[Smt-talk] Classical Form and Recursion

Fred Lerdahl awl1 at columbia.edu
Mon Apr 6 15:48:47 PDT 2009

In response to Dmitri:

> The reservation I've always had about this paper [Lerdahl & Krumhansl, 
> 2007] is that these results are only as strong as the particular 
> nonhierarchical model you happened to choose.  It remains possible 
> that tension is perceived nonhierarchically, but the model used in the 
> paper isn't the best one.  It's certainly not the one I would 
> construct.

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.

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.

I should add that one part of the tension model, the amount of 
attraction between pitches and chords, is indeed calculated 
sequentially. This part of the model represents expectancy-tension 
(Margulis's term).

> 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.

Krumhansl's and my general procedure was to go back and forth between 
model and data, checking limited options within the theory to deduce 
explanations of the data and to find weaknesses in the theory. It is 
typical of the hard sciences for there to be a fruitful interaction 
between theory and experiment, and that is what we sought in our 
project. There was no stacking of the deck. As for the non-hierarchical 
model, I don't know how to vary it, since pitch-space distances 
themselves do not vary whether hierarchically treated or not.

> There's a more general issue, which is that the TPS system has an 
> awful lot of moving parts -- essentially, you have several hundred 
> pages of book (or books, since GTTM is in there too) which are being 
> boiled down to a single tension value.  Now there have been some 
> interesting critiques of various small parts of the TPS system (such 
> as Richard Randall's) and it remains possible that these will affect 
> bottom-line tension values.

The assumption is that listeners experience tension holistically. The 
individual contributors to tension, however, are factored out in the 
analysis using multiple regression. As for improving the GTTM/TPS 
theory, yes it is complicated because music is complicated, and yes it 
can be improved.

> BTW, I always thought an interesting project for a grad. student would 
> be to try to build a really good, totally nonhierarchical model of 
> tension.  Even if it was not cognitively accurate, it would strengthen 
> the sorts of comparisons that occur in that paper.
>> 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.

Fred Lerdahl
Columbia University
awl1 at columbia.edu

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