Near-top val basis

Suppose we take the set of vals belonging to a wedgie. We can take the
TOP tunings of the vals and the TOP tuning of the wedgie, weight them
in the usual way, subtract and take the maximum difference. In other
words, we can look at the distance, in val space, between the TOP
tuning of the wedgie and the TOP tunings of the vals belonging to it.
If we multiply the distance for the val v by v[1], and look only at
vals such that v[1]>0, then we can pick the two smallest by this
measure. They will be equal temperament vals which have TOP tunings
which are close (relative to the N of the equal temperament) to the
TOP tuning for the temperament.

For example, consider 7-limit pajara. The val <146 232 339 412| has a
TOP tuning which is essentially indistinguishable from the TOP tuning
for pajara, with a val space distance of 0.0087 cents; multiplied by
146 that is 1.2742, which seems by far the minimal value. Second
appears to be <90 143 209 254|, and together these give pajara. Note
that the stretched TOP/Kees tuning for pajara has pure major thirds,
and the pure-octave tuning for the above 146 val has very nearly pure
major thirds, and this sort of system also works if we start from a
Kees point of view.

On 2/18/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> Suppose we take the set of vals belonging to a wedgie. We can take the
> TOP tunings of the vals and the TOP tuning of the wedgie, weight them
> in the usual way, subtract and take the maximum difference. In other
> words, we can look at the distance, in val space, between the TOP
> tuning of the wedgie and the TOP tunings of the vals belonging to it.
> If we multiply the distance for the val v by v[1], and look only at
> vals such that v[1]>0, then we can pick the two smallest by this
> measure. They will be equal temperament vals which have TOP tunings
> which are close (relative to the N of the equal temperament) to the
> TOP tuning for the temperament.

Can you guarantee they won't be contorted?

Graham

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> Can you guarantee they won't be contorted?

I'd be interested to see an example.