[Smt-talk] Classical Form and Recursion
Thomas Noll
noll at cs.tu-berlin.de
Tue Mar 24 17:27:43 PDT 2009
If we regard syntax as a part of the semiotic system it should be the
goal of the theoretical presentation to provide access to its
constitution. Therefore I think in each case the answer should be
given by the experts of the semiotic system under investigation.
In the concrete case of harmonic progressions and - in particular -
secondary dominants the question arises, what kind of semiotic system
we deal with [Remember Dmitri's claim that secondary dominants seem
to be genuinely recursive]. Do we speak about syntax and/or about
semantics? Shall we think of harmonic progressions as being modeled
by a formal language? Is this language finite or regular or is it
even non-regular? In the latter case the theoretical need for
recursion would be more serious. Shall we think of expressions like D
(D(D(...(X)..))) with a variable depth of embedding, as David Lewin
used them sometimes? What is the scope of the predicate "Dominant of"?
There is an interesting argument in Riemann's "Ideen zu einer Lehre
von den 'Tonvorstellungen'" where he discusses an instance of V/ii in
C-major. He rejects Moritz Hauptmann's characterization of the d-
minor triad in the C-major context as a 'mixtum compositum' of
dominant and subdominant and argues: If the d-minor chord would not
be a consonant triad we would not be able to conceive its dominant.
This means: To Riemann only prime chords are permitted as arguments
in the scope of D(...). But does this include expressions, whose
evaluation eventually amounts to a prime chord?
An affirmative answer provokes a curious exchange between syntax and
semantics with our normal expectations. Normally we might tend to say
that the tonal functions should be meanings of chords. Carl Dahlhaus
was puzzled by this question and didn't find a clear answer in
Riemann's writings. But to accept expressions like D(D(D(C))) under
Riemann's restricted concept of the scope of D, we would be forced to
say that prime-chord-imaginations (such as an A-major triad) are the
meanings(!) of tonal functions (such as D(D(D(C)))). Thus the
communicated objects are meanings and these are interpreted in terms
of expressions which eventually recover these meanings. This is
neither connotation nor meta-signs in the strict sense of Louis
Hjelmslev, but in way it is an extreme combination of both. And
second curiosity of this view is that the harmonic functions behave
like mathematical functions defined on the chord-tonnetz (a position
that Guerino Mazzola envisaged in some discussions and writings)
Sincerely
Thomas Noll
> The Tonnetz would appear to me as an example of recursion, in that
> it can be taken to represent pitches (as in Euler), or chords (as
> at times in Riemann), or tonalities (Schoenberg's regions). This
> exemplifies the assumption that functions are the same or similar
> at these three embedded levels. Some may remember that at one of
> the early OxMac conferences (in the late '80s, I think), Leonard
> Meyer forcefully questioned this assumption.
>
> One question: does recursion belong to the language or the semiotic
> system itself, or is it mainly a matter of its theoretical
> presentation? Is Chomsky's idea of linguistic recursion merely that
> (Chomsky's idea), or is it a property of language?
>
> Nicolas Meeùs
> nicolas.meeus at paris-sorbonne.fr
> http://www.plm.paris-sorbonne.fr
>
>
> Wayne Slawson a écrit :
>>
>> In response to Gogins and Dmitri:
>>
>> I can believe that recursion can be admitted into a geometric
>> model; the question for me is whether that would be a plausible
>> way of modeling recursion in classical music. But this may be a
>> good way for an old guy to get into this new stuff. I'll think
>> over Dmitri's suggestions:
>>
>>> I don't see any special problem. You can talk about voice-
>>> leading distance on the chord-to-chord level, or between the
>>> start of each sequential units -- as when a theorist like Caplin
>>> says that the descending fifths sequence is really descending by
>>> step.
>>>
>>> More generally, Schenkerians talk about voice-leading
>>> relationships at various levels of the recursive hierarchy; each
>>> level could be modeled geometrically.
>>
>> Regarding the first two measures of the Goldberg V. 25: Yes, one
>> could call this a sequence because the surface details from the
>> first measure are repeated a whole-step down in the second. And,
>> yes, one could say that sequences are either examples of
>> recursion---along the lines of my suggestion earlier---or
>> conversely just moving a bunch of music up or down. These two
>> measures can be heard either way (I guess). But the Saraband
>> Bass is expressed as small, well-formed progressions _throughout
>> the variation_. At bars 9--11, for example, where the bass
>> repeats the first three bass notes of the beginning, there is no
>> sequence, but there are the lower-level progressions. The one on
>> F is now a closing phrase:
>>
>> B-flat5,3/F6,3/G7,3(natural)/C7,3(natural)/F3("flat" then natural
>> in the figuration)].
