[Smt-talk] Classical Form and Recursion
Dmitri Tymoczko
dmitri at Princeton.EDU
Tue Apr 7 06:40:41 PDT 2009
Hi Fred,
Let me say that there's a world of differences between thinking TPS
is great book (which I do) and thinking that it's been proven
correct. My feeling was that your initial email exaggerated the
degree of scientific confirmation of the theory, but that doesn't
mean that I disrespect your work -- far from it. For one thing, I
think it's one of the only books out there that tries to think
simultaneously about both the level of the chord and the level of the
scale. More generally, it's filled with exciting theoretical ideas,
interesting analyses, and it has a kind of synoptic vision that I
find truly inspiring.
That doesn't mean I think it's totally right.
For me, the single biggest issue with TPS is that it seems to operate
with the assumption of complete perceptual accuracy. Tension values
(etc.) are calculated on the basis of complex relations found in
scores -- suggesting that the unconscious mind is accurately
perceiving 100% of the (relevant) musical information aurally. (It
has to have this information in order to calculate in the way the
book describes.) As someone with a fair amount of experience with
introductory music students, I find this implausible. We lose an
awful lot of information when listening to music. It can be
genuinely hard, for introductory students, to tell whether they're
hearing a V chord. (To say nothing about returning to the tonic key
at the end of a long piece.) This is an area, I think, where the
model of linguistics has not had a salutary influence -- the
assumption of perceptual accuracy is much more plausible in language
than in music.
So, as I see it, it's really an open question how much of the TPS
formalism would survive a more realistic appraisal of the perceiver's
limitations. Maybe this issue of inaccuracy isn't such a big deal;
maybe it is. We simply don't know right now.
> The tension model in question addresses tonal tension, which is
> induced purely by pitch relations, and ignores for the present
> other kinds of tension, such as those caused by speed, loudness,
> timbral characteristics, or rhythm. Probably some of these
> contributors to tension are less hierarchical than is tonal tension.
The assumption that there is a notion of tonal tension independent of
all these factors, and that it contributes a substantial part of the
overall experienced tension in a musical work, is a big one. (For
what it's worth, I doubt this would be true if we looked at non-
classical works such as Palestrina or even Nirvana's "Smells Like
Teen Spirit"; I'd actually be interested in the results of knob-
turning experiments on nonclassical pieces such as these.) You can't
test this assumption by creating a model of tonal tension and showing
it correlates with overall tension values derived from experiments.
What needs to be done is to show that there don't exist other models
that are better.
If I were trying to build a robust nonrecursive model of tension, I'd
feature rhythm, loudness, timbre, dissonance, register, schematic
expectations ala Gjerdingen, etc. I'd have the model expect cadences
at certain temporal intervals. Would it do better than TPS? We just
don't know -- which is the point I'm trying to make. Until we know
this, we are not in the position to conclude that recursive models of
music perception have been shown to be correct.
> Sequential (non-hierarchical) tension is calculated mostly by pitch-
> space distances from one event to the next. The model's distances
> correlate with empirical data on distances, and it is clear that
> our subjects took the hierarchical alternative. I don't know what
> other sequential model Dmitri would construct.
When you get into the details, you uncover some interesting questions
about how TPS relates to empirical data. As we've talked about
privately, the model's "key distances" do not correlate to the
results of experiments that directly test key distances -- they
correlate to Krumhansl's *calculated* key distances, which are not
the direct product of experiments. (I happen to think there's a
better approach to key distance; one that correlates to modulation
frequencies in actual pieces.) Similarly, Richard Randall shows that
it's easy to come up with alternative chord distances that correlate
better with the empirical data.
TPS, in my view, walks a fine line between being a theory book -- and
trying to come up with theoretically interesting models -- and being
responsive to the data. That's part of its attraction -- if you just
adopted the empirical results and went from there, it would not be as
interesting. Instead, you try to come up with plausible theoretical
models that generate the empirical data reasonably well. But
sometimes these two goals pull in opposite directions.
>> Another issue is that (as I recall) the prolongational analyses
>> used in the paper are chosen to fit the data, rather than being
>> derived from theoretical principles -- so in essence, the
>> conclusion is that there *is* a prolongational hearing that models
>> tension better than the particular nonhierarchical model you've
>> chosen. But this is sort of stacking the deck, since you didn't
>> vary the nonhierarchical model to fit the data.
>
> On the contrary, all of the prolongational analyses are generated
> by rule. There is a limited degree of wiggle room in the analyses
> because different rules can conflict, so that the overall result
> depends somewhat on rule weightings. In the one such case we
> discussed in detail (the Grail theme in Parsifal), the conflict was
> between pitch-space distance and branching balance on one hand and
> sequential parallelism on the other. The analysis of one event hung
> in the balance. It was apparent from the data that listeners
> favored the parallelism factor. Other cases of rule conflict (in a
> Bach chorale and a Chopin prelude) were mentioned more briefly.
I'm not sure we're disagreeing -- the prolongational analyses are
indeed chosen to fit the data, from among the sets consistent with
the various weightings of the rules. If there were not a reasonably
wide range of possibilities, you could easily have chosen the *worst*
fit, in order to make your results as strong as possible. You chose
the best fit instead, thereby biasing the results in favor of the
recursive approach.
So what you've shown is that there does exist some (derivable-
through-TPS) recursive hearing that improves upon the nonrecursive
portions of the TPS system. This is really not the same thing as
showing that we actually do hear recursively.
>>> We have already shown that the perception of hierarchical
>>> structures goes deeper in music than what is usually supposed for
>>> language.
>>
>> Respectfully, I think it's fair to consider this issue still to be
>> open, given the issues discussed previously. Isn't it correct
>> that a simple opposition between diatonic and octatonic (using Ian
>> Quinn's Fourier-based method) predicts the Messiaen tension values
>> about as well as TPS?
>
> I agree that this issue between language and music remains open.
> Krumhansl's and my study was really just a beginning. As for
> Quinn's Fourier balances, Krumhansl tried to make this work, but
> the project fizzled out. The data fit was poor, and the approach
> didn't make much theoretical sense. It would be worthwhile, though,
> to have a contrasting tension model as competitor. A well-
> articulated alternative, even if unsuccessful, would help set the
> issues in relief.
To me, there's a world of difference between "we have already shown
that the perception of hierarchical structures goes deeper in music
than what's usually supposed for language" and "I agree that this
issue between language and music remains open." As long as we agree
on the latter, I'm happy.
DT
Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)
http://music.princeton.edu/~dmitri
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