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<div><blockquote type="cite"><div>I think Dimitri ...</div></blockquote><div><br></div>Actually "Dmitri" with no "i" between the "D" and "m." You wouldn't believe the trouble this causes with telephone-operator types.<br><div><br></div><blockquote type="cite"><div>rightly cites secondary dominants as evidence of "hierarchical" recursion, but there's more (as they say in those TV ads). I think it's very hard indeed to analyze Goldberg Var. 25 without a kind of recursion that "rewrites" a single unit from the underlying bass of the Saraband---a well-formed harmonic progression itself, of course---as an entire well-formed progression on the level of that unit. For example, the A-flat in the second bar is a pretty convincing suggestion that we're now starting a one-bar "Opening Phrase (I to V)" here in F-minor (i.e., the F that is the second unit of the bass in its minor version). A similar spot in bar 10-11(also on an F from the bass) is even more elaborate, with, incidently, many secondary dominants. We seem here to have at least two levels of recursion.</div></blockquote><div><br></div><div>Two general points:</div><div><span class="Apple-tab-span" style="white-space:pre"> </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></body></html>