new work on Rothenberg's model

This paper assumes familiarity with Rothenberg's model of melodic
perception. For those of you who aren't familiar with the model, try the
tuning dictionary, or go to...

http://www.egroups.com/group/tuning/

...and put message number "4044" in the "jump to" box. If anything's still
unclear, post your question to the list.

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First, let's recall the following exchange, which took place here in March
of this year...

[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'd like now to clarify the response I gave on March 9th, in a message
titled "barely improper" (egroups message #9175).

First off, yes, Paul, I believe you were representing Rothenberg correctly.
But I think we can go one better than old R here. With the "show data"
command in Scala 1.6, we see "Lumma stability", followed by "impropriety
factor". What are they?

Imagine a log-frequency ruler whose total length is the interval of
equivalence ("formal octave") of our periodic scale. Take all of the
unique intervals in the scale's interval matrix and mark them off on the
ruler. Now, draw line segments on the ruler with colored pencil, using the
marks as endpoints. Connect all marks belonging to the same scale degree
with a single line, using a different color for each scale degree. "Lumma
stability" is the portion of the ruler that has no pencil on it -- the
portion of the octave that is not covered by scale degrees. "Impropriety
factor" is the portion of the ruler that is more than singly covered -- the
part where different colors overlap. 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 propriety. Lumma stability measures how well-
distinguished the non-overlapping degrees are.

In my earlier post, I correctly characterize my measure as being a version
of Rothenberg's, applied to logarithmic pitch space rather than 'scale
degree rank space'. I say that both versions are useful, begging the
questions: "Which is more useful?", "When is one better than the other?",
and so on.

To answer, this paper will contribute "rank standard deviation/range".
This measure will tell us, for a given scale, how effective "equivalence"
is, and thus how pertinent any of the measures which are "invariants of
equivalence" will be to the given scale.

What is "equivalence"? It is the process which converts an interval matrix
into a rank-order matrix. Which measures are "invariants of equivalence"?
Rothenberg-stability, Efficiency, Constant Structures, are all invariant
under equivalence, since they can be defined on the rank-order matrix
alone. But Lumma-stability is _not_, since scales which share the same
rank-order matrix may have different Lumma-stability values. Therefore,
"rank standard deviation" and range will tell us when my version is better,
along with a few other things...

Rank standard deviation and rank range are designed to measure the
complexity of a scale's interval matrix. The higher these values, the more
difficult it will be for the listener to construct the matrix, and the more
likely it will be that he uses a matrix he has already learned instead
(thus hearing the scale in question as a re- or mis-tuning of a scale he
already knows). The measures are easy to calculate. Again, we need our
log-frequency ruler, and again, we'll take all of the unique intervals in
our scale's interval matrix and mark them off on the ruler. But this time,
instead of involving scale degrees, we simply find the ratios between all
pairs of consecutive values marked on the ruler. The measures are then
the standard deviation and the range of the decimal values of these ratios.

For example, take the major 7th chord in 12-tet. Its interval matrix:

0 4 7 11 12
0 3 7 8 12
0 4 5 9 12
0 1 5 8 12

Unique intervals: 0 1 3 4 5 7 8 9 11 12

Distances between consecutive marks: 1 2 1 1 2 1 1 2 1

Range: 1 (this should be expressed as a fraction of the formal octave,
so: 1/12).

Standard deviation: Well, you get the idea (any measure of central tendency
should work here).

The idea is that since the interval matrix works by ranking intervals by
size, the easiest matrices to use will be ones in which the sizes are most
evenly distributed. Since for periodic scales the intervals between the
intervals (ratios between consecutive marks on the ruler) must sum to the
formal octave, the mean ratio occurs between every pair of marks in the
ideal case, and the standard deviation will be in direct proportion to the
complexity. Range should work in a similar way, but may differ by a
scaling factor from the standard deviation? I'm not sure which is more
intuitive. Perhaps someone with some knowledge of statistics can help me
out here.

So there you have it. I should go over the Rothenberg papers again, to
refresh myself on what he says about the tolerance of perception. My
feeling was, that since he was working with ETs, and especially since he
used a relatively small ET (12-tet) for testing his measures, he didn't
run into many close calls. From speaking with him, I got the impression
that he considers equivalence very strong -- scales are gestalts which hold
their character through a great deal of mistuning. From his point of view,
any instability is undesirable, and scales would be ruled out long before
close calls were an issue. In other words, Rothenberg is really only
interested in scales for which my rank deviation is very small.

But I, for one, don't frown so much on instability, especially if it isn't bad
in pitch space, and we may arrive at close calls for harmonic reasons.
Rothenberg was interested only in melodic scales -- many of us are interested
in harmony, too, and don't have the luxury of re-tuning scales until they fit
neatly into a rank order matrix.

-Carl