[Smt-talk] Classical Form and Recursion
Thomas Noll
noll at cs.tu-berlin.de
Mon Mar 23 08:35:57 PDT 2009
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
>>>>>
>>>>> *********************************************************
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> _______________________________________________
>>>>> Smt-talk mailing list
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>>>>> http://lists.societymusictheory.org/listinfo.cgi/smt-talk-
>>>>> societymusictheory.org
>>>>
>>>>
>>>> -----------------------
>>>>
>>>> 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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>
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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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