[Smt-talk] Classical Form and Recursion

[Ioannis Rammos]

On  25-Mar-09, at 04:19AM, William Caplin wrote: 

> 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. 

At the risk of being charged with terminological misappropriation, I would venture to say that this difference of "defining criteria" between hierarchical levels does not preclude recursion. To express a musical structure recursively, e.g. 

Beginning(Middle(Middle(Beginning(X))))

does not require that the two instances of the object-type Beginning (or the two instances of Middle)be internally (algorithmically) identical. If I may momentarily trespass on the domain of data structures and algorithm design, "Object Oriented Programming" (OOP) allows the construction of functions (e.g. "Beginning", "Middle" or "Ending") which, unlike the functions of old-school "declarative" programming,  can detect the type of their supplied argument X (including e.g. the structural level X resides in and its "defining criteria") and handle it accordingly. Thus, Beginning(X) would return the presentation, the exposition, or the 1st movement, given respectively a Sentence X, a Sonata X or an entire Concerto X. In technical terms, the definition of Beginning is "overridden" for each different type of argument X. Although suspicious at first sight, such "overridden" definitions are actually robust and unambiguous. And in terms of elegance, object-oriented modeling tends to
 be unparalleled as it masks procedural details and obviates checks for data type matching. Most programmers in LISP -- the emblematic recursive programming language -- would attest to this.

As a corollary, and perhaps more to the point, recursion really is a property of the representation and not of the music at hand. That some musical styles (and analytical intents) are more natural candidates for recursive representation is equally undeniable, of course. As inspiring material towards a framework for such comparative assessments (and taxonomies) one could recall William Benjamin's "Models of Underlying Tonal Structure: How Can They Be Abstract, and How Should They Be Abstract?" (Music Theory Spectrum 4 (1982): 28-50). For instance, tonal theories that refer to specific pitch events within the musical structure (e.g. Komar's Theory of Suspensions) rather than abstract or implied "tones" (as in Schenker) may be more resistant to recursive modeling -- though this is merely a spontaneous hunch. Benjamin's distinction between prolongational and progressional ("concatenational") models also invites similar hypotheses, the former being probably more conducive to recu
rsive formalization than the latter.

Sincerely,

Ioannis Rammos
Doctoral Candidate
New York University