Duals to ems optimization

For any set of consonances C we want to do an rms optimization for, we
can find a corresponding Euclidean norm on the val space (or
octave-excluding subspace if we are interested in the odd limit) by
taking the sum of terms

(c2 x2 + c3 x3 + ... + cp xp)^2

for each monzo |c2 c3 ... cp> in C. If we want something corresponded
to weighted optimization we would add weights, and if we wanted the
odd limit, the consonances in C can be restricted to quotients of odd
integers, in which case c2 will always be zero. We then may form the
symmetric matrix corresponding to the quadratic form we get from the
above sum and invert it; using this inverted matrix to define a
quadratic form on monzos and taking the square root gives us a
Euclidean norm on monzo space. We can of course normalize either norm
by multiplying by any positive constant.

If we do this in the 5-limit using {3, 5, 5/3} for C, we get

sqrt(x2^2 - x2 x3 + x3^2)

for the norm on the val (sub)space, and correspondingly

sqrt(e2^2 + e2 e3 + e3^2)

for the norm on octave classes--the triangular lattice norm.

Similarly, if C = {3, 5, 7, 5/3, 7/3, 7/5} we get

sqrt(3(x3^2 + x5^2 + x7^2) - 2(x3 x5 + x3 x7 + x5 x7))

as the norm on vals, and the familiar

sqrt(e3^2 + e5^2 + e7^2 + e3 e5 + e3 e7 + e5 e7)

as the norm on octave classes.

In the 11-limit and beyond, of course, things become more complicated
because we will want to introduce ratios of odd numbers which are not
necessarily primes. If we take ratios of odd numbers up to 11 for our
set of consonances, we get

sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-6x3x11)

as our norm on vals, and correspondingly,

sqrt(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11+62e7^2+58e11e7+62e11^2)

as our norm on octave classes. This norm is not altogether
satisfactory; for instance it gives a length of sqrt(44) to 5/3 and
6/5, and a length of sqrt(62) to 5/4. This suggests to me that there
is something a little dubious in theory about using unweighted rms
optimization, at least in the 11 limit and beyond. An alternative rms
optimization scheme would be to use dual of the norm I've been using
on octave classes as the norm for a weighted rms optimization.

In the 5-limit, this norm on octave classes is

sqrt(p3e3^2 + p3e3e5 + p5e5^2)

where p3 = log2(3), p5 = log2(5). The dual norm on vals is

sqrt(p5x3^2 - p3x3x5 + p3x5^2)

These norms will weigh lower prime errors a little higher than higher
prime errors, which of course is also what TOP does. Now I need a
catchy name for them.

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

> In the 11-limit and beyond, of course, things become more
complicated
> because we will want to introduce ratios of odd numbers which are
not
> necessarily primes. If we take ratios of odd numbers up to 11 for
our
> set of consonances, we get
>
> sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-
6x3x11)
>
> as our norm on vals, and correspondingly,
>
> sqrt
(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11+62e7^2+58e11e7+62e
11^2)
>
> as our norm on octave classes. This norm is not altogether
> satisfactory; for instance it gives a length of sqrt(44) to 5/3 and
> 6/5, and a length of sqrt(62) to 5/4. This suggests to me that there
> is something a little dubious in theory about using unweighted rms
> optimization, at least in the 11 limit and beyond.

Gene, note that I've always counted 3/1 and 9/3, etc., separately in
these optimizations. If you use that "weighting", do things look less
dubious? (The weight is proportional to the number of ways the
interval class can be represented by a ratio of odd numbers within
the limit.)

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

> Gene, note that I've always counted 3/1 and 9/3, etc., separately in
> these optimizations. If you use that "weighting", do things look less
> dubious? (The weight is proportional to the number of ways the
> interval class can be represented by a ratio of odd numbers within
> the limit.)

