[Smt-talk] Classical Form and Recursion

Brian Robison brian.c.robison at gmail.com
Sun Mar 22 09:07:21 PDT 2009

Thanks to Thomas Noll for introducing this topic. It's a long time since I
investigated these ideas formally, so I can't cite chapter and verse, but
I'll share the following, largely anecdotal summary:

My instinctive response is to agree with Bierwisch, because his conclusion
matches what I found in the 1990s when I investigated fractal geometry in
music: viz., that traditional music is neither strictly recursive (as in
classical mathematical fractals such as Koch snowflake, Sierpinski gasket,
Julia sets), nor strictly random (as in L-systems and the quasi-botanical
forms generated algorithmically in *The Algorithmic Beauty of Plants,* by
Przemyslaw Prusinkiewicz, Aristid Lindenmayer et al., Springer 1996).

Some attempts to claim deep recursion have gotten considerable press, but
they don't hold up to specialist scrutiny. Almost 20 years ago, Kenneth J.
Hsu and Andreas Hsu published an article, 'Self-Similarity of the "*1/f
Noise*" Called *Music*.' in *Proceedings of the National Academy of
Sciences,* Vol. 88, No. 8. (15 April 1991), pp. 3507-3509. At first glance,
their graphs of J.S. Bach's C-major Invention and various mechanistic
reductions of it look mighty impressive... but on closer inspection, one
sees that they've indicated rests as occurrences of a pitch one semitone
below the lowest pitch in the piece, rather than as gaps in the lines.
Redraw the graph more accurately, and the images are far less striking. And
as of about 10 years ago, David Cope's *Experiments in Musical
Intelligence*(or EMI, for short) generated music in established styles
used a strictly
recursive application of harmonic formulas (or at least, that's how it was
explained by Douglas R. Hofstadter in a public lecture); Cope's results
sounded convincing within any one measure or two, but tended to wander
incoherently at larger levels (in particular, randomly returning to the
tonic key area long before it was formally appropriate to do so).

When I was an undergraduate first learning Schenker's ideas and techniques,
I was skeptical (and even irked) by the different harmonic rhythms of
different time-scales; wouldn't the theory make better sense if patterns
were more consistent, from the musical surface all the way through to its
deepest structures? But in light of the Hsus' work, and Cope's, it seems
that these discrepancies embody precisely the insight suggested above: that
music is self-*similar*, not self-*identical*.

Of course, I would be delighted to learn of more precise examples of
recursion; if any of my former students in tonal theory were to read the
above, they might be taken aback, as I've always emphasized the value (for
students who haven't yet internalized the traditional norms) of generating
long musical phrases via the (quasi-)recursive application of short
formulas, rather than trying to compose coherently in real time, one chord
after another. (I've tried to attach a PDF example to this message; I'm
guessing that the list software will strip it out, so if you're dying to see
it, please write me off-list and I'll send it.)

All best,

Brian Robison
Visiting Assistant Professor
Department of Music
Middlebury College
Middlebury VT 05753

2009/3/21 Thomas Noll <noll at cs.tu-berlin.de>

> 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 <http://flp.cs.tu-berlin.de/%7Enoll>
> 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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Brian Robison

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