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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="-webkit-text-stroke-width: -1; ">I take it that recursion is hard to fit into a geometric model of distance, or am I missing something?</span></div><div></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-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><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 href="http://music.princeton.edu/~dmitri">http://music.princeton.edu/~dmitri</a></div><div><br class="khtml-block-placeholder"></div><div><br class="khtml-block-placeholder"></div><br class="Apple-interchange-newline"></span><br class="Apple-interchange-newline"> </div><br></blockquote></div><br></div></div></body></html>