Scale-temperament metrics

Suppose we have a rank 3 p-limit temperament, which will therefore
have a comma basis consisting of n-3 commas, where n=pi(p). Suppose we
also have a p-limit scale (gamut, set, what have you) of m p-limit
intervals, reduced to an octave, s[i]. Then we can define a metric for
the scale and temperament in question as follows: for each scale
element s[i], form the monzo matrix for 2, s[i], and the comma basis.
This will be a square integral matrix, and we may take the absolute
value of the determinant. Take the maximum of these over all the scale
elements s[i]; this is the metric. Aside from just being nifty, it
seems to me it might be a useful way to find the convex closure of the
scale s with respect to the temperament. You can also think of this in
terms of wedge products, which I did at first, but using matricies we
don't need them.

Gene,

You wrote:
> Suppose we have a rank 3 p-limit temperament, which will therefore
> have a comma basis consisting of n-3 commas, where n=pi(p). Suppose we
> also have a p-limit scale (gamut, set, what have you) of m p-limit
> intervals, reduced to an octave, s[i]. Then we can define a metric for
> the scale and temperament in question as follows: for each scale
> element s[i], form the monzo matrix for 2, s[i], and the comma basis.
> This will be a square integral matrix, and we may take the absolute
> value of the determinant. Take the maximum of these over all the scale
> elements s[i]; this is the metric.

Pretty neat!

> Aside from just being nifty, it
> seems to me it might be a useful way to find the convex closure of the
> scale s with respect to the temperament.

Umm ... how exactly does your procedure generate the non-scale notes in
the closure?

> You can also think of this in
> terms of wedge products, which I did at first, but using matricies we
> don't need them.

Thinking in terms of wedge products has not become second-nature
for me ... yet.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.322 / Virus Database: 266.11.16 - Release Date: 24/5/05

--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Umm ... how exactly does your procedure generate the non-scale notes in
> the closure?

It's not a proceedure; it's just a thought it might be useful for some
proceedure. However, you can use it for related purposes--by using
this metric and then going out to the minimal distance needed to
include all the intervals you start from, you may well go far enough
from the unison to include other intervals. This would be a convex
scale but not necessarily minimal.