[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.


Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)

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