It weighs 3 more, but it still seems weird. The norm on vals is

sqrt(-6x3x11-2x5x11-2x5x7-6x3x7-6x3x5-2x7x11+21x3^2+5x5^2+5x7^2+5x11^2)

and the corresponding norm on pcs is

sqrt(3e3^2+6e3e5+6e3e7+6e3e11+11e5^2+10e5e7+10e5e11+11e7^2+10e11e7+11e11^2)

This gives us ||3|| = ||3/2|| = ||4/3|| = sqrt(3), ||9/8||=2sqrt(3),
||5/4|| = ||7/4|| = ||11/8|| = sqrt(11). What seems more dubious
is ||11/6|| = ||7/6|| = ||6/5|| = 2sqrt(2). I think my idea of using
the dual norm to my "geometric" norm makes more sense.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Gene, note that I've always counted 3/1 and 9/3, etc., separately
in
> > these optimizations. If you use that "weighting", do things look
less
> > dubious? (The weight is proportional to the number of ways the
> > interval class can be represented by a ratio of odd numbers
within
> > the limit.)
>
> It weighs 3 more, but it still seems weird. The norm on vals is
>
> sqrt(-6x3x11-2x5x11-2x5x7-6x3x7-6x3x5-
2x7x11+21x3^2+5x5^2+5x7^2+5x11^2)
>
> and the corresponding norm on pcs is
>
> sqrt
(3e3^2+6e3e5+6e3e7+6e3e11+11e5^2+10e5e7+10e5e11+11e7^2+10e11e7+11e11^2
)
>
> This gives us ||3|| = ||3/2|| = ||4/3|| = sqrt(3),

Perfect . . .
(1.7321)

> ||9/8||=2sqrt(3),

Excellent . . .
(3.4641)

> ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).

OK . . .
(3.3166)

> What seems more dubious
> is ||11/6|| = ||7/6|| = ||6/5|| = 2sqrt(2).

(2.8284)
Doesn't seem unduly dubious, though, given that all the lengths are
about equal here, except 3 is shorter. Pretty much in accordance with
what we put in.

Now what if we apply 'odd-limit-weighting' to each of the intervals,
including 9:3 which is treated as having an odd-limit of 9? Try
using 'odd-limit' plus-or-minus 1 or 1/2 too.

> I think my idea of using
> the dual norm to my "geometric" norm makes more sense.

Why is that?

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

> Now what if we apply 'odd-limit-weighting' to each of the intervals,
> including 9:3 which is treated as having an odd-limit of 9? Try
> using 'odd-limit' plus-or-minus 1 or 1/2 too.

Is the weighting by multiplying or dividing by the log of the odd
limit? Presumably mutliplying will make more sense. Do we square and
then multiply, since we will be taking square roots?

> > I think my idea of using
> > the dual norm to my "geometric" norm makes more sense.
>
> Why is that?

It's more or less reasonable to start with. We have

||3/2|| = log2(3), ||9/8|| = 2log2(3), ||5/4|| = log2(5),
||6/5|| = log2(5), ||7/6|| = ||7/5|| = ||7/4|| = log2(7),
||11/6|| = ||11/7|| = ||11/8|| = ||11/10|| = log2(11),
||9/5|| = sqrt(2log2(3)^2 + log2(5)^2)
||9/7|| = sqrt(2log2(3)^2 + log2(7)^2)
||11/9|| = sqrt(2log2(3)^2 + log2(11)^2)

which isn't too bad.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Now what if we apply 'odd-limit-weighting' to each of the
intervals,
> > including 9:3 which is treated as having an odd-limit of 9? Try
> > using 'odd-limit' plus-or-minus 1 or 1/2 too.
>
> Is the weighting by multiplying or dividing by the log of the odd
> limit? Presumably mutliplying will make more sense.

Divide. As in TOP, errors of more complex intervals are divided by
larger numbers.

> Do we square and
> then multiply, since we will be taking square roots?

No, we want to apply the weighting directly to the errors, before
deciding how overall error is calculated from the individual weighted
errors.

> > > I think my idea of using
> > > the dual norm to my "geometric" norm makes more sense.
> >
> > Why is that?
>
> It's more or less reasonable to start with. We have
>
> ||3/2|| = log2(3), ||9/8|| = 2log2(3), ||5/4|| = log2(5),
> ||6/5|| = log2(5), ||7/6|| = ||7/5|| = ||7/4|| = log2(7),
> ||11/6|| = ||11/7|| = ||11/8|| = ||11/10|| = log2(11),
> ||9/5|| = sqrt(2log2(3)^2 + log2(5)^2)
> ||9/7|| = sqrt(2log2(3)^2 + log2(7)^2)
> ||11/9|| = sqrt(2log2(3)^2 + log2(11)^2)
>
> which isn't too bad.

