Maximizing categorical independence - weighted range

For any MOS, there are always three potential ways that intervals
could become ambiguous with one another:
1) If s=0, this is another equal temperament
2) If L=s, meaning that c=0; this is an equal temperament
3) If s=c, meaning that L/s = 2/1; this is what Rothenberg meant by
ambiguous intervals

The goal should be, for any MOS, to find the generator that is
maximally far from every one of these disastrous outcomes, weighted by
the number of ambiguous intervals that could potentially exist in each
case. 31 doesn't apply if there's only one "s" interval per period of
the MOS.

- The potentially ambiguous intervals in #1 differ by s, the small
step. The larger the small step is, the better we are at
distinguishing the intervals in #1.
- The potentially ambiguous intervals in #2 differ by c, or the
chroma. The larger the chroma is, the better we are at distinguishing
the intervals in #2.
- The potentially ambiguous intervals in #3 differ by s-c, or the
diesis. The larger the diesis is, the better we are at distinguishing
the intervals in #3.

Now let's weight each factor. It obviously doesn't matter as much how
big the diesis is in something like the diatonic scale, where it only
appears for one interval - one A4 vs one P5 - vs something like the
Pajara SPM scale, where it appears between the four M3s and the four
m4's, and also between the six M4s and two m5's. We'll weight each
factor by the total number of ambiguous intervals presented in each
case. So for diatonic, #1 would be weighted by 2, and for SPM, it
would be weighted by 16.

We hence want to maximize the following equation, where w1, w2, and w3
are the number of intervals falling into #1, #2, and #3 respectively,
and CI is categorical independence:

CI = w1(s) + w2(c) + w3(s-c)

It's 4 AM and I have to sleep, but I'll come back to this. I stumbled
on this concept after playing with Pajara standard pentachordal major
in 22-equal, and hating, as I always do, how the minor fourths, set at
327 cents, and the major thirds, set at 273 cents, sound so close to
one another. Experimenting with Pajara in 32- and 42-equal, where the
difference between these intervals is 75 and 86 cents, respectively,
blew my mind. I urge you all to try it, focusing more on categorical
perception than purity of intonation.

I think that once you get past looking at MOS's the concept is going
to end up being, more or less, a weighted version of Carl's "rank
range" concept.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> It's 4 AM and I have to sleep, but I'll come back to this. I stumbled
> on this concept after playing with Pajara standard pentachordal major
> in 22-equal, and hating, as I always do, how the minor fourths, set at
> 327 cents, and the major thirds, set at 273 cents, sound so close to
> one another.

Not kidding about being sleepy, I see. :)

On Sep 6, 2011, at 2:52 PM, "genewardsmith" <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> It's 4 AM and I have to sleep, but I'll come back to this. I stumbled
> on this concept after playing with Pajara standard pentachordal major
> in 22-equal, and hating, as I always do, how the minor fourths, set at
> 327 cents, and the major thirds, set at 273 cents, sound so close to
> one another.

Not kidding about being sleepy, I see. :).

That's right, isn't it? I was trying to use decatonic scale categories for
that.

Trump card, blam!

-Mike

On Tue, Sep 6, 2011 at 3:39 AM, Mike Battaglia <battaglia01@...> wrote:
>
> It's 4 AM and I have to sleep, but I'll come back to this. I stumbled
> on this concept after playing with Pajara standard pentachordal major
> in 22-equal, and hating, as I always do, how the minor fourths, set at
> 327 cents, and the major thirds, set at 273 cents, sound so close to
> one another. Experimenting with Pajara in 32- and 42-equal, where the
> difference between these intervals is 75 and 86 cents, respectively,
> blew my mind. I urge you all to try it, focusing more on categorical
> perception than purity of intonation.

I spent some time thinking about this today, and I think that it is in
fact going to be sort of a weighted rank range. I think it's closer to
stability, however, than anything. Perhaps it's a new type of thing,
for which "categorical independence" would be the best word for it.

Here's one possible way to do it: lay all of the intervals out on a
ruler, as Carl did. Work out the meta-intervals between each interval,
as with Carl's approach. Now take a weighted average of the size of
all of these meta-intervals, weighted by the number of times that they
appear in the scale (e.g. the total number of times that the intervals
on either side of the meta-interval appear).

I think that scales in which this average is maximized will be easier
to learn than those that aren't. If the average is maximized, each
interval will be positioned maximally far away from the other ones,
which will lead to less confusion. Another way to think of this is
that it will generate the "most unequal" tuning for an MOS, which is
what I was originally looking for when I started this whole thing.
There may be a few nits to pick in the above definition to get it to
work smoothly, but from the perspective of my caffeine-fueled brain at
9 AM, it looks pretty good.

For MOS's, the L/s value at which the average is maximized should vary
depending on the MOS. For intervals with less ambiguous intervals,
e.g. those differing by the |s-c| diesis, the value should be closer
to 2/1. For intervals with more ambiguous intervals, it should be
further from that, although I'm not sure if it'll end up closer to 3/2
or 3/1.

A problem presents itself for irregular tunings that deliberately have
a lot of intervals that center around some general size, but vary
slightly from it. This measure will only consider the meta-intervals
between adjacent intervals, so these slight intonational differences
will turn up as meta-intervals that are supposed to be maximized, and
hence the weighting will end up going screwy. I think a good approach
for this is to consider the average difference between every interval
and ALL other intervals in the scale, even those on the other
half-octave of it, but I want to start with the above first and work
from there.

I'm moving a lot slower these days than I used to, but I'll crunch
some numbers soon.

-Mike