Arbitrary RMS errors

I've added some files to my website:

http://x31eq.com/summary.pdf

http://x31eq.com/lattice.pdf

These outline temperament class finding problems in the simplest mathematical terms. You can try waving them in front of mathematicians to get their interest.

For today, the important thing is that I put the weighting in explicitly. So look at summary.pdf and you should recognize M as the unweighted mapping, H as the list of prime intervals, and W as the weighting. Equation (6) is a kind of complexity and Equation (7) is a kind of badness. They're normalized differently to primerr.pdf but badness/complexity still gives the TOP-RMS error.

To find the optimal RMS error for an arbitrary set of intervals, define a matrix C expressing the composite intervals in terms of the primes. Then W can be the weights for each interval. To use these intervals instead of the primes you replace M with MC and H with C^T H (A^T is the transpose of A). The result is that everywhere you see W^2 (W squared) in equations 6 and 7 you replace it with C W^2 C^T.

That matrix looks like a metric, so let's call it G.

G = C W^2 C^T

Now, however many intervals you use the metric G is exactly the same size: d by d for d prime intervals. It's always symmetric because of the way it's constructed. Also, because it occurs in both the numerator and denominator of the optimal error formula, you get the same result by multiplying G by a scalar.

It follows that whatever composite intervals you look at, and whatever weighting you use, the problem is of the same form and the same complexity. If you're interested in all 7-limit ratios with a product less than 100 weighted equally, there's a particular G that does it.

This being the case, there should also be a G that corresponds to all the infinite number of notes in an prime limit. I believe that this G is simply the W^2 for Tenney weighting. It'll take me some time to prove it, but I think it works. It treats 9:3 and 3:1 separately.

For a finite number of intervals between harmonics, the RMS error seems to be a combination of the RMS and STD errors of the harmonics. The form is similar to the parametric badness I use in lattice.pdf and maybe there's a deep connection. The more harmonics you consider the more it looks like an RMS. (Hence the RMS is the limit as the number of harmonics tends to infinity.)

Graham

At 08:09 PM 2/20/2008, you wrote:
>I've added some files to my website:
>
>http://x31eq.com/summary.pdf
>
>http://x31eq.com/lattice.pdf
>
>These outline temperament class finding problems in the
>simplest mathematical terms. You can try waving them in
>front of mathematicians to get their interest.

What's the difference between these files (I can see a
difference, but what does it ammount to)?

-Carl

Carl Lumma wrote:
> At 08:09 PM 2/20/2008, you wrote:
>> I've added some files to my website:
>>
>> http://x31eq.com/summary.pdf
>>
>> http://x31eq.com/lattice.pdf
>>
>> These outline temperament class finding problems in the >> simplest mathematical terms. You can try waving them in >> front of mathematicians to get their interest.
> > What's the difference between these files (I can see a
> difference, but what does it ammount to)?

The lattice version is about minimizing a single function that takes a free parameter. The first one is also about minimizing badness, but within a complexity constraint.

The "lattice" one is hopefully a real lattice problem for which there might be a ready-made solution. You can at least do searches knowing that the best equal temperaments with a given epsilon are the most likely to produce the best temperaments of a higher rank. This follows from geometric considerations. Where there's a complexity constraint it isn't so straightforward because you need to compare the complexities of different rank temperaments.

Epsilon has dimensions of error. It's something to do with the highest error you want do deal with. Strict in the sense of being higher than what you expect but fuzzy because you might still get something higher.

Graham

Carl Lumma wrote:
> At 08:09 PM 2/20/2008, you wrote:
>> I've added some files to my website:
>>
>> http://x31eq.com/summary.pdf
>>
>> http://x31eq.com/lattice.pdf
>>
>> These outline temperament class finding problems in the >> simplest mathematical terms. You can try waving them in >> front of mathematicians to get their interest.
> > What's the difference between these files (I can see a
> difference, but what does it ammount to)?

Incidentally, one reason for keeping summary.pdf is that I can't prove that the problem in lattice.pdf is properly bounded. So lattice.pdf is simpler and that's the one to show a mathematician. But if they come back and say that it leads to infinities, show them summary.pdf, which is closer to the real problem we want solved.

Graham

At 09:08 PM 3/3/2008, you wrote:
>Carl Lumma wrote:
>> At 08:09 PM 2/20/2008, you wrote:
>>> I've added some files to my website:
>>>
>>> http://x31eq.com/summary.pdf
>>>
>>> http://x31eq.com/lattice.pdf
>>>
>>> These outline temperament class finding problems in the
>>> simplest mathematical terms. You can try waving them in
>>> front of mathematicians to get their interest.
>>
>> What's the difference between these files (I can see a
>> difference, but what does it ammount to)?
>
>Incidentally, one reason for keeping summary.pdf is that I
>can't prove that the problem in lattice.pdf is properly
>bounded. So lattice.pdf is simpler and that's the one to
>show a mathematician. But if they come back and say that it
>leads to infinities, show them summary.pdf, which is closer
>to the real problem we want solved.
>
>
> Graham

Maybe they should be combined into a single file, along
with this note...

-Carl

Carl Lumma wrote:

> Maybe they should be combined into a single file, along
> with this note...

Yes, but it's supposed to be simple, and you don't make something simpler by adding to it.

Graham