[Smt-talk] Classical Form and Recursion

Thomas Noll noll at cs.tu-berlin.de
Sun Mar 22 11:03:18 PDT 2009

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.

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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>>> Smt-talk at societymusictheory.org
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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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>> Smt-talk mailing list
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Thomas 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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