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*.