[Smt-talk] Classical Form and Recursion

[Panayotis Mavromatis]

Hi Dmitri,


On Mar 25, 2009, at 10:20 AM, Dmitri Tymoczko wrote:

> I think discussions of the issue of recursion could stand to pay  
> more attention to the distinction between intrinsic structure and  
> our psychological organization of a structure.

One important distinction must be made: recursive vs non- recursive  
is a property of the grammar (i.e. the model), not of the corpus.   
Any finite corpus can be modeled with a non-recursive grammar, even  
if it contains strings like

ABCDCBA

This is because, if the corpus is finite, you'll never have to deal  
with infinite depth of recursion.  Of course, if recursive structures  
keep occurring with high frequency and at deep levels of embedding,  
it will be rather forced and unnatural to insist on a non-recursive  
model.  The real question therefore becomes one of the model's  
economy, explanatory power, ability to relate to other disciplines  
(e.g. cognition) etc.

To summarize, in my mind recursive vs non-recursive is a question of  
model selection, not a question of what the music is intrinsically.

For example, if you claim that Roman numeral progressions can be  
modeled quite adequately using a non-recursive finite-state model,  
you are explicitly or implicitly making certain model-selection  
commitments.  More specifically, you systematically avoid  
interpretations of the corpus that involve higher levels of  
embedding, such as

( ( ( ( (I ii V I) ii V I) ii V I) ii V I) ii V I) ii V I

and instead claim that an interpretation of

I (ii V I) (ii V I) (ii V I) (ii V I) (ii V I) (ii V I)

is just as good.  In other words, you are making modeling choices by  
balancing qualities such as model simplicity vs goodness-of-fit to  
the data.  ("If a complex model only gives us minor improvements in  
accuracy over a simpler one, we might as well just go with the  
simpler model.")

These are all modeling decisions which can be ultimately quite  
subjective.  Even if not everybody agrees on a single set of model  
selection criteria, at least these criteria have to be made explicit,  
and it is precisely these criteria that we should be debating.

Best,
Panos


% ===================================================
%  Panayotis Mavromatis
%  Music Theory Director,  Music and Performing Arts
%  The Steinhardt School, New York University
%  35 West 4th Street, Room 777
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