>> In F: IV I V of V
>> V I
>>
>> It's not possible to _prove_ that recursion is required in cases
>> like these, but it seems the most plausible, and simplest
>> explanation. We're in exactly the same situation as the
>> linguists in mid-century. Chomsky couldn't prove that a Markov
>> grammar was inadequate to explain nested relative clauses; but he
>> could come up with hard cases. (I wish I could dredge up his
>> famous eleven-level-of-recursion sentence whose sentenceship was
>> so hard to confirm that some thought it a counter-case to his
>> argument.)
>>
>> Recursive structures tend to be hard to understand in both
>> language and music. It's not so hard to compose them, but the
>> burden on the listener/reader can be pretty heavy. I think this
>> makes multi-level recursion fairly rare. It might account for
>> the failure of Dmitri's search for harmonic recursion in Mozart's
>> Piano Sonatas. Mozart may have wanted to keep that aspect of his
>> music simple---in contrast to Papa Bach. (I think there are
>> intimations of recursive-like structures in places, however:
>> emphasis on the melodic sixth-degree in K. 333 is reflected in a
>> dominant pedal on V of VI near the end of the Development. There
>> are other instances throughout that piece.) Deep recursion's
>> relative rarity does not mean that it can be ignored, and, in any
>> case, the simplest cases, secondary dominants, are ubiquitous.
>>
>> I agree with Dmitri that music-theoretical recognition and
>> description of recursion in music would be an important and
>> interesting effort. Success would decisively generalize the
>> cognitive status of features widely held to be language-specific.
>> A heady prospect, indeed!
>>
>> And, yes, I've noticed the connection to the Waldstein. Do you
>> think the move to III# is a kind of reflection of the opening?
>>
>> Wayne Slawson
>>
>>
>>
>>
>>
>>>
>>>
>>> 1. I've mainly been thinking about recursion in the harmonic
>>> grammar; I leave open the question about whether there is
>>> recursion in other domains.
>>>
>>> 2. Variations structures, or places where one passages of music
>>> rewrites another, may be a special circumstance. Philip Johnson
>>> Laird addresses this issue (with specific reference to Chomsky
>>> and recursion) in "How Jazz Musicians Improvise" (Music
>>> Perception, 2002).
>>>
>>> About your specific example, I'm not sure I quite follow. Are
>>> you making a claim over and above the fact that this is a
>>> sequence? If the idea is that this is a sequential pattern that
>>> elaborates the theme's descending bass, I agree that it forms a
>>> potential example of recursion.
>>>
>>> The question about whether sequences are recursive is a
>>> complicated one.
>>>
>>> One the one hand, someone might say: sequences aren't
>>> necessarily recursive, are they? You just take a chunk of music
>>> and repeat it, transposed by some interval. Eventually you
>>> stop. It's not clear that we need a recursive grammar to explain
>>> this.
>>>
>>> On the other, it's clear that sequences involve a hierarchical
>>> structure, and that you can't explain them with a simple chord-to-
>>> chord (first-order Markov) model. This is a point Salzer
>>> expresses quite forcefully at the start of Structural hearing.
>>>
>>> Note, BTW, that this Bach passages is one of my stepwise
>>> descending VL sequences: (G, Bb, D)->(F#, A, D)->(F, Ab, C)->(E,
>>> G, C) ... Basically the Waldstein with mode changes.
>>>
>>>> I take it that recursion is hard to fit into a geometric model
>>>> of distance, or am I missing something?
>>>
>>>
>>> I don't see any special problem. You can talk about voice-
>>> leading distance on the chord-to-chord level, or between the
>>> start of each sequential units -- as when a theorist like Caplin
>>> says that the descending fifths sequence is really descending by
>>> step.
>>>
>>> More generally, Schenkerians talk about voice-leading
>>> relationships at various levels of the recursive hierarchy; each
>>> level could be modeled geometrically.
>>>
>>> DT
>>>
>>> Dmitri Tymoczko
>>> Associate Professor of Music
>>> 310 Woolworth Center
>>> Princeton, NJ 08544-1007
>>> (609) 258-4255 (ph), (609) 258-6793 (fax)
>>> http://music.princeton.edu/~dmitri
>>>
>>>
>>>
>>>
>>>
>>
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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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