LLL reduction pairs revised list

My last list allowed in some duplicates; the following list of 81
pairs is from an improved version of the program:

{135/128, 36/35}
{36/35, 200/189}
{405/392, 25/24}
{81/80, 126/125}
{4000/3969, 1728/1715}
{64/63, 245/243}
{25/24, 21/20}
{28/27, 50/49}
{36/35, 225/224}
{160/147, 25/24}
{50/49, 81/80}
{5103/5000, 49/48}
{28/27, 49/48}
{64/63, 126/125}
{1728/1715, 3136/3125}
{128/125, 50/49}
{672/625, 28/27}
{81/80, 3136/3125}
{3200/3087, 28/27}
{1029/1024, 4375/4374}
{245/243, 2048/2025}
{3136/3125, 4375/4374}
{225/224, 1029/1024}
{50/49, 64/63}
{49/48, 392/375}
{28/27, 256/245}
{126/125, 245/243}
{50/49, 245/243}
{50/49, 875/864}
{50/49, 6144/6125}
{25/24, 49/48}
{245/243, 225/224}
{50/49, 4375/4374}
{25/24, 64/63}
{25/24, 49/45}
{126/125, 1029/1024}
{225/224, 4375/4374}
{25/24, 2401/2400}
{81/80, 6144/6125}
{81/80, 1728/1715}
{64/63, 686/675}
{36/35, 1323/1280}
{49/48, 3136/3125}
{64/63, 4375/4374}
{28/27, 126/125}
{49/48, 225/224}
{3200/3087, 25/24}
{81/80, 225/224}
{245/243, 1029/1024}
{245/243, 3136/3125}
{36/35, 50/49}
{28/27, 6144/6125}
{25/24, 243/224}
{4000/3969, 245/243}
{50/49, 525/512}
{36/35, 64/63}
{28/27, 1029/1024}
{1029/1024, 3136/3125}
{25/24, 81/80}
{49/48, 81/80}
{2401/2400, 3136/3125}
{81/80, 2401/2400}
{49/48, 126/125}
{64/63, 875/864}
{36/35, 1029/1000}
{25/24, 28/27}
{28/27, 3125/3024}
{2401/2400, 6144/6125}
{36/35, 5625/5488}
{36/35, 21/20}
{81/80, 875/864}
{36/35, 16/15}
{25/24, 360/343}
{126/125, 1728/1715}
{36/35, 15/14}
{225/224, 1728/1715}
{64/63, 225/224}
{64/63, 3136/3125}
{4375/4374, 6144/6125}
{2401/2400, 4375/4374}
{28/27, 21/20}

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

> {126/125, 1728/1715}

I selected the above at random from my list, and came to the
following conclusions:

(1) It should be discussed in conjunction with the 31+27;58 and
27+4;31, and

(2) It should be discussed in conjunction with a corresponding
11-limit linear temperament.

Is the world of JI ready for this, I wonder? Paul made the music
theory world sound a bit like the big-endians vs the little-endians.

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

>Paul gave a list previously which was a little more inclusive:

>25/24, 28/27, 36/35, 49/48, 50/49, 64/63, 81/80, 126/125, 245/243,
>225/224, 1029/1024, 1728/1715, 2401/2400, 3136/3125, 4375/4374,
>6144/6125

Well, Gene, the first four entries were intended to be used only as
chromatic unison vectors.

Actually, we should be taking three at a time, not two. If all three
(in the Minkowski reduced basis, hopefully) are commatic UVs, we get
an ET. If one is chromatic, we get an MOS. If two are chromatic, a
planar temperament. If three are chromatic, a JI block.

I think the UVs in the reduced basis have to be in Kees's list or
something like it -- if they aren't, then it seems likely that some
step in the scale will be smaller than one of the commatic unison
vectors -- which we shouldn't allow.

Just some imprecise thoughts . . .

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

> Actually, we should be taking three at a time, not two. If all
three
> (in the Minkowski reduced basis, hopefully) are commatic UVs, we
get
> an ET. If one is chromatic, we get an MOS. If two are chromatic, a
> planar temperament. If three are chromatic, a JI block.

This will prevent you from considering a lot of interesting
temperaments.

> I think the UVs in the reduced basis have to be in Kees's list or
> something like it -- if they aren't, then it seems likely that some
> step in the scale will be smaller than one of the commatic unison
> vectors -- which we shouldn't allow.

Where is Kees's list? I was thinking of producing a list based on
some mathematical conditions, and using that, but I wondered if these
lists already represent such an effort.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Actually, we should be taking three at a time, not two. If all
> three
> > (in the Minkowski reduced basis, hopefully) are commatic UVs, we
> get
> > an ET. If one is chromatic, we get an MOS. If two are chromatic,
a
> > planar temperament. If three are chromatic, a JI block.
>
> This will prevent you from considering a lot of interesting
> temperaments.

Such as?

> > I think the UVs in the reduced basis have to be in Kees's list or
> > something like it -- if they aren't, then it seems likely that
some
> > step in the scale will be smaller than one of the commatic unison
> > vectors -- which we shouldn't allow.
>
> Where is Kees's list? I was thinking of producing a list based on
> some mathematical conditions, and using that, but I wondered if
these
> lists already represent such an effort.

Start with http://www.kees.cc/tuning/perbl.html and follow the link
to http://www.kees.cc/tuning/s2357.html. What Kees doesn't tell you
(he told me) is that the unison vectors not in parentheses are the
smallest (in octaves or cents) for their level of expressibility
(i.e., for their length). The ones in parentheses are included so
that you're guaranteed to have the three smallest for any level of
expressibility. Perhaps Kees would like to chime in here?

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

> > This will prevent you from considering a lot of interesting
> > temperaments.

> Such as?

If you take 2401/2400 and 4375/4374 by themselves, you get
ennealimmal temperament. If you try to add 25/24, 28/27, 36/35 you
get garbage. Besides, there's no point in it that I can see--why add
anything when two commas are all you need to define a 7-limit linear
temperament?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > > This will prevent you from considering a lot of interesting
> > > temperaments.
>
> > Such as?
>
> If you take 2401/2400 and 4375/4374 by themselves, you get
> ennealimmal temperament.

I know you love that one.

> If you try to add 25/24, 28/27, 36/35 you
> get garbage.

These are not the only canditates. You left out 49/48. Also, 50/49
and 64/63 can be used as _either_ commatic _or_ chromatic -- I don't
know if I mentioned that but I think so. Other candidates may emerge
once we have a solid foundation for all this.

Anyway, I'd just like to set some reasonable bounds within which we
can flesh out the possibilities. If someone wants to use 36/35 as a
commatic unison vector, I'm all for that, but I'd like to start with
a digestible array of possibilities just for the sake of presentation.

> Besides, there's no point in it that I can see--why add
> anything when two commas are all you need to define a 7-limit
linear
> temperament?

The point is not so much infinite temperaments but rather finite
scales. At least in the approach I envision in this paper. Another
paper could more specifically address infinite temperaments.

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

> The point is not so much infinite temperaments but rather finite
> scales.

Given two generators, you are set to make scales; the LLL reductions
tend to give an interval of equivalence and a generator, which is
perfect. What I like about starting from two commas is your idea to
use this as a canonical scheme for classification, but certainly one
can go on to scales.

At least in the approach I envision in this paper. Another
> paper could more specifically address infinite temperaments.

The scales will presumably be in one of three things: a temperament,
an et, or a p-limit. It seems to me you can't get away from
addressing one or more of these if you are going to work with scales.
Scales are also less easy to classify than temperaments, because
there are more reasonable possibilities. Moreover, if you are willing
to restrict yourself to et scales, my a;n+m notation already does
classify them.

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

> the LLL reductions
> tend to give an interval of equivalence and a generator, which is
> perfect.

What do you mean, exactly? What do the LLL reductions give that an
unreduced basis for a linear temperament don't, in the way of an
interval of equivalence and a generator?
>
> > At least in the approach I envision in this paper. Another
> > paper could more specifically address infinite temperaments.
>
> The scales will presumably be in one of three things: a
temperament,
> an et, or a p-limit.

An ET is a temperament -- but, as I said, some might be in a linear
temperament, some might be in a planar temperament, etc.

> It seems to me you can't get away from
> addressing one or more of these if you are going to work with
scales.

Right -- but that doesn't mean that I have to talk about ennealimmal
temperament if it doesn't give me a scale with a reasonable number of
notes (as I said, I have to delimit this project somewhere!)

