32201 temps

Gene, your zip file is corrupted.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Gene, your zip file is corrupted.
>
> -C.

Thanks, I uploaded again.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > Gene, your zip file is corrupted.
> >
> > -C.
>
> Thanks, I uploaded again.

Thanks, Gene. I finally got this into Matlab using 'importdata' and
then

for j=1:32201;
s(j,:)=str2num(cell2mat(sy(j)));
end

I graphed it, and I see the same constellations as before, but
there's "stuff" in the region where there is nothing. This is what I
wanted. Thanks.

I noticed that the great majority of the temperaments in the list had
extremely high complexity, but virtually none -- a tiny, tiny number -
- of them had very high error.

Comments welcome.

-Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> I noticed that the great majority of the temperaments in the list
had
> extremely high complexity, but virtually none -- a tiny, tiny
number -
> - of them had very high error.

One would not expect to get an infinite number of high-error
temperament candidates, so why is this a surprise?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > I noticed that the great majority of the temperaments in the list
> had
> > extremely high complexity, but virtually none -- a tiny, tiny
> number -
> > - of them had very high error.
>
> One would not expect to get an infinite number of high-error
> temperament candidates, so why is this a surprise?

I would expect that, at high levels of complexity, one would see a
large number of high-error temperament candidates.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> I would expect that, at high levels of complexity, one would see a
> large number of high-error temperament candidates.

Why? The number of larger commas is limited on any list, the number
of small ets, the same. What method is going to produce for you a
large number of high-error garbage temperaments? On the other hand,
high complexity garbage temperaments I would expect, a priori, out
the wazoo.

Hi Gene.

> > I would expect that, at high levels of complexity, one would see
a
> > large number of high-error temperament candidates.
>
> Why? The number of larger commas is limited on any list,

Larger in terms of their sizes in cents? I don't see why that is
necessarily limited on any list. I had no trouble finding myriad
ridiculously large 'commas' in the 5-limit case I posted extensively
on.

> the number
> of small ets, the same.

It's true that there are a limited number of small ETs. But there
aren't a limited number of large, garbage ET breeds (vals).

> What method is going to produce for you a
> large number of high-error garbage temperaments?

Using ridiculously large commas, for example, or using garbage ET
breeds (vals).

I do realize that each of the items in your kitchen sink
was 'limited' in the ways you seem to be claiming 'any' list would be
(maybe you actually mean 'any reasonable' and not 'any'?), but I was
still surprised as to the extremely small number of large-error
temperaments in your list of 32201 -- and less so, by the vast that
the vast majority of these had extremely large complexity.

Thanks again for doing this work,
Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Using ridiculously large commas, for example, or using garbage ET
> breeds (vals).

Using garbage commas or garbage vals will definately create garbage
results, but why would anyone want garbage results? If you simply
want all possibilities, those can be obtained by requiring the six
integers of the wedgie to be in appropriate reduced form and satify a
single algebraic conditions (wedgies are all rational points on a
projective variety in mathspeak.)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Using ridiculously large commas, for example, or using garbage ET
> > breeds (vals).
>
> Using garbage commas or garbage vals will definately create garbage
> results, but why would anyone want garbage results?

No one would. That's why we're calling them "garbage". It just
strikes me that near the logarithmic error and complexity of 7-limit
blackwood, we hit the boundary of your search.

> If you simply
> want all possibilities, those can be obtained by requiring the six
> integers of the wedgie to be in appropriate reduced form and satify
a
> single algebraic conditions (wedgies are all rational points on a
> projective variety in mathspeak.)

I would have thought the six integers would need to have a GCD of 1,
and with that restriction alone you'd get two and only two
occurrences of each temperament. But I take it it's more complicated
than that?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> I would have thought the six integers would need to have a GCD of
1,
> and with that restriction alone you'd get two and only two
> occurrences of each temperament. But I take it it's more
complicated
> than that?

You can normalize by making the GCD 1, and then normalizing the sign,
for instance by making the first nonzero coefficient positive.
Another approach is simply to take any vector of six rational numbers
equivalent to any other under multiplication by a nonzero scalar;
this makes the five-dimensional projective space over the rationals.
Either way, that is still not enough conditions, if
<<x1 x2 x3 x4 x5 x6|| is our wedgie, we also must have that

x1x6 - x2x5 + x3x4 = 0

This defines a 4D quadric in 5D (real or complex) projective space;
it has an infinity of rational points, each of which corresponds to a
wedgie. There's some math due to Klein connected to all of this.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Either way, that is still not enough conditions, if
> <<x1 x2 x3 x4 x5 x6|| is our wedgie, we also must have that
>
> x1x6 - x2x5 + x3x4 = 0

Aha! This is new.

> This defines a 4D quadric in 5D (real or complex) projective space;
> it has an infinity of rational points, each of which corresponds to
a
> wedgie. There's some math due to Klein connected to all of this.

So the space of 7-limit "linear" temperaments is essentially only 4-
dimensional? I was thinking 6-dimensional but then 1 dimension is
redundant because of torsion considerations. So in musical terms,
what consideration brings us down another dimension?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > Either way, that is still not enough conditions, if
> > <<x1 x2 x3 x4 x5 x6|| is our wedgie, we also must have that
> >
> > x1x6 - x2x5 + x3x4 = 0
>
> Aha! This is new.
>
> > This defines a 4D quadric in 5D (real or complex) projective
space;
> > it has an infinity of rational points, each of which corresponds
to
> a
> > wedgie. There's some math due to Klein connected to all of this.
>
> So the space of 7-limit "linear" temperaments is essentially only 4-
> dimensional?

Correct; they correspond to what a number theorist or algebraic
geometer would call the "Q-rational points" on the "Klein quadric",
which is the x1x6-x2x5+x3x4 thingie; Q being the rational numbers. So
we have a 4D quadric and an infinity of rational points on it, in 1-1
correspondence to 7-limit linear temperaments in some sense. I say in
some sense because they can be arbitarily crappy.

I was thinking 6-dimensional but then 1 dimension is
> redundant because of torsion considerations. So in musical terms,
> what consideration brings us down another dimension?

In musical terms, you ask? That bears thinking about. The main thing
is anything else doesn't give you an actual temperament.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> I was thinking 6-dimensional but then 1 dimension is
> > redundant because of torsion considerations. So in musical terms,
> > what consideration brings us down another dimension?
>
> In musical terms, you ask? That bears thinking about. The main
thing
> is anything else doesn't give you an actual temperament.

In geometric terms?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > I was thinking 6-dimensional but then 1 dimension is
> > > redundant because of torsion considerations. So in musical
terms,
> > > what consideration brings us down another dimension?
> >
> > In musical terms, you ask? That bears thinking about. The main
> thing
> > is anything else doesn't give you an actual temperament.
>
> In geometric terms?

Here's an on-line math book ("Projective and Polar Spaces", by Peter
Cameron, revised on-line edition) with some relevant information:

http://www.maths.qmul.ac.uk/~pjc/pps/

Chapter eight is "The Klein quadric and triality", which discusses
the Klein quadric (obviously) and the Klein correspondence. Chapter
ten is "Exterior powers and Clifford algebras" which might be another
place to find out about these.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> >
> > > I was thinking 6-dimensional but then 1 dimension is
> > > > redundant because of torsion considerations. So in musical
> terms,
> > > > what consideration brings us down another dimension?
> > >
> > > In musical terms, you ask? That bears thinking about. The main
> > thing
> > > is anything else doesn't give you an actual temperament.
> >
> > In geometric terms?
>
> Here's an on-line math book ("Projective and Polar Spaces", by
Peter
> Cameron, revised on-line edition) with some relevant information:
>
> http://www.maths.qmul.ac.uk/~pjc/pps/
>
> Chapter eight is "The Klein quadric and triality", which discusses
> the Klein quadric (obviously) and the Klein correspondence. Chapter
> ten is "Exterior powers and Clifford algebras" which might be
another
> place to find out about these.

Thanks. I've bookmarked this for later reference. Time to finish the
paper.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Thanks. I've bookmarked this for later reference. Time to finish
the
> paper.

It doesn't look that digestible; I may write a Wikipedia article.