[Smt-talk] Classical Form and Recursion

Dmitri Tymoczko dmitri at Princeton.EDU
Mon Mar 30 08:08:12 PDT 2009

On Mar 30, 2009, at 7:59 AM, Olli Väisälä wrote:

> Instead of discussing Bach, I shall again present simple examples  
> for illustrating that there may be varying degrees of evidence for  
> prolongational structuring. Consider the following progressions in  
> 4/4 time (soprano tones in parentheses; / = barline).
> Progression (1). I (1^) – V (2^) – I (1^) – V (2^) / etc. / etc.
> Progression (2). I (3^) – V (2^) – I (3^) quarter rest / V (2^) – I  
> (1^) – V (2^) q.r. / I (1^) – V (7^) – I (1^) q.r.
> In Progression (1), there is no empirical evidence that the  
> "composition" is affected by any prolongational patterning beyond  
> the most immediate level in which each V prolongs surrounding Is  
> (as supported by the meter). Deeper-level models have no  
> explanatory power in this case. (Trying to identify deeper levels  
> would be as nonsensical as hierarchicizing the tones in a trill.)
> In Progression (2), there is considerable empirical evidence that  
> the composition is affected by a larger prolongational pattern,  
> since meter, grouping, and regular design bring out archetypal  
> patterns both for the harmony (I–V–I) and the top voice (3^–2^–1^).  
> The prolongational model has considerable explanatory power for the  
> way in which the music is shaped. Of course, one cannot "prove"  
> that the composition manifests prolongation. The features that seem  
> to support the 3^–2^–1^ shape might be a chance products or better  
> explained by some alternative model. Given the number of such  
> features, however, the chance explanation does not seem too likely,  
> and it is not easy to see ways in which the prolongational model  
> could or should be bettered. At the very least we can say that  
> restricting ourselves to a purely concatenational model would  
> create a high risk of losing a crucial compositional aspect.

I realize now that I wasn't being clear enough.  In the example I gave:

	ABABA ... = (ABA)(BAB)... = ABAB "on a higher level"

There were really a pair of issues.  One is grouping -- getting from  
ABAB... to (ABA) ...  But the other is reduction -- getting from  
(ABA) to A.  The point of the "Americans care only about Americans"  
example was that this latter process is also problematic: the mere  
presence of ABA (as in "Americans care ...") does not automatically  
license or motivate a reduction to A ("Americans").

Your distinction between Progression 1 and Progression 2 addresses  
the grouping issue but not the reduction issue.  Here, it is clear  
that the harmonies should be grouped (I-V-I)(V-I-V)... But it is  
still not clear why we should "reduce" each unit to a single chord.   
You simply assert that we should -- which is fine, but it's basically  
presupposing an answer to the very question I'm trying to ask.

The point I'm trying to get at is that may be perfectly reasonable to  
reject a certain notion of prolongation in Progression 2.  Consider  
two different interpretations:

	A. Progression 2 consists of a melodic down-up-down pattern,  
harmonized with I and V, and repeated a step lower each time.
	B. Each "unit" of Progression 2 "stands for" or "represents" a  
single chord.

Statement (A) I think is unproblematic; statement (B) is much more  
so.  The point is that there is no need to postulate a relation of  
"standing for" or "representing" in order to explain the descending  
melodic line, or the hierarchical relations among the units, etc.  In  
fact, I think it's interesting to really try to see how far you can  
get without "representation" in the sense of statement B.  My  
preliminary conclusion is that it's quite possible to be a good  
musician -- to write interesting music, to appreciate the Great  
Works, to perform well on an instrument -- without it.

There is, furthermore, an interesting-but-subtle methodological  
issue.  Suppose you have a first-order harmonic grammar, in which  
each chord has a certain number of "allowable" destinations.  Then,  
given a sequence such as ABA CDC EFE ..., you can always get a  
grammatical sequence by "reducing" each 3-chord unit, for example to  
produce ACE... The reason is that each local transition in the latter  
sequence is legal, because it appears as a local transition in the  
original.  So if ABA CDC is legal than so is AC.  (Which is just to  
say it's a first-order harmonic grammar, in which the acceptability  
of transitions is a local matter.)  So what might *appear* to be  
evidence of recursive structure in a particular sequence (in this  
case the grammaticality of ACE) is really just a reflection of basic  
features of the underlying, nonrecursive harmonic grammar.

So in this particular case, you might think that, given

I (3^) – V (2^) – I (3^) quarter rest / V (2^) – I (1^) – V (2^)  
q.r. / I (1^) – V (7^) – I (1^) q.r.

It's a really interesting and remarkable fact about this particular  
sequence that

I (3^) V (2^) I (1^)

Is legal.  But it's not -- it's just a feature of the fact that  
certain local transitions are commonly acceptable in tonal music.

Now it might be you're not interested in this sort of issue -- for  
you, perhaps, it's obvious that Progression (2) has recursion in the  
sense of statement B.  That's fine, but it just means you're not  
interested in the question I'm interested in.  I believe it's really  
important and interesting to ask what evidence we have for recursion  
in the sense of statement B.  To really subject statement B to  
skeptical challenges, to see how well it stands up.  Just repeating  
that we do have evidence for statement B doesn't really speak to this  

> I am not sure whether I understand your concerns here. Tonal  
> tradition is rich in practices in which basic skeletal progressions  
> are subjected to diminution, variation, embellishment etc. I do not  
> suppose you mean that you have difficulties in grasping this  
> concept intuitively—in cases such as Progression 2 above, for  
> example. (Naturally, when one proceeds to higher structural levels,  
> involving more extended temporal spans, such concepts become less  
> self-evident for intuition, but this does not seem to be your point.)

The question is really about the legitimacy of extending certain  
sorts of unproblematic notions of embellishment to other cases.  I  
have no problem saying that, given CEG in the harmony and a melody C- 
D-C, the D is a neighboring tone and that there is an "underlying" C  
major chord that persists throughout.  I have a much harder time  
understanding an analogous claim where we start with a perfectly  
syntactic progression, like I-ii-V-I-V-I, and we say that it  
embellishes another syntactic question, say I-V-I.  The difference is  
that in the one case, we transform a nonsyntactic object (like the  
"harmony" CEGD) into a syntactic one.  In the latter case, we're  
trading one syntactic progression for another, and the justification  
for this is far less clear.  This is because we cannot use the fact  
of nonsyntacticality as a trigger for the need for reduction.

Again, it may be that you're not interested in the almost- 
philosophical question "what does it mean to say one passage of music  
represents another?"  Perhaps you're not at all motivated by the  
thought that some of our assumptions might be on shaky ground.   
Perhaps you want to do your reductions and get on with things.  All  
of this is perfectly fine.  But there are, I think, interesting  
questions here -- even if theorists often proceed as if the answers  
were obvious.

> Instead of trying to describe "what it means to say" that something  
> is prolonged, it might be more useful here to concentrate on the  
> consequences of prolongation. If prolongational readings are  
> capable of revealing larger patterns that seem likely to be  
> pertinent to a certain composition (as in Progression 2 above),  
> this should suffice for justifying the concept.

Oh, but this justification works just as well for the reduction  
ABABAB = (ABA)(BAB) = "AB on a higher level."  It's just to say,  
roughly, that recursive interpretation is something *we* do to music,  
and if we like doing it, then that's justification enough.  This is  
the Cook/Boretz line.


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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