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

> My last list allowed in some duplicates; the following list of 81
> pairs is from an improved version of the program:

I'm going to keep the ones where both UVs appear on my original list,
until we have a firmer basis for making such a list. Commas like
4000/3969 seem like unlikely choices, since there are three or more
commas that are both shorter vectors _and_ smaller musical intervals.
Meanwhile, the superparticulars with smaller numbers than 50:49 lead
to too much tempering for my taste.

> {81/80, 126/125}
> {64/63, 245/243}
> {50/49, 81/80}
> {64/63, 126/125}
> {1728/1715, 3136/3125}
> {81/80, 3136/3125}
> {1029/1024, 4375/4374}
> {3136/3125, 4375/4374}
> {225/224, 1029/1024}
> {50/49, 64/63}
> {126/125, 245/243}
> {50/49, 245/243}
> {50/49, 6144/6125}
> {245/243, 225/224}
> {50/49, 4375/4374}
> {126/125, 1029/1024}
> {225/224, 4375/4374}
> {81/80, 6144/6125}
> {81/80, 1728/1715}
> {64/63, 4375/4374}
> {49/48, 225/224}
> {81/80, 225/224}
> {245/243, 1029/1024}
> {245/243, 3136/3125}
> {1029/1024, 3136/3125}
> {2401/2400, 3136/3125}
> {81/80, 2401/2400}
> {2401/2400, 6144/6125}
> {126/125, 1728/1715}
> {225/224, 1728/1715}
> {64/63, 225/224}
> {64/63, 3136/3125}
> {4375/4374, 6144/6125}
> {2401/2400, 4375/4374}

Still a whopping 34 possibilities! Who can be quickest to the draw to
give the generator, period, and 7-limit complexity for all 34 linear
temperaments? If Gene did this right, they should all be
distinct . . . and this very well might be all the "interesting" ones
for my present purposes.

I wrote,

> If Gene did this right, they should all be
> distinct . . .

Well, the least-squares generators should be distinct . . . some of
the minimax generators might turn out to be the same (?)

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

> I'm going to keep the ones where both UVs appear on my original
list,
> until we have a firmer basis for making such a list. Commas like
> 4000/3969 seem like unlikely choices, since there are three or more
> commas that are both shorter vectors _and_ smaller musical
intervals.

However, 4000/3969 shows up as part of the reduced basis for 41, 53
and 68, which are all important ets. I conclude therefore it's
significant, and in any case this gives us a way of picking them.

The alternative approach, of course, is to start from pairs of ets; I
think perhaps we should do it both ways as a check.

> Meanwhile, the superparticulars with smaller numbers than 50:49
lead
> to too much tempering for my taste.

I think drawing the line between 50/49 and 49/48 is a little absurd;
why not between 49/48 and 36/35? The schisma showed up in the reduced
basis for 171; perhaps we should include that, then look at 130 and
140 and call it a day?

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

> I think drawing the line between 50/49 and 49/48 is a little
absurd;
> why not between 49/48 and 36/35? The schisma showed up in the
reduced
> basis for 171; perhaps we should include that, then look at 130 and
> 140 and call it a day?

I get

130: <2401/2400, 3136/3125, 19683/19600>
140: <2401/2400, 5120/5103, 15625/15552>

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I'm going to keep the ones where both UVs appear on my original
> list,
> > until we have a firmer basis for making such a list. Commas like
> > 4000/3969 seem like unlikely choices, since there are three or
more
> > commas that are both shorter vectors _and_ smaller musical
> intervals.
>
> However, 4000/3969 shows up as part of the reduced basis for 41, 53
> and 68, which are all important ets.

Maybe not for the conditions I have in mind.

> I conclude therefore it's
> significant, and in any case this gives us a way of picking them.
>
> The alternative approach, of course, is to start from pairs of ets;
I
> think perhaps we should do it both ways as a check.
>
> > Meanwhile, the superparticulars with smaller numbers than 50:49
> lead
> > to too much tempering for my taste.
>
> I think drawing the line between 50/49 and 49/48 is a little
absurd;
> why not between 49/48 and 36/35?

I wouldn't have a big problem with that -- I'm trying to keep the set
of outcomes as small as possible, without introducing a larger number
of conditions than is reasonable.