Rothenberg question

I'm using David Rothenberg's pitch perception calculus in a thesis I'm
working on and am having a little difficulty with exactly what property of
scales the efficiency quantity is meant to capture. Since some of you seem
to be pretty familiar with the system, I thought you might have an answer
to this question. He introduces the idea of efficiency by saying that an
efficient musical language is one which can form the greatest number of
distinct words (ordered sequences of tones) given the number of notes in
the scale. The way I interpret this comment is that, for instance, two
augmented triads, C E G# and D F# A# (both 444), are not distinct words.
The calculation succeeds at what it is clearly meant to do: distinguish
asymmetrical proper scales with high stability such as the diatonic scale,
from symmetrical scales with higher efficiency (although much less common
in practice) such as the whole tone and octatonic scales. However, it
gives the highest efficiency rating to scales which seem intuitively closer
to symmetrical scales, such as C E F# G# A#. This scale contains
indistinct 4 note subsets E F# G# c and G# A# c E (2244), indistinct 3 note
subsets C E A#, E F# A# and F# G# c (246), subsets E F# c, G# A# e, and A#
c f#, (264) and so on. But it is because of this property of the scale
that it recieves a high efficiency rating (it always takes 5 notes to
distinguish the scale from one of its transpositions), relative to the
anhemitonic pentonic, which contains more distinct 2, 3 and 4 note subsets.
I must be interpreting "distinct words" incorrectly, in which case the
question becomes, how should we think of the distinctness of pitch
sequences as Rothenberg defines it, and what is the value of that way of
looking at it?

jason

Hello, Jason!

>I'm using David Rothenberg's pitch perception calculus in a thesis I'm
>working on and am having a little difficulty with exactly what property of
>scales the efficiency quantity is meant to capture.

I'm a casual observer of Rothenberg's material. I'll try my best to
communicate my view of things. I'm doing this from memory, but at the very
least it should be something to chew on.

>He introduces the idea of efficiency by saying that an efficient musical
>language is one which can form the greatest number of distinct words
>(ordered sequences of tones) given the number of notes in the scale.

Efficiency is defined as the average length of a scale's sufficent sets
(in terms of the length of the entire scale). The greater this value, the
greater the number of distinct words which contribute _key information_ to
the listener.

>The way I interpret this comment is that, for instance, two augmented
>triads, C E G# and D F# A# (both 444), are not distinct words.

? Are not distinct words of what scale? Perhaps you meant "sufficient
sets" instead of "distinct words"?

>The calculation succeeds at what it is clearly meant to do: distinguish
>asymmetrical proper scales with high stability such as the diatonic scale,
>from symmetrical scales with higher efficiency (although much less common
>in practice) such as the whole tone and octatonic scales.

One could explain efficiency as a measure of this type of symmetry.
Overtly, efficiency is a measure of how much information the listener needs
to determine what key (and thus, what mode) of the scale a melody is in.
For scales where this symmetry is low, the listener may find his place
after only a few scale members are heard. As this symmetry goes up, more
and more motivic material is required for the listener to locate the
frequencies he hears on the rank-order matrix. Rothenberg implies that
some of the fun of melody is playing around in this area -- melodies may
make puns as to which key they belong, and so forth. And eventually, says
R., when the symmetry is total, it's entirely up to the melody to guide
the listener into an interpretation of what tonic is being used.

>However, it gives the highest efficiency rating to scales which seem
>intuitively closer to symmetrical scales, such as C E F# G# A#.

Exactly.

In fact, since sufficient sets are not defined for scales with perfect
modal symmetry (like the whole tone scale), neither is efficiency defined,
and I can see an argument for calling it zero in these cases. But by R.'s
reasoning (above), 1.0 is probably more intuitive: seeing efficiency as how
much work the melody will have to do to fix the tonic.

>This scale contains indistinct 4 note subsets E F# G# c and G# A# c E >(2244), indistinct 3 note subsets C E A#, E F# A# and F# G# c (246),
>subsets E F# c, G# A# e, and A# c f#, (264) and so on. But it is because
>of this property of the scale that it recieves a high efficiency rating
>(it always takes 5 notes to distinguish the scale from one of its
>transpositions), relative to the anhemitonic pentonic, which contains
>more distinct 2, 3 and 4 note subsets.

I assume you meant "insufficient" instead of "indistinct".

>I must be interpreting "distinct words" incorrectly, in which case the
>question becomes, how should we think of the distinctness of pitch
>sequences as Rothenberg defines it, and what is the value of that way of >looking at it?

Substituting "sufficient" for "distinct", you're pretty much on target.

Importantly, says R., proper scales can participate both in this
key-guessing fun, which results from having lots of insufficient words,
and in scale-degree tracking fun. But improper scales can only participate
in scale-degree tracking fun, and so will be most desirable when efficiency
is _low_ (so scale-degree tracking can begin as soon as possible). Proper
scales will be best when efficiency is high, but still functional when it
is low.

I think it's important to keep in mind that with Rothenberg's model, we
address everything in terms of the _entire scale_. We're used to this
kind of thinking, with 5- and 7-tone scales found the world over, where
all the tones are 'on hand', and the entire scale functions as a unit.
But improper scales have proper subsets! I believe that worthwhile
composition can also be done on large, microchromatic scales, where
melodies are played locally, and the sense of a global source set is
non-existent. In fact I believe this has been done with 12-tET in this
century. Rothenberg's model still applies, to local events, but the
usefulness of the model at this scale (ahem!) is questionable, IMO.

Let me know if you can make sense out of any of this,

-Carl