[Smt-talk] Classical Form and Recursion

[Olli Väisälä]

On the basis of Dmitri's latest post, it seems that Dmitri and I  
agree on most issues. Some clarifying comments might be in order,  
however.

>
> 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.

I am, indeed, of the opinion that music – complex art music, at least  
– typically accommodates different and even "conflicting" aspects of  
organization. The relationship between prolongational (recursive) and  
"concatenational" aspects of common practice harmony and voice  
leading is one example of this phenomenon. For example, in a recent  
paper on Bach's music, I discuss the following recurring pattern:

strong II (4^ in top voice)– weak V7 (or VII6 etc.) – weak I (3^)–  
strong II6 (2^)– strong V (or some inversion) – strong I

(The "strength" above derives from factors such as meter, design,  
register, cadential emphasis etc.)

 From the "concatenational" point of view we have II–V–I–II–V–I  
circulation. From the prolongational (recursive) point of view we  
have a II harmony prolonged by voice exchange before proceeding to  
the higher-level V and I, ie., II–(V–"I")–II6–V–I. (The quotes around  
I are relevant for the latter but not for the former aspect.) I find  
both aspects crucial for both Bach's compositional patterning and  
adequate listener psychology. On the one hand, the connection between  
the two II chords would not work if the chord sequence did not  
observe "concatenionist" grammar. On the other hand, the first V and  
I are too weak to satisfy the harmonic implications of the strong  
initial II; there is a need to return to the II for making a stronger  
progression to V and I; this is the aspect described by the  
prolongational model.

>
> 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.

Let us state, in somewhat simplistic terms, that Schenker made the  
following two claims. (1) Prolongation—or multilevel patterning of  
harmony and voice leading—is an important aspect of organization in  
tonal masterworks. (2) It is the all-embracing aspect to which all  
other aspects are strictly subordinate. Now (1), of course, does not  
imply (2). I think many if not most present-day Schenkerians (such as  
me) would approve of (1) but not of (2).

>
>
>> 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.
>

I did not want to deny the significance of psychology. There are wide  
variances in what we perceive or fail to perceive depending on  
circumstances, predispositions, conscious efforts etc. What I wanted  
to point out is that the ways in which chords and tones relate with  
other parameters (meter, design, register, etc.) (1) offer empirical  
evidence of the ways in which composers perceived the relationships  
between the chords and tones (2) affect the probabilities of the ways  
in which the listeners will perceive these relationships.

Above, I have described the listener psychology that corresponds with  
the II–(V–"I")–II6–V–I prolongational model. Of course, this  
psychology does not necessarily describe the hearing of all  
listeners: it is "entirely possible to hear (or conceive)" such  
passages simply as six "separate-but-equal" harmonies, but such  
hearing or conception risks neglecting a significant aspect of Bach's  
organization.

Olli Väisälä
Sibelius Academy
ovaisala at siba.fi