[Smt-talk] Classical Form and Recursion

Ildar Khannanov solfeggio7 at yahoo.com
Fri Mar 27 12:48:02 PDT 2009

Dear Michele,
I think, very much so. The paradigmatic aspect resists grouping or any kind of allignment. In Jakobson's view the paradigmatic is perpendicular to the line of the syntagmatic. And ancient Greek etymology of paradigmatic tells us about para-deiknymi, pointing out. Yet, these obects at which it points may create their own syntax, the secondary syntax, the order of a diffent plan. Recently, I have been working with Algirdas Greimas's fonction narratif, which is an attempt to describe the order of modalities and essentially paradigmatic events in parallel with syntax. I completely agree with and enjoy Thomas' idea that tonal functions are paradigmatic, while the realisation of their power, the progression, certain voice leading, is syntagmatic. Tonal functions are such not because of their placement, but beacause... they are such! They are terminal, they determine what other things will be.. The Dominant is the Dominant because it is the Dominant (CIRCULAR
 DEFINITION!). It is a phenomenon. It is alpha and omega of music. There are also Tonic and Subdominant. They are not determined by syntax, rather they give the law of syntax. The V represents D? No, the D can be represented by V (or by vii, or by VII) or by (048) (the latter is in Schoenberg op. 15 No 14).
By the way, formal functions are also paradigmatic. At least, that is how Russian theorists beginning with Asafiev see it. Arkhon, medzon kai eskhatos are things in themselves. As such, they are a-temporal. Their presence in music constitutes time, kneeds and molds time in a certain way. Very often the eskhatos, or, in Asafiev's term, terminus, appears in the beginning of a piece, thus wreaking havoc on its temporal organization. Schubert op. 89 begins with the ending function. This tells us that the time has ended even before the winter journey has begun. It is over. 
The use of recursion to grasp the essence of the paradigmatic is interesting. However, as many have already pointed out, the term resursion is difficult. I think, it is such not because it is poorly defined in mathematics, but because its application to music is tricky.
In fact, recursive function is y= f(x) where x icludes the y. This can be good or bad, it depends because in order to determine the y, we must already know what y is! This is not logic itself, it is just a trick of logic or an ostensive definition.  The essence of logic is not recursion, but inference. We have to know what y is in relation to x operated upon by f. If x is such and f does so and so, then the y is such. Y is inferred from x by means of action f. Tertium non datur. Recursive function may cause circulatio in adjectum, which is very bad for logic! What helps is that most of recursive functions can be expressed in a non-recursive form.
Another aspect of recursion is the method of insertion by analogy (Russ. podstanovka). 
1 1 2 3 5 ....? Of course, 8 and 13 etc. Tristan and Isolde, Pelleas and Melisande, Ruslan and.........? Of course, Ljudmila. A.B. Marx provides a similar example in the 3rd volume of his Lehre,  termed "the Maxim": C.D,E,F..........C' ? of course, there must be G and A. However, in the first case, the purpose of having this recursion is important, it is COUNTABLITY. If there is no recursion, the row in not countable. This is the purpose of recursive function in mathematics. But the second example is much more trivial. It does not constitute the logic of natural language. Yes, threre is the sense of rhythm and grouping, but it is superficial. It is quite disappointing, how the useful mathematical rule is sold cheap here. In fact, Chomsky's recursion is a metaphor, it is the  recursion "so to speak." If we apply  this "linguistic recursion" to music, it would be the metaphor of metaphor, "so to speak" in second power. The events A B C D are
 countable and conform with the initial function y=f(x(y)). What have we proven? The original purpose, the definition of whether the numeric row is countable, will be completely lost. And A.B. Marx does not describe his Maxim as a recursion. It is a tree-like structure powered by motivation of the motive. 
Ildar Khannanov
Peabody Conservatory
solfeggio7 at yahoo.com

--- On Thu, 3/26/09, Michele Ignelzi <m.ignelzi at tin.it> wrote:

From: Michele Ignelzi <m.ignelzi at tin.it>
Subject: Re: [Smt-talk] Classical Form and Recursion
To: "smt-talk" <smt-talk at societymusictheory.org>
Date: Thursday, March 26, 2009, 6:52 PM

Dear Ildar, Thomas, Dmitri and all,

Does this approach have anything in common with Jakobson's projection "of the principle of equivalence from the axis of selection into the axis of combination," which was the basis for Ruwet's and Nattiez's paradigmatic analysis?


On Mar 26, 2009, at 02:14 AM, Ildar Khannanov wrote:

> Something similar has been done in literary analysis under the term "juxtaposition of metaphor on metonymy."

Michele Ignelzi
State Conservatory of Music, Fermo, Italy
m.ignelzi at tin.it
Smt-talk mailing list
Smt-talk at societymusictheory.org

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