OK, but I think my proposal above will make even more sense than
this. The two should agree for 7-odd-limit and below.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > Now what if we apply 'odd-limit-weighting' to each of the
> intervals,
> > > including 9:3 which is treated as having an odd-limit of 9? Try
> > > using 'odd-limit' plus-or-minus 1 or 1/2 too.
> >
> > Is the weighting by multiplying or dividing by the log of the odd
> > limit? Presumably mutliplying will make more sense.
>
> Divide. As in TOP, errors of more complex intervals are divided by
> larger numbers.
>
> > Do we square and
> > then multiply, since we will be taking square roots?
>
> No, we want to apply the weighting directly to the errors, before
> deciding how overall error is calculated from the individual
weighted
> errors.
>
> > > > I think my idea of using
> > > > the dual norm to my "geometric" norm makes more sense.
> > >
> > > Why is that?
> >
> > It's more or less reasonable to start with. We have
> >
> > ||3/2|| = log2(3), ||9/8|| = 2log2(3), ||5/4|| = log2(5),
> > ||6/5|| = log2(5), ||7/6|| = ||7/5|| = ||7/4|| = log2(7),
> > ||11/6|| = ||11/7|| = ||11/8|| = ||11/10|| = log2(11),
> > ||9/5|| = sqrt(2log2(3)^2 + log2(5)^2)
> > ||9/7|| = sqrt(2log2(3)^2 + log2(7)^2)
> > ||11/9|| = sqrt(2log2(3)^2 + log2(11)^2)
> >
> > which isn't too bad.
>
> OK, but I think my proposal above will make even more sense than
> this. The two should agree for 7-odd-limit and below.

>For any set of consonances C we want to do an rms optimization for, we
>can find a corresponding Euclidean norm on the val space (or
>octave-excluding subspace if we are interested in the odd limit) by
>taking the sum of terms
>
>(c2 x2 + c3 x3 + ... + cp xp)^2
>
>for each monzo |c2 c3 ... cp> in C. If we want something corresponded
>to weighted optimization we would add weights, and if we wanted the
>odd limit, the consonances in C can be restricted to quotients of odd
>integers, in which case c2 will always be zero.
//
>In the 11-limit and beyond, of course, things become more complicated
>because we will want to introduce ratios of odd numbers which are not
>necessarily primes. If we take ratios of odd numbers up to 11 for our
>set of consonances, we get
>
>sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-6x3x11)
>
>as our norm on vals, and correspondingly,
>
>sqrt(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11//
>
>as our norm on octave classes. This norm is not altogether
>satisfactory; for instance it gives a length of sqrt(44) to 5/3 and
>6/5, and a length of sqrt(62) to 5/4. This suggests to me that there
>is something a little dubious in theory about using unweighted rms
>optimization, at least in the 11 limit and beyond. An alternative rms
>optimization scheme would be to use dual of the norm I've been using
>on octave classes as the norm for a weighted rms optimization.
>
>In the 5-limit, this norm on octave classes is
>
>sqrt(p3e3^2 + p3e3e5 + p5e5^2)
>
>where p3 = log2(3), p5 = log2(5). The dual norm on vals is
>
>sqrt(p5x3^2 - p3x3x5 + p3x5^2)
>
>These norms will weigh lower prime errors a little higher than higher
>prime errors, which of course is also what TOP does. Now I need a
>catchy name for them.

Have you checked that this weighted version is ok at the 11-limit
(ie doesn't make 5/4 shorter than 5/3)?

Also: If you leave in the c2 term does this optimize over "all the
intervals" like TOP?

It seems to me a comparison of...

(1) TOP
(2) rms-TOP
(3) odd-limit TOP
(4) rms odd-limit TOP

...has not been done.

I'm happy to give up on unweighted TOPs.

-Carl

>Also: If you leave in the c2 term does this optimize over "all the
>intervals" like TOP?

Or do you just get integer limit?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >For any set of consonances C we want to do an rms optimization
for, we
> >can find a corresponding Euclidean norm on the val space (or
> >octave-excluding subspace if we are interested in the odd limit) by
> >taking the sum of terms
> >
> >(c2 x2 + c3 x3 + ... + cp xp)^2
> >
> >for each monzo |c2 c3 ... cp> in C. If we want something
corresponded
> >to weighted optimization we would add weights, and if we wanted the
> >odd limit, the consonances in C can be restricted to quotients of
odd
> >integers, in which case c2 will always be zero.
> //
> >In the 11-limit and beyond, of course, things become more
complicated
> >because we will want to introduce ratios of odd numbers which are
not
> >necessarily primes. If we take ratios of odd numbers up to 11 for
our
> >set of consonances, we get
> >
> >sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-
6x3x11)
> >
> >as our norm on vals, and correspondingly,
> >
> >sqrt(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11//
> >
> >as our norm on octave classes. This norm is not altogether
> >satisfactory; for instance it gives a length of sqrt(44) to 5/3 and
> >6/5, and a length of sqrt(62) to 5/4. This suggests to me that
there
> >is something a little dubious in theory about using unweighted rms
> >optimization, at least in the 11 limit and beyond. An alternative
rms
> >optimization scheme would be to use dual of the norm I've been
using
> >on octave classes as the norm for a weighted rms optimization.
> >
> >In the 5-limit, this norm on octave classes is
> >
> >sqrt(p3e3^2 + p3e3e5 + p5e5^2)
> >
> >where p3 = log2(3), p5 = log2(5). The dual norm on vals is
> >
> >sqrt(p5x3^2 - p3x3x5 + p3x5^2)
> >
> >These norms will weigh lower prime errors a little higher than
higher
> >prime errors, which of course is also what TOP does. Now I need a
> >catchy name for them.
>
> Have you checked that this weighted version is ok at the 11-limit
> (ie doesn't make 5/4 shorter than 5/3)?

I don't know of

>
> Also: If you leave in the c2 term does this optimize over "all the
> intervals" like TOP?

No.

> (3) odd-limit TOP

We have several different ideas of what this means floating around.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >For any set of consonances C we want to do an rms optimization
> for, we
> > >can find a corresponding Euclidean norm on the val space (or
> > >octave-excluding subspace if we are interested in the odd limit)
by
> > >taking the sum of terms
> > >
> > >(c2 x2 + c3 x3 + ... + cp xp)^2
> > >
> > >for each monzo |c2 c3 ... cp> in C. If we want something
> corresponded
> > >to weighted optimization we would add weights, and if we wanted
the
> > >odd limit, the consonances in C can be restricted to quotients
of
> odd
> > >integers, in which case c2 will always be zero.
> > //
> > >In the 11-limit and beyond, of course, things become more
> complicated
> > >because we will want to introduce ratios of odd numbers which
are
> not
> > >necessarily primes. If we take ratios of odd numbers up to 11
for
> our
> > >set of consonances, we get
> > >
> > >sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-
> 6x3x11)
> > >
> > >as our norm on vals, and correspondingly,
> > >
> > >sqrt(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11//
> > >
> > >as our norm on octave classes. This norm is not altogether
> > >satisfactory; for instance it gives a length of sqrt(44) to 5/3
and
> > >6/5, and a length of sqrt(62) to 5/4. This suggests to me that
> there
> > >is something a little dubious in theory about using unweighted
rms
> > >optimization, at least in the 11 limit and beyond. An
alternative
> rms
> > >optimization scheme would be to use dual of the norm I've been
> using
> > >on octave classes as the norm for a weighted rms optimization.
> > >
> > >In the 5-limit, this norm on octave classes is
> > >
> > >sqrt(p3e3^2 + p3e3e5 + p5e5^2)
> > >
> > >where p3 = log2(3), p5 = log2(5). The dual norm on vals is
> > >
> > >sqrt(p5x3^2 - p3x3x5 + p3x5^2)
> > >
> > >These norms will weigh lower prime errors a little higher than
> higher
> > >prime errors, which of course is also what TOP does. Now I need a
> > >catchy name for them.
> >
> > Have you checked that this weighted version is ok at the 11-limit
> > (ie doesn't make 5/4 shorter than 5/3)?
>
> I don't know of

Whoops . . . I don't know if you followed the thread this far:

/tuning-math/message/8692

>> ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).
>
>OK . . .
>(3.3166)

How is that ok?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).
> >
> >OK . . .
> >(3.3166)
>
> How is that ok?
>
> -C.

It's not surprising given how Gene set it up: with the same weighting
that gives equilateral triangles and tetrahedra in the 5-limit and 7-
limit lattices . . .

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

> It's not surprising given how Gene set it up: with the same weighting
> that gives equilateral triangles and tetrahedra in the 5-limit and 7-
> limit lattices . . .

Actually, isosceles triangles. The fifth gets a length of log(3) (or
cents(3) or whatever log you are using) and the major and minor thirds
have the same length, log(5). The septimal consonances are even
farther out, 7/6, 7/5 and 7/4 all having the same length. Carl
presumably would like the symmetrical lattices even less.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > >> ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).

Oops. I see it was symmetrical.