[Smt-talk] Classical Form and Recursion

Mon Mar 23 11:10:55 PDT 2009

Dear Thomas,

I agree that both functional model and thematic-motivic extraction model will lead to discoveries in the area of recursion. Sorry for an unwanted advertisement, but my article on Bobrovsky's formal functions which will be published in Theoreia this summer addresses the issue of a ladder-like structural ascension of levels of functional elements initio-movere-terminus. This is done by Asafiev, Sposobin and Bobrovsky, I am just a messenger.

As for thematic-motivic work, although I agree that there is no distinct universal dictionary of musical ideas, I find it facinating how, say, on the first page of Beethoven's Sonata op. 2.no1/I everything seems to work in recursive logical way. The iambic idea in the fist two measures is not just a theme of a composition, but something which amounts to a law, or constitution. It cannot be translplanted to other works, but locally it provides the law & order for the whole movement. I would also call this Grundgestalt a functional recursion because it defines itself using its own characteristics. Everything else is produced from this iambic motive by well-formulated and rigorous operations (extraction, expansion, extension, contraction, inversion, transpostion, permutation, etc), embedded in the first two measures. I wonder if it is possible to formulate this first page in symbols used in logic?

Best,

Ildar Khannanov
Peabody Conservatory
JHU
solfeggio7 at yahoo.com

--- On Mon, 3/23/09, Thomas Noll <noll at cs.tu-berlin.de> wrote:

From: Thomas Noll <noll at cs.tu-berlin.de>
Subject: Re: [Smt-talk] Classical Form and Recursion
To: "smt-talk Talk" <smt-talk at societymusictheory.org>
Date: Monday, March 23, 2009, 10:35 AM

An interesting domain where recursion seems to be a well-applicable concept is human behavior, in particular human action (Presumably there are formal models available in cognitive science, but I'm not familiar with the literature). The semiotician Charles William Morris  approached the semiotics (of Charles Sanders Peirce) from a behavioristic perspective. And thereby  he connected the phases of action with dimensions of sign, and dimensions of value and also distinguished different types of discourse along those lines.  

With respect to the subject which I chose for this thread, namely "classical form and recursion" I like (to flirt with) the idea to draw a connection between the phases of action on the one hand: (1) orientation, (2) modification, (3) consumption and the three formal functions which are central to William Caplin's approach to classical form: (1) presentation, (2) continuation, (3) cadence. 
My feeling is that, in search for recursion in "thematisch-motivische Arbeit" on the one hand and formal functions on the other we would be faced with different theoretical problems and would probably arrive with different results, notwithstanding the numerous meeting points.
Technically the (not yet digested) idea would - at first sight - lead to derivation trees with labels "presentation", "continuation", "cadence" (and some more) for non-terminals and concrete units from the score for terminals. Admittedly this is brute force from a grammatical point of view. But it takes into consideration that there is no dictionary of musical ideas in the narrow sense. 
Sincerely
Thomas Noll

One major difference between linguistics and music theory, with respect to recursion, is that linguistic syntax deals with phrases (sentences) only, not with discourses as does music semiotics. It is to be expected that a semiotic system based on such limited functionality as that of phrases will evidence a higher level of (formal) recursion than a system of discourses. It could be argued that natural languages evidence so to say no recursion at the level of discourses (except, perhaps, in some poetic discourses; but those probably would be more adequately treated by musical than by linguistic theory). 

To state that "music – unlike language – lacks instances of true recursion" begs several questions, not only that of the definition of 'recursion', but also that of the extent, and the conditions, to which any language evidences recursion. Some of the messages posted in this thread explicitly or inplicitly refer to intertextuality, which indeed may be considered a form of external (or extrinsic) recursion. Some stress Schenkerian theory as a theory of recursion – which it is, in my opinion, to a larger (or, at least, to a more interesting) extent than GTTM.

The whole discussion is biased, in my opinion, by the fact that it started with an assumption that natural languages evidence *true* recursion: if natural languages must be the true yardstick of [semiotic] recursion, then indeed music lacks it and the discussion is not worth pursuing. If, however, one engages the discussion without preconcieved idea of where the "truth" lies, then a cross-disciplinary approach might become possible. This is a recurrent problem with general semiotics, that it claims cross-disciplinarity, but hardly can free itself from the model of natural languages.

I don't know whether all natural languages evidence recursion, but I would easily agree with the idea that not evey type of music in the world does – or, to say it otherwise, I am not sure that all types of music can be considered semiotic. But I do believe that our Occidental music, probably one among the most semiotized ones in the world, must be included in any thorough reflexion about [true?] recursion.

Yours,

Nicolas Meeùs
nicolas.meeus at paris-sorbonne.fr
http://www.plm.paris-sorbonne.fr

Thomas Noll a écrit : 

Dear Richard, dear all
many thanks for the vivid resonance to my posting and for the particular links, examples and hints. 
it was the intention of the organizer of the workshop in Berlin to have concepts like iteration, repetition, hierarchy, recursion, self-similarity etc.. on the table and to consider their manifestations in various fields with participants from various fields. The challenge was of course to search for related meanings in different fields.

