<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">
Hi Folks,<div><br></div><div>I think Dimitri 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><div><br></div><div>To be clear about this one would need a system of rules that include re-writing rules---or, as Chomsky called them, transformational rules---that generate these kinds of things. John Peel and I gave a try at this in our review of Lerdahl and Jackendoff's book published in JMT 28, 271--294 (1984). Bob Morris offers something like this as well: pp 339-340 in PNM</div><div>33:1&2 (1995). This kind of recursion is not unlike, say, your usual C-language recursive subroutine, in which a simple test---here some kind of end-of-phrase marker---backs you out of the otherwise infinite process.</div><div><br></div><div>Chomsky insisted (or at least he did around the time of "Syntactic Structures") that the infinite embedding of relative clauses has to be allowed by the syntax, even if it never occurs, of course, in the actual utterance ("performance") of anyone, no matter how long-winded. One way of approaching a (sympathetic) critique of Schenker's ideas would be to attempt to compose a corpus of phrase-structure and transformation rewriting rules that would generate "correct", say, middlegrounds, along the lines of Michael Kassler's work in 1975 (cited in ftnt 2 of the Peel/Slawson review).</div><div><br></div><div>I take it that recursion is hard to fit into a geometric model of distance, or am I missing something?</div><div><br></div><div>Wayne Slawson</div><div><br></div><div>On Mar 22, 2009, at 10:51 AM, Dmitri Tymoczko wrote:<div><br class="Apple-interchange-newline"><blockquote type="cite"><div>Over the last few years, I've been working on the problem of trying to figure out whether the rules of functional harmony are recursive, and if so, to what extent. This is one reason I enlisted a group of theorists to produce a corpus of Roman Numeral analyses of the Mozart sonatas -- that is, to ask whether we need recursive rules to describe the chord progressions we find in the music. So far, the data seems to suggest that the answer is "no." Simple rules like "ii goes to V but not vice versa" account for the data pretty well. One important exception is the practice of secondary dominants, which permits V/x->x to be embedded inside I->...->V->I in a different key. This seems genuinely recursive.</div><div><br></div><div>Some early work on these ideas is summarized here:</div><div><br></div><div><a href="http://music.princeton.edu/~dmitri/tonaltheories.pdf">http://music.princeton.edu/~dmitri/tonaltheories.pdf</a></div><div><br></div><div>Published (in French, in Nicholas Meeus's excellent translation) in Musurgia. More is coming in my book.</div><div><br></div><div>To make further progress on this topic, one needs -- as Richard Hermann suggests -- to define terms more precisely. In particular, one needs to distinguish</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>1. Psychological theories, which claim we need recursion to account for our *perception* of music, from grammatical theories, which claim we need recursion to account for the formal structures in pieces, whether perceived or no.<br></div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>2. Hierarchical theories, which suggest (incontrovertibly) that different sorts of rules govern different levels of musical structure, from recursive theories, which claim that tonal structures are built up by embedding musical units within other units of the same type. (For instance, ii/ii->V/ii->ii embedded within I->ii->V->I.) <br></div><div><br></div><div>It's certainly true that many theorists -- including Schenker, Sadai, Lerdahl, and Jackendoff -- have claimed tonal harmony is recursive. Figuring out whether this is in fact true is, to my mind, a very deep and important problem. It speaks to fundamental issues about the human mind: about the extent to which recursion is basic to human cognition, and about the extent to which music and language are similar.</div><div><br></div><div>Interestingly, this is also a hot topic in linguistