Re: [tuning] Re: question about 24-tET

>> I just continued a thread on tuning-math in which Gene apparently
>> demonstrated odd-limit TOP...
>
>Nothing particularly odd-limit about it. You just need a set of
>consonances, with weights (or multiplicities) if you choose.

Can we get generators for 5-limit meantone, 7-limit schismic,
and 11-limit miracle for each of:

(1) TOP
(2) odd-limit TOP
(3) rms TOP (or can you only do integer-limit rms TOP?)
(4) rms odd-limit TOP

Thanks,

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Can we get generators for 5-limit meantone, 7-limit schismic,
> and 11-limit miracle for each of:
>
> (1) TOP
> (2) odd-limit TOP
> (3) rms TOP (or can you only do integer-limit rms TOP?)
> (4) rms odd-limit TOP

I can do the TOP. What's the definition for the others? If I'm doing
rms analogs of TOP, don't I need a list of intervals and maybe weights
for them in order to cook up a Euclidean metric? I think Paul wanted
something like that, and I could do it if I could remember exactly
what it was.

>> Can we get generators for 5-limit meantone, 7-limit schismic,
>> and 11-limit miracle for each of:
>>
>> (1) TOP
>> (2) odd-limit TOP
>> (3) rms TOP (or can you only do integer-limit rms TOP?)
>> (4) rms odd-limit TOP
>
>I can do the TOP. What's the definition for the others?

You know what (1) is.

I thought you just posted something about doing (2) & (4) by
leaving out the 2-terms in a certain formula. Here:

>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 (2) I mean the tuning that gives minimax error over all odd-limit
consonances (try the 9-limit). As far as weighting for this, I'd try
the usual Tenney weighting as in (1), and Paul's odd-limit weighting
suggestions:

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

For (4) it's the tuning that gives minimum rms error over the 9-limit
consonances. All weighting suggestions apply.

For (3) it's the tuning that gives minimum rms over all intervals
with Tenney weighting as in (1).

>If I'm doing rms analogs of TOP, don't I need a list of intervals
>and maybe weights for them in order to cook up a Euclidean metric?
>I think Paul wanted something like that, and I could do it if I
>could remember exactly what it was.

See above.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

I used the 45 (counting multiplicities) 10-limit intervals to define a
norm, and the result clearly did not make sense as a way of ranking
musical intervals. I could add weighting, but there already is heavy
weighting for the lower primes automatically.

I think Paul's theory about this is wrong, and mine was right--we are
better off starting from a norm we know works reasonably well, like the
sqrt(sum log(p)log(q)x_p x_q) norm.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> I used the 45 (counting multiplicities) 10-limit intervals to
define a
> norm, and the result clearly did not make sense as a way of ranking
> musical intervals. I could add weighting, but there already is heavy
> weighting for the lower primes automatically.
>
> I think Paul's theory about this is wrong, and mine was right--we
are
> better off starting from a norm we know works reasonably well, like
the
> sqrt(sum log(p)log(q)x_p x_q) norm.

I wish I knew what you were talking about.