[Smt-talk] Classical Form and Recursion

[Dmitri Tymoczko]

Olli Väisälä raises a bunch of good points:

> I would like to make some points that I find crucial for the  
> discussion about recursion in prolongational (Schenkerian)  
> structures and about pertinent empirical evidence.  First of all, I  
> do not think this issue can be adequately treated merely on the  
> basis of symbol sequences, as suggested by Dmitri Tymoczko's post:

I wasn't trying to suggest that the issue can be comprehensively  
treated in this way.  I was just trying to point out a distinction  
that is important, and that is rarely discussed.

> Musical events are not adequately determined in terms of such  
> sequences, since each occurrence of "A" and "B" may relate  
> differently with aspects such as meter, design, register, emphasis,  
> etc. , and these aspects are crucial for both the expression (by  
> the composer) and perception (by the listener) of prolongational  
> relationships. Consider a very simple example in 4/4 time:
>
> bar 1: I–V–I–quarter rest
> bar 2: V–I–V–quarter rest
> bar 3: I–V–I–quarter rest
>
> If this occurs in a composition, we may speak of empirical evidence  
> for the hypothesis that the recursive interpretation, (IVI)(VIV) 
> (IVI) = IVI, models a pattern relevant for composition. The model  
> has considerable explanatory power for the way in which the  
> composer has related the chords with  features such as meter and  
> grouping.

In the one place where I've published on this (http:// 
music.princeton.edu/~dmitri/tonaltheories.pdf), I mention exactly  
this possibility.  The point is that a supporter of recursion is  
going to have to rest their case on rhythm, register, etc., rather  
than on the Roman-numeral harmonies themselves.

On this view, the harmonic organization and the "recursive"  
organization will in general conflict.  For instance, there's an  
extremely well established, purely harmonic grammar, that claims that  
tonal music organizes itself into (speaking loosely) tonic- 
(predominant)-dominant-tonic cycles.  And indeed Olli's example does  
exactly this: it's a series of overlapping I-V-I cycles.

Presumably the supporter of recursion will agree that this  
organization-into-harmonic-cycles is an important feature of the  
music.  Which means that they will claim that we need to conceive  
music in a disunified way: on the one hand, nonrecursively, as a  
series of concatenated cycles; on the other hand, recursively, as a  
series of recursively nested harmonies.

To my mind, this raises interesting questions vis-a-vis Schenkerian  
ideas about musical unity.  Clearly, we have a disunified picture  
here, with the harmony and register/rhythm articulating very  
different organizational schemes.  The issue is analogous to that  
discussed by Rick Cohn, with respect to motive.

> As my example clarifies in a very simple way, a sequence may be  
> "intrinsically nonrecursive," but perceiving it recursively may be  
> supported by features that are intrinsic to the music.

I would say: the sequence of harmonies is intrinsically nonrecursive,  
but the music as a whole may suggest recursive structure.

> Whether the "recursive potential" of a sequence becomes realized  
> does not depend only or primarily on the listener's psychology, but  
> on the composer's acts: on the ways in which he/she supports (or  
> fails to support) certain elements and relationships through  
> parameters such as meter, design, register etc.

Here I would probably disagree: it's still a psychological matter  
whether one perceives the simple example Olli has described as  
recursive.  We, in our post-Schenkerian era, tend to take this for  
granted, but it's entirely possible to hear (or conceive) this sort  
of sequence non-recursively.  Just because you have I-V-I,  
articulated by rhythm, doesn't mean you have to hear this as  
"prolonging" I; you could hear three separate-but-equal harmonies.   
(When I take a trip from Philadelphia to New York and back, my trip  
does not "prolong" the state in which I stay at Philadelphia.)

So I'd say there's a lot of psychology here, regardless.

DT

Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)
http://music.princeton.edu/~dmitri





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