[Smt-talk] Classical Form and Recursion

Olli Väisälä ovaisala at siba.fi
Mon Mar 30 04:59:26 PDT 2009

> I think the deep question is this: is "prolongational structure" a  
> quasi-syntactical, more-or-less objective feature of the music, the  
> way TSDT harmonic structure (arguably) is, or is it something we  
> (perceivers, analysts, etc.) impose on the music from outside -- as  
> when we interpret ABABAB... as (ABA)(BAB) = ABABA...

Yes. This is, indeed, a crucial question.

I do not think tonal music is necessarily (deeply) prolongational,  
but it has the potential to be structured (and perceived)  
prolongationally, and I have been suggesting that there is empirical  
evidence that some composers—such as Bach—have utilized this  
potential extensively. This empirical evidence relies on the way in  
which features such as design, register, meter, and rhetoric/gestural  
emphasis support prolongational structures in his music. In other  
words, prolongational models have great explanatory power for the  
arrangement of such compositional features in his music.

[I have touched upon this issue in two recent (forthcoming) studies  
on Bach, and have drafted another one in which I address it more  
expressly. I can send the draft to anyone interested, but at present  
it is only a draft that I have already started to rewrite. I also  
wrote on this issue in this list under the rubric "Why I am a  
Schenkerian"—last summer, I think.]

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.

My work on Bach relies on an argument similar to that on Progression  
(2) above, even though the music and issues involved are, of course,  
much more complex.

> 	1. Representation.  I understand why someone would say that the  
> melody (C4-E4-D4-C4)-(D4-F4-E4-D4) has a hierarchical, quasi- 
> recursive structure: a single motive is repeated at two different  
> pitch levels, and similar concepts ("up by step") apply at both.  I  
> am much less clear about what it means to say that (C4-E4-D4-C4)  
> "represents" or "stands for" (or "prolongs" or "embellishes") a  
> single note on "a higher structural level."  (This gets at a point  
> raised by Ioannis in a previous email.)

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.)

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.

> 	2. Surface accent vs. Depth.  Schenker emphasized that chords  
> could be accented and emphasized by the musical surface in all  
> sorts of ways, and yet not belong to deeper levels of structure.   
> Olli's claim seems to be that accent (meter, design, register) in  
> general provide a definitive recursive interpretation.

The relationship between emphasis and structural weight is complex.  
The relationship between different aspects of emphasis (meter,  
design, register, rhetoric/gestural emphasis) is also complex. There  
are crucial syntactical rules such as dissonance treatment, that may  
override aspects of emphasis. A dissonance is subordinate to its  
resolution even if it is more emphasized metrically or by other  
means. While the issue of analytical criteria is, indeed, complex, I  
think much more clarity about it can be acheived than what is often  
evident in Schenkerian literature (and I have tried to offer some  
contributions in this respect in my recent work).

> 	Personally, I don't see why differences in strength and weakness  
> necessarily imply the need for recursive interpretation.   [...]  
> The question is -- do we think that differences in stress and  
> accent create a need for recursive interpretation of music?  To  
> take your example, suppose someone tells us they hear a particular  
> I-ii-V-I-ii-V-I passage as a sequence of chords, with some strong  
> and some weak, some short and some long, but no "embedding" or  
> "representing" or "prolongation?"  Can we really say that this  
> hearing is *wrong* -- that it misses objective syntax-like features  
> of the music?

I totally agree that differences in strength and weakness do not by  
alone imply the need for recursive interpretation. My argument is  
based on the notion that insofar as recursive (prolongational)  
interpretations based on such aspects reveal meaningful larger  
patterns, this points to the recursive model's compositional  
significance. An example of a "meaningful larger pattern" is the 3^– 
2^–1^ on I–V–I in the above Progression 2.


While I first thought that Bach examples may be too complex for this  
mail, it might be useful, after all, to close it by a brief  
discussion of a real Bach example. Since I mentioned the II–(V–I)–II6– 
V–I pattern in my previous post, I choose an example featuring this  
pattern, Invention in G Major. The pattern is to be found in mm. 20– 
27 (–29).

The initial II (m. 20) is marked by an outstanding textural element,  
the right-hand trill, and opens a four-bar passage with the chord  
succession II–V7–I–VI. The weakness of the three latter chords is  
implied by (1) the absence of change of texture or design, (2)  
hypermetric weakness, (3) higher register of bass tones. After this  
four-bar passage, a relatively strong change in design marks the II6  
(with 7-6 suspension) in m. 24, as the trill transfers to the left  
hand, which also returns to the register close to that in m. 20.  
These features unequivocally bring out the II (m. 20) and the II6 (m.  
24) as the two most prominent chords within span; moreover the  
correspondence between the trills on C in m. 20 (r.h.) and m. 24  
(l.h.) suggest a special connection between the top voice of II and  
bass of II6. There are thus several compositional features that  
clearly could be explained on the basis of a prolongational model in  
which the II with 4^ in its top voice is prolonged by a II–II6 voice  
exchange pattern. For more evidence that this pattern had  
compositional significance for Bach, let us look how it relates with  
larger contexts.

The II6 is followed by V4/2 (m. 26), marked by a relatively strong  
design change. The V4/2, in turn, resolves to I6 (m. 27), which is  
highlighted by a crucial event in the overall design, the beginning  
of the recapitulatory closing section. A hypermetrically  
straightforward I6–V6–I progression follows, with top voice  
emphatically reaching the high B in m. 29. This, of course, suggests  
prolongation of I, with 3^ transferring from the bass to the top  
voice. Such an event supports the compositional significance of the  
preceding II–II6 voice exchange, since the transference of 4^ to the  
bass (II–II6) is followed, after a 4^–3^ resolution (V4/2–I6), by a  
reciprocal transferrence of 3^ back to the top voice (I6–I),  
completing a neat, strongly highlighted 4^–3^ top-voice pattern (mm.  

Further support for the 4^–3^ pattern's compositional significance  
can be gained in two ways:

First, within this Invention, it participates in the overall top- 
voice progression 5^–4^–3^–2^–1^. (5^ is introduced in bar 1 and  
confirmed by its linear relationship with the neighboring 6^ in m. 4,  
which returns to 5^ through a E5–G4 [mm. 4–9] F#4–D5 [10–14]  
unfolding; the points in this structural framework are clearly marked  
by changes of design. The final 3^–2^–1^ progression occurs in mm. 31– 
32; observe how the B5 in m. 29 relates with the B4 in m. 31 through  
similar r.h. figuration.)

Second, more or less similar prolonged 4^–3^ motions occur very often  
in Bach in approximately similar points of form. The recurrence of  
such a pattern speaks to its compositional significance; we need the  
prolongational model for discussing such patterns. Moreover, as I  
discuss in my forthcoming Invention article under the heading "The  
Predictive Power fo the Urlinie", it is characteristic of Bach that  
in pieces in which he first establishes the 5^ as the governing top- 
voice tone (Kopfton), he later underlines a 4^ that leads to 3^ (and  
further to 2^ and 1^) in some special way, in this case the trills.

I hope this suffices to give an idea why I think that restricting  
oneself to "concatenational" hearing may, indeed, risk missing  
objective syntax-like features in Bach.

Olli Väisälä
Sibelius Academy
ovaisla at siba.fi

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