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The <i>Tonnetz</i> 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.<br>
<br>
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?<br>
<br>
Nicolas Meeùs<br>
<a class="moz-txt-link-abbreviated" href="mailto:nicolas.meeus@paris-sorbonne.fr">nicolas.meeus@paris-sorbonne.fr</a><br>
<a class="moz-txt-link-freetext" href="http://www.plm.paris-sorbonne.fr">http://www.plm.paris-sorbonne.fr</a><br>
<br>
<br>
Wayne Slawson a écrit :
<blockquote
cite="mid:C72E9D5E-AB6E-4387-89AF-314384D471CD@yankgulchmusic.com"
type="cite">In response to Gogins and Dmitri:
<div><br>
</div>
<div>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:</div>
<div><br>
</div>
<blockquote type="cite">
<div>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. </div>
<div><br>
</div>
<div>More generally, Schenkerians talk about voice-leading
relationships at various levels of the recursive hierarchy; each level
could be modeled geometrically.</div>
</blockquote>
<div><br>
<div>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:</div>
<div><br>
</div>
<div><span class="Apple-tab-span" style="white-space: pre;"> </span>B-flat5,3/F6,3/G7,3(natural)/C7,3(natural)/F3("flat"
then natural in the figuration)]. </div>
<div><span class="Apple-tab-span" style="white-space: pre;"> </span>In
F:<span class="Apple-tab-span" style="white-space: pre;"> </span><span
class="Apple-tab-span" style="white-space: pre;"> </span> IV
I V of V V I</div>
<div><br>
</div>
<div>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.) </div>
<div><br>
</div>
<div>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. </div>
<div><br>
</div>
<div>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!</div>
<div><br>
</div>
<div>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?</div>
<div><br>
</div>
<div>Wayne Slawson</div>
<div> </div>
<div><br>
</div>
<div> </div>
<div><br>
</div>
<div><br>
<div>
<blockquote type="cite">
<div>
<div><span class="Apple-tab-span" style="white-space: pre;"><font
class="Apple-style-span" color="#000000"><span class="Apple-style-span"
style="white-space: normal;"><br>
</span></font> </span><br>
</div>
<div><span class="Apple-tab-span" style="white-space: pre;"> </span>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.<br>
</div>
<div><br>
</div>
<div><span class="Apple-tab-span" style="white-space: pre;"> </span>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).</div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<div>The question about whether sequences are recursive is a
complicated one.</div>
<div><br>
</div>
<div><span class="Apple-tab-span" style="white-space: pre;"> </span>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.</div>
<div><br>
</div>
<div><span class="Apple-tab-span" style="white-space: pre;"> </span>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.</div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<blockquote type="cite">
<div><span class="Apple-style-span" style="">I take it that
recursion is hard to fit into a geometric model of distance, or am I
missing something?</span></div>
</blockquote>
</div>
<div><br>
</div>
<div>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. </div>
<div><br>
</div>
<div>More generally, Schenkerians talk about voice-leading
relationships at various levels of the recursive hierarchy; each level
could be modeled geometrically.</div>
<div><br>
</div>
<div>DT</div>
<div><br>
</div>
<div> <span class="Apple-style-span"
style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div>Dmitri Tymoczko</div>
<div>Associate<span class="Apple-converted-space"> </span>Professor
of Music</div>
<div>310 Woolworth Center</div>
<div>Princeton, NJ 08544-1007</div>
<div>(609) 258-4255 (ph), (609) 258-6793 (fax)</div>
<div><a moz-do-not-send="true"
href="http://music.princeton.edu/%7Edmitri">http://music.princeton.edu/~dmitri</a></div>
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</blockquote>
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