With regard to the connection with computer programming and music let me mention the rhythmic trees in the software "OpenMusic" (as they occur in the voice-objects of OM) as a nice instance: The very elegant(!) implementation is based on a recursive concept of musical duration. Simply speaking, a duration is a pair (d L), where d is a number and L a (possibly empty) sequence of durations. I like this example because it embodies essential syntactic aspects of traditional rhythmic notation in terms of a context free Chomsky LL1-grammar, namely the programing language LISP. I even tend to believe that the OM developers noticed that traditional rhythmic notation with embedded beams intellectually predates(!) the invention of derivation trees by the 20th century grammarians.

it seems useful, but is not that easy, to formulate satisfactory conditions of fulfillment for what might be called "true recursion" in music.
In order to agree upon an instance of recursion it is perhaps necessary to enter a formalized metalanguage. In a second phase of the discussion different applications of recursive definitions to musical objects would have to be compared and distinguished. A quite general and flexible kind of concept with embodied recursion is what Guerino Mazzola in "Topos of Music" (section 6.7) calls circular denotators. The above definition of rhythmic tree in OM is circular, for example. If we take Mazzola's attempt seriously we should say: Whenever a music theorist deliberately uses a circular denotator in order to identify a musical object, he/she takes responsibility for involving an instance of recursion. But which ones are the "truly" interesting cases for a cross-disciplinary discourse? That's my question in this thread. 

Following a common subdivision of syntactics we may decide to distinguish between paradigmatic and syntagmatic recursion. David Lewin's K-Net-tutorial (on the "Choral" in Schoenbergs Op.11 No.2 is an instance of paradigmatic recursion, while formalized approaches to Schenkerian Analysis (a recent one was presented by Alan Marsden and Geraint Wiggins, CIM08, Thessaloniki) are candidates for instances of syntagmatic recursion. 
I fear a comparison with natural language cannot bypass the differences in the constitution of paradigms and syntagms and the particularities of their interaction.

Sincerely,
Thomas Noll   

Dear all, 

I find this thread less interesting than it might be because the term "recursion" has many different meanings attached to it in different fields. My first exposure was in computer programming, and descriptions of "recursion" in Schenkerian thought made little sense to me at first because of my initial exposure to the term. In such circumstances, it is easy to pull out a definition that will tank someone else's position. I believe philosophers call this "equivocation of terms."
Thomas, how was this problem addressed at the conference? It must have been very stimulating.

Best,

Richard Hermann, Prof. of Music
Univ. of New Mexico

On Mar 22, 2009, at 5:19 AM, Dave Headlam wrote:

Dear Thomas:   I would suggest a few things, of couRsE starting with not a CURSery romp through the Inimitable VolumE “Godel Escher Bach” by Hofstadter:

In an article I wrote for Spectrum 7 I noted the affinity between the opening Rhythm of Beethoven’s opus 59, no. 1 / II and the exposition, both of which (I now think) are sentence-structure based in a (IMO) clear case of recursion — the expos “tries” to get to the secondary key three times, and gets bogged down the first two times in a stubborn D minor, reflecting the rhythm’s inability to break free until it’s third measure.  (It was Ed Hantz who pointed out the “Satz” connection to me when I arrived in Rochester.)

If you are versed in Knets and Gretchen Foley’s writings, you will find that recursive sum / difference combinations are the essence of George Perle’s music — even Michael Buchler would be satisfied with the rationale for recursion in this setting, I would wager.

Dave Headlam

On 3/21/09 6:40 PM, "Thomas Noll" <noll at cs.tu-berlin.de> wrote:

Dear Colleagues,
last summer I participated in a cross-disciplinary workshop on "Recursion in Logics, Language and Art" in Berlin, organized by the logician Ingolf Max..
One participant was the well-recognized linguist Manfred Bierwisch, who argued in favor of a particular difference between natural language and music in the light of the concept of recursion.
He said that music exhibits repetition in a variety of ways, but – unlike language – it lacks instances of true recursion. My feeling is that Bierwisch has a point. But I nevertheless feel the obligation to challenge this assertion. 
My own contribution to this workshop addressed a transformational approach to the theory of well-formed modes, and thereby implied a potential counter-argument on a mathematical level. But I started to think of other possible counter-arguments to Bierwisch's denial of recursion in music. 20th century fractal composition techniques come to mind, but they are still music-theoretical wall-flowers and wouldn't easily overthrow Bierwisch's position with respect to common practice repertoire. Event hierarchies in the sense of Lerdahl and Jackdoff's GTTM are candidates for recursive structures, but their music-theoretical meaning cannot compete with the grammatical meaning of derivation trees in linguistics. In the workshop I spontaneously summarized William Caplin's analysis (Classical Form, p. 149) of the core of the development of the 1st movement of Beethoven's F-minor sonata (Op. 2, No.1). Recall that Caplin interprets formal syntagmatic units with
 formal functions, such as presentation, continuation, cadence (closing function). If we understand the core in terms of a loosely organized "super-sentence", we find units with the functions presentation and continuation in recursive embedding - even if only with depth 2. In particular the presentation of the model involves a large portion of the secondary theme (including its presentation phrase and the first bars of its continuation phrase). 
I would be glad to share this discussion with the list and to later forward the thread to the participants of the workshop.
Sincerely
Thomas Noll      

*********************************************************
Thomas Noll
http://flp.cs.tu-berlin.de/~noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona 
Departament de Teoria i Composició 
Tel (priv.):   +34 93 268 75 19
Tel (mobil): +34 66 368 12 02

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-----------------------

Dave Headlam
Professor of Music Theory

Eastman School of Music
26 Gibbs St
Rochester, NY 14604
(585) 274-1568 office
dheadlam at esm.rochester.edu
http://theory.esm.rochester.edu/dave_headlam

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Thomas Noll
http://flp.cs.tu-berlin.de/~noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona 
Departament de Teoria i Composició 
Tel (priv.):   +34 93 268 75 19
Tel (mobil): +34 66 368 12 02

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Thomas Noll
http://flp.cs.tu-berlin.de/~noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona 
Departament de Teoria i Composició 
Tel (priv.):   +34 93 268 75 19
Tel (mobil): +34 66 368 12 02

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