--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> >Number 82
> >
> >[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
> >TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
> >TOP generators [601.7004928, 230.8749260]
> >bad: 79.825592 comp: 4.619353 err: 3.740932

...

> When I plug 10 and 16 into the temperament finder, this is what I
> end up with.
>
> 5/13, 229.4 cent generator
>
> basis:
> (0.5, 0.191135896755)
>
> mapping by period and generator:
> [(2, 0), (2, 3), (5, -1), (6, -1)]
>
> mapping by steps:
> [(16, 10), (25, 16), (37, 23), (45, 28)]
>
> highest interval width: 4
> complexity measure: 8 (10 for smallest MOS)
> highest error: 0.014573 (17.488 cents)

This comparison of different outputs for the same temperament shows up
the need to correctly normalise the new weighted error and complexity
figures so they actually have units we can relate to. i.e. cents for
the error and gens per interval for the complexity.

This should be simple to do.

I think the correct normalisation of a weighted norm is the one where,
if every individual value happened to be X then the, the norm would
also be X, irrespective of the weights.

e.g. if the individual errors are E1, E2, ... En, and the respective
weights are W1, W2, ... Wn (all positive), I think the p-norm should
not be

[(|W1E1|**p + |W2E2|**p + ... |WnEn|**p)/n]**(1/p)

but instead

[(|W1E1|**p + |W2E2|**p + ... |WnEn|**p)/(W1**p + W2**p + ...
Wn**p)]**(1/p)

i.e. n is replaced by (W1**p + W2**p + ... Wn**p)

However it bothers me slightly that for minimax (p -> oo), this is
equivalent to

Max(|W1E1|, |W2E2|, ... |WnEn|)/Max(W1, W2, ... Wn)

It seems like I'd rather have

Max(|W1E1|, |W2E2|, ... |WnEn|)/Mean(W1, W2, ... Wn)

but I guess that would be inconsistent.