odds vs. primes - Graham

Hi Graham,

Wondering if you have a quick, ready answer to the
following: You've been known to use the average
tenney-weighted error of primes. Why is this better
than doing the same with with odds?

-Carl

Carl Lumma wrote:
> Hi Graham,
> > Wondering if you have a quick, ready answer to the
> following: You've been known to use the average
> tenney-weighted error of primes. Why is this better
> than doing the same with with odds?

The quick, simple answer is -- no.

Another simple answer -- that was the easiest way of defining Paul E's TOP.

A deceptively simple answer -- 2 is a prime but not an odd.

I did look at only the primes for odd limits. You can calculate the worst odd-limit error by only looking at the errors of odd harmonics. Optimizing it isn't as simple and I'm not clear about how the RMS would work.

There are certainly ways of relating the averages of primes to odd limits but I haven't looked into it yet. You'd need other odd numbers, like 15, in there to do it properly. Putting in all the odds explicitly should be equivalent to using different weights.

Graham

Carl Lumma wrote:
> Hi Graham,
> > Wondering if you have a quick, ready answer to the
> following: You've been known to use the average
> tenney-weighted error of primes. Why is this better
> than doing the same with with odds?

A more considered answer :-

The main motivation is that the code is simpler by not having to worry about factorizing numbers. You always have to think about the primes for reasons of consistency. Only thinking about primes is simpler than thinking about both primes and odds. The time taken for the calculation shouldn't be greatly affected, however, because there aren't many more odds than primes for the kind of limits we're interested in (whereas there are quadratically as many intervals between odds).

A theoretical argument is that looking at Tenney-weighted primes treats all intervals equally according to their complexity. Looking at odd numbers favors composite odds, as 15:1 is directly considered but 5:3 only indirectly. This is the exact opposite of what I, at least, want, which is to favour intervals between harmonics.

The answer, like with octave-equivalent prime measures, is to look at a dispersion rather than an average. In this case I don't see any problems. Remember that 1 is an odd number and has to be included.

As I said before, the range of the deviations of odd harmonics is the worst odd-limit error. So there's nothing new about this.

You could always try taking the standard deviation of the unweighted deviations of odd harmonics. I didn't do this when I was working with odd limits because I hadn't thought of using standard deviations then. It should relate to the RMS odd-limit error. Maybe there should be a different weighting to take account of small odd numbers being more likely as factors of small even numbers. Or you can choose a subjective weighting.

Graham