[Smt-talk] Classical Form and Recursion

William Caplin caplin at music.mcgill.ca
Tue Mar 24 17:27:51 PDT 2009

Dear Colleagues,

On 21-Mar-09, at 6:40 PM, Thomas Noll wrote:

>  In the workshop I spontaneously summarized William Caplin's  
> analysis (Classical Form, p. 149) of the core of the development of  
> the 1st movement of Beethoven's F-minor sonata (Op. 2, No.1). Recall  
> that Caplin interprets formal syntagmatic units with formal  
> functions, such as presentation, continuation, cadence (closing  
> function). If we understand the core in terms of a loosely organized  
> "super-sentence", we find units with the functions presentation and  
> continuation in recursive embedding - even if only with depth 2. In  
> particular the presentation of the model involves a large portion of  
> the secondary theme (including its presentation phrase and the first  
> bars of its continuation phrase).

Seeing as some of my ideas were invoked at the start of this thread, I  
would like to add some remarks on the topic of recursion in classical  
phrase structure.  As has already been pointed out, recursion in music  
more obviously appears in the domains of harmony and counterpoint, and  
are most fully developed (as noted by Jeff Perry and others) in  
Schenkerian approaches. Identifying recursion has enormous appeal due  
largely to general aesthetic and theoretical goals (typically  
associated with various forms of organicist thought) of unity,  
coherence and the like. I would, however, urge some caution when  
extending such notions to classical phrase structure and form.

But first, I must acknowledge that in one important sense, formal  
organization is highly recursive.  (Warning: shameless plug to  
follow.) In my essay, “What are Formal Functions?” which appears in  
the recently published book Musical Form, Forms & Formenlehre: Three  
Methodological Reflections, co-authored by myself, James Hepokoski,  
and James Webster (ed. Peter Bergé; University of Leuven Press, 2009),  
I present a number of “tree diagrams” (somewhat in the manner of L&J’s  
GTTM) showing how the generalized temporal functions of “beginning,”  
“being-in-the-middle,” and “ending” can be seen at every level in a  
formal hierarchy (except the very top level, which comprises the  
entire temporal extent of the musical work), and in so doing draw upon  
Kofi Agawu’s notion of introversive semiosis.  In this (highly  
general) sense, formal organization is largely recursive. I then  
propose that the various formal functions defined in my book Classical  
Form are manifestations of these temporal functions, an obvious case  
being the three functions of the sentence form: presentation (a  
beginning), continuation (a middle), cadential (an end).  Seeing as  
the various “formal” functions represent, or express, the generalized  
“temporal” functions, a sense of formal recursion is manifest.  But to  
the extent that the various formal functions are based on widely  
diverse parametric criteria, the sense of recursion is less palpable.

For example, from bottom to top, a “basic idea,” a “presentation,” a  
“main theme,” and an “exposition” each express the general notion of  
temporal beginning.  And their being embedded within each other  
(again, from bottom to top) seems to be a perfect example of recursion  
(when their accompanying middles and ends are included in the picture  
as well).  But the actual structure and defining elements of these  
various functions are significantly different.  The structure of a  
presentation has little in common with the structure of a main theme,  
which itself has little in common with an entire exposition.  Each  
formal function at various hierarchical levels has its own set of  
defining criteria, and as such, it is not clear that the notion of  
recursion is applicable in these cases.  Indeed, I have argued  
elsewhere (in my JAMS article on “The Classical Cadence”) that many  
misconceptions of cadence arise precisely because theorists tend to  
associate any type of “closure” (a generalized formal/temporal  
function) at any level with the notion of “cadence,” a more specific  
formal function, one that (in my opinion) is restricted to certain  
middle-ground formations.

My sense is that the composers of the classical style tend to avoid  
form-functional recursion at successive hierarchical levels.  Thus,  
while it is very common for an 8-bar sentence type (presentation,  
continuation, cadential) to be embedded within a 16-bar period  
(antecedent=sentence, consequent=sentence), and while it sometimes  
occurs that a periodic design can be found within the presentation of  
a sentence (esp. the 8-bar presentation of the compound 16-bar  
sentence), it is rare for a period to be embedded within a period, or  
a sentence, within a sentence.  The case cited by Thomas Noll, where  
the “model” of a developmental “core” comprises a sentence form (as in  
Op. 2/1), seems to be an exception in that  a sentence-like unit is  
embedded within another sentence-like unit.  But although we can  
appreciate that the “grouping structure” of a core indeed resembles  
that of a sentence, marked differences between them must be  
acknowledged as well: in this case, the repetition of the “model” is  
harmonically sequential, whereas the repetition of a basic idea within  
a presentation phrase is normally tonic prolongational.  These  
differences in fundamental harmonic organization are not trivial, but  
go to the heart of why a “core” is situated where it is in the formal  
organization of a work, as opposed to a more genuine “sentence,” which  
has its normative placements (as main theme, or subordinate theme)  
within a movement’s structure.

In short, I believe that certain elements of classical phrase  
structure are regularly recursive, but a view that overemphasizes this  
aspect has the potential of blurring important distinctions among  
units as to their precise formal functionality.  I am thus more  
inclined to recognize the non-recursive aspects of classical form, and  
in that sense I have been strongly influenced by Leonard Meyer (as  
cited in an earlier posting by Nicolas Meeùs), whose notion of  
“hierarchic non-uniformity” seems highly applicable to classical form.

Finally, I would welcome any feedback on these ideas from members of  
the list, especially since I'm venturing here into terrain (i.e.,  
recursion theory) that is not in my area of expertise.

William Caplin

William E. Caplin
James McGill Professor of Music Theory

Schulich School of Music
McGill University
555 Sherbrooke Street West
Montreal, Quebec
Canada H3A 1E3

office: (514) 398-4535 x00279
home: (514) 488-3270

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