barely improper? (was RE: constant structure)

[Dan Stearns wrote...]
>OK, this makes sense to me as well (and I was indeed thinking along
>these lines in the three 3L & 4s examples I gave as well), but what if
>the intervals in question are something like a 17/12 and a 24/17
>though? Strictly proper but oh so close to not being so... is their
>some agreed upon practical range to draw the line with scales that
>might recognize these tiny commatic differences that would perhaps
>only trivially distinguish them from being strictly proper or just
>proper?

[Paul Erlich wrote...]
>Dan, much as with consistency, propriety does involve a very sharp boundary,
>and crossing that boundary by a very small amount, while possibly having no
>audible effect, can amount to a change from a strictly proper scale to a
>proper one to an improper one. I was once bothered by the fact that the
>Pythagorean diatonic scale was improper, but it turns out that Rothenberg
>believed that as long as the worst impropriety (in this case, a Pythagorean
>comma) was to small to be perceived (since you're comparing B-f with f-b in
>this case, it would be hard to perceive the difference), the scale is
>essentially perceived as proper. Carl, am I representing Rothenberg
>correctly here?

I admit I don't have a good grasp of Rothenberg's treatment of these types of cases, throughout his work. I addressed the problem, as regards propriety, by creating measures that work on logarithmic pitch space rather than the number of intervals involved. They are implemented in the "show data" section of Scala 1.6 as "Lumma stability" and "impropriety factor". They are explained in the following excerpt from an un-published paper I wrote, where they are called "Lumma stability" and "Lumma propriety"...

"Lumma propriety and Lumma stability are also easy. Again, we'll use our log-frequency ruler, and again, we'll make a list of all the intervals in the interval matrix and mark them off on the ruler. This time ... we'll draw line segments on our ruler with colored pencil, using the marks as endpoints. We connect all marks belonging to the same scale degree with a single line, using a different color for each scale degree. Lumma propriety is then the portion of the ruler that has no pencil on it -- the portion of the octave that is not covered by scale degrees. Lumma stability is the portion of the ruler that is more than singly covered -- the part where different colors overlap -- subtracted from 1. The idea being that when two scale degrees overlap, the listener will not be able to distinguish them in all cases -- that's a loss of stability. Lumma propriety measures how well-distinguished the non-overlapping degrees are."

My measures are not intended to replace Rothenberg's -- both are useful.

-Carl