thinking at least of posting the example of 12 and 34.

Graham expressed some confusion as to what you were doing with pairs
of ETs and I share his confusion. Perhaps we could attempt to begin
directly from the list of unison vectors, drawing up a list of ETs
and corresponding LLLs (where the "orthogonality" is above some
threshold, if that makes any sense) that result from triplets of UVs,
and a list of linear temperaments and their LLLs (where
the "orthogonality" is above some threshold, if that makes still
makes any sense) that result from pairs of UVs. Hopefully we can use
the Tenney lattice, with a Euclidean metric if need be, or preferably
a taxicab one. We will then be able to see certain cases where the
result of combining two ETs is quite clear -- they'll unambiguously
have two UVs in common.

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

> Is it? I think something of the sort, with pretty color graphics
and
> lots of musical explanations, would make a great paper for XH18
> (coming out in Februrary). Would you like to co-author such a paper
> with me, or perhaps write one paper upon which I'd draw in one of
my
> own . . . ?

I'm not likely to produce pretty color graphics on my own, and my
experiences with attempted publication in non-mathematical forums
suggest to me that I might need a coauthor. If you understood
everything a paper said, and I thought nothing it said was wrong,
that would be a great start. On the other hand there are probably
lots of things we have discussed which could be the basis of such a
project, so are you sure this is the right one to start out with?

Presumably, we don't need to define LLL-reduction, but can cite a
reference.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Is it? I think something of the sort, with pretty color graphics
> and
> > lots of musical explanations, would make a great paper for XH18
> > (coming out in Februrary). Would you like to co-author such a
paper
> > with me, or perhaps write one paper upon which I'd draw in one of
> my
> > own . . . ?
>
> I'm not likely to produce pretty color graphics on my own,

I've already produced some for a few linear temperaments . . . I
thought I'd simply make more.

> If you understood
> everything a paper said, and I thought nothing it said was wrong,
> that would be a great start. On the other hand there are probably
> lots of things we have discussed which could be the basis of such a
> project, so are you sure this is the right one to start out with?

Virtually all of my discussions with you have been geared toward this
one project -- start by motivating periodicity blocks, continue by
motivating temperament of smaller unison vectors, and conclude by
motivating ETs and linear temperaments. In the process one supplies a
few spelled-out examples and a fairly comprehensive list of
possibilities, the list obtained with as few conditions as one can
manage.

> Presumably, we don't need to define LLL-reduction, but can cite a
> reference.

Absolutely.

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

> Virtually all of my discussions with you have been geared toward
this
> one project -- start by motivating periodicity blocks, continue by
> motivating temperament of smaller unison vectors, and conclude by
> motivating ETs and linear temperaments.

Or we could start from ets and detemper to linear and planar
temperaments. I think it's good to point out the duality between ets
and unison vectors.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Virtually all of my discussions with you have been geared toward
> this
> > one project -- start by motivating periodicity blocks, continue
by
> > motivating temperament of smaller unison vectors, and conclude by
> > motivating ETs and linear temperaments.
>
> Or we could start from ets and detemper to linear and planar
> temperaments.

Many of the readers, coming from a JI background, would be most
unhappy if we started from ETs rather than from JI. I'd like the
paper to be capable of convincing one or two people as to the value
of ETs and LTs. Meanwhile, most modern academic microtonal music
theory starts from ETs and MOSs, and do not motivate this starting
point correctly or at all. I'd like to provide that foundation.

Oh yeah, did I mention the Hypothesis would be part of this paper?

> I think it's good to point out the duality between ets
> and unison vectors.

The duality might be a bit too abstract for musicians to grasp or
even care about. And many of the readers, coming from a JI
background, would be most unhappy if we started from ETs rather than
from JI. I'd like the paper to be capable of convincing one or two
people as to the value of ETs. Meanwhile, most modern academic
microtonal music theory starts from ETs, and do not motivate this
starting point correctly or at all.

Gene:
> > I had in mind a similar project, which was to go
> > through pairs of ets in the 11-limit, and LLL reduce to linear
> > temperaments. Sometimes there is more that one good way to do this;
> I
> > was thinking at least of posting the example of 12 and 34.

Note that this project is what my temperament finder is trying to do. Thanks
for the definition of LLL, Gene. I'll read it and see if I can implement it.
So far that's the main weak point of the program.

> Graham expressed some confusion as to what you were doing with pairs
> of ETs and I share his confusion.

The confusion was because he was coming up with unique results for
inconsistent temperaments. Firstly, he's said that he takes the nearest
prime approximations, which should clear up the confusion although I'm not
sure our results agree. Secondly, this is exactly the problem he mentions in
that paragraph!

Graham

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

> The confusion was because he was coming up with unique results for
> inconsistent temperaments. Firstly, he's said that he takes the
nearest
> prime approximations, which should clear up the confusion although
I'm not
> sure our results agree. Secondly, this is exactly the problem he
mentions in
> that paragraph!

The uniqueness was in connection with generators in ets. In my
notation, 12+34 uniquely defines a generator, since we add 12 and 34,
get 46, and take the nearest primes *for 46*.