Re: Weighting Rothenberg/Lumma stability by where in the scale the ambiguous intervals occur

On Fri, Apr 1, 2011 at 12:14 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Has anyone ever worked with a metric like this before, before I spend
> time trying to implement it?

I should also add that this sort of thing could be used to model the
"stability" of even improper scales: something like 22-tet superpyth
diatonic, for example, has the improper interval hidden between the
aug4 and the dim5, so you barely ever notice it. On the other hand,
22-tet superpyth melodic minor has the diminished fourth set at 6/5
and the major third at 9/7, which is more noticeable. 22-tet superpyth
harmonic minor is more noticeable still, since the aug2 is 5/4 and the
minor third is 7/6.

22-tet locrian major is a complete nightmare, where the diminished
third is 109 cents and the major second is 218 cents. The aug4 and
dim5 in superpyth diatonic also differ by the same amount, but it's
much less noticeable because the impropriety is hidden away in the
fourth and fifth. So maybe this could be used to come up with some
kind of combined stability/impropriety coefficient.

-Mike

Hi all,

I was playing around with the Rothenberg scale search when I noticed
something about proper scale perception: the larger the interval class
in which the ambiguous intervals in a proper scale occur, the less
they matter. I notice that the cohesion of the scale really takes a
hit if the ambiguity is present between the seconds and thirds,
whereas if it's "hidden" between the fourths and fifths like in the
diatonic scale, it's not as noticeable.

For example, the 12-tet diatonic scale has one ambiguous interval
between the augmented fourth and the diminished fifth, which is really
not that noticeable - when I talk about propriety to my jazz friends,
I usually have to point it out to them where exactly the ambiguity in
the diatonic scale lies. The melodic minor scale has more ambiguities,
and this time present between interval classes 3-4, 4-5, and 5-6;
these are a bit more noticeable. The harmonic minor scale has an
ambiguity in every class - 2-3, 3-4, 4-5, 5-6, and 6 - but the one
really obvious ambiguous interval, that everyone who has ever heard
the scale is aware of, is the augmented second and the minor third.

Locrian major in 12-tet, one mode of which is C D E F# G# A# B C, has
ambiguous intervals in every interval class, but being as most of the
seconds are ambiguous the whole thing to me sounds like a whole tone
scale with an extra note thrown in there.

Whether it's that ambiguities in smaller interval classes are more
noticeable, or that an ambiguity in interval class 2 is just
particularly noticeable, is something I'm not sure of. Either would be
a decent way to model this. Neither Rothenberg nor Lumma stability
seems to model this aspect of scale perception - here are some
stability ratings for some common scales:

Diatonic scale (2212221)
Rothenberg stability 0.952381 (20/21)
Lumma stability 0.500000

Melodic minor (2122221)
Rothenberg stability 0.714286 (5/7)
Lumma stability 0.333333

Harmonic minor (2122131) and Harmonic major (2212131)
Rothenberg stability 0.476190 (10/21)
Lumma stability 0.166667

Locrian major (1222221)
Rothenberg stability 0.285714 (2/7)
Lumma stability 0.166667

Tcherepnin's scale (121121121)
Rothenberg stability 0.750000 (3/4)
Lumma stability 0.500000

So by these two measures, Tcherepnin's scale ranks higher than melodic
minor, harmonic minor/major, and Locrian major. By Lumma stability
it's on par with the diatonic scale, by Rothenberg stability it's
about on par with the melodic minor scale. By my ears it's about on
par with Locrian major because of all of the conflicting major seconds
and diminished thirds.

Has anyone ever worked with a metric like this before, before I spend
time trying to implement it?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Apr 1, 2011 at 12:14 AM, Mike Battaglia <battaglia01@...> wrote:
> >
> > Has anyone ever worked with a metric like this before, before I spend
> > time trying to implement it?
>
> I should also add that this sort of thing could be used to model the
> "stability" of even improper scales: something like 22-tet superpyth
> diatonic, for example, has the improper interval hidden between the
> aug4 and the dim5, so you barely ever notice it. On the other hand,
> 22-tet superpyth melodic minor has the diminished fourth set at 6/5
> and the major third at 9/7, which is more noticeable. 22-tet superpyth
> harmonic minor is more noticeable still, since the aug2 is 5/4 and the
> minor third is 7/6.
>
> 22-tet locrian major is a complete nightmare, where the diminished
> third is 109 cents and the major second is 218 cents. The aug4 and
> dim5 in superpyth diatonic also differ by the same amount, but it's
> much less noticeable because the impropriety is hidden away in the
> fourth and fifth. So maybe this could be used to come up with some
> kind of combined stability/impropriety coefficient.
>
> -Mike
>
I see you work alot with 22-tET. Do you consider (mostly) 7 note scales here? Thanks --- is there any reason you choose 22 over, like
31, 41, 43 or 53 etc? (meantones or diaschismics)--- PGH