[Smt-talk] Classical Form and Recursion

Fred Lerdahl awl1 at columbia.edu
Wed Apr 8 08:34:56 PDT 2009

Dmitri writes:

> 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.

This issue concerns the competence/performance distinction in 
linguistics and, more generally, the status of scientific models. The 
world is messy. To make progress in understanding it, one must  break 
the object of study into manageable parts and adopt idealizations that 
simplify it. Physicists say they build models. That is my attitude, and 
I think that most music theorists share it implicitly. One can't study 
the loss of information because of wandering attention while listening 
to music unless one has already developed a theory about what that 
information is and the principles by which the information is received.

The study of musical tension is attractive partly because intuitions 
about it are so spontaneous. The average listener cannot name a V or a 
I chord but nevertheless responds to a progression in terms of degrees 
of tension. Put another way, a chord has a particular location in tonal 
space; the listener has implicit knowledge of the space and feels 
motion and force as patterns of tension.

>> 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.

Again, to make progress one must idealize and break apart the 
phenomena. In the project with Krumhansl, we equalized speed, loudness, 
and timbre, and we neutralized rhythm as best we could, given the 
current lack a theory of rhythmic tension. For the most part, then, we 
were able to present the subjects with materials that isolated tonal 
tension from other kinds of tension. As for comparisons, only a 
philosopher would demand that a proposed model succeed better than all 
other hypothetical models. Scientific inquiry doesn't work that way. 
One makes a theory to explain and predict the phenomena in question. If 
there are alternative theories, one tries to find out which one works 
better. In the case of musical tension, an alternative theory does not 
yet exist, so how is one to compare?

> 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.

TPS already includes a dissonance component that appears to succeed 
empirically, and it makes a start in modeling schematic expectations. 
For the rest, yes, please make a theory so a comparison can be made. I 
agree that loudness, for example, is not hierarchical.

In TPS I developed both sequential and hierarchical calculations of 
tension, anticipating that listeners rely on both in more or less equal 
measure. The evidence so far, however, indicates that listeners hear 
tension patterns hierarchically more than sequentially.

> 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.

In our project, Krumhansl isolated the contributing variables to 
pitch-space distance and found a high correlation with the empirically 
derived distances from her earlier work. Thus the tension project 
provided independent confirmation of that work. The alternative that 
you propose, to study the statistical occurrences of modulations in a 
broad corpus of tonal pieces, is also valuable. Perhaps the probability 
of a given modulation correlates better with your model than mine. If 
so, this raises the interesting question of the relationship between 
compositional practices and listening schemas.

I haven't studied the Randall alternative.

> the prolongational analyses are indeed chosen to fit the data, from 
> among the sets consistent with the various weightings of the rules.

You exaggerate. The choices are few.

> 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.

There is no contradiction in my two statements. The evidence so far 
suggests that hierarchy is deeper in music than in the syntax of 
sentences. But this is a complex topic, and of course the issue is 

Fred Lerdahl
Columbia University
awl1 at columbia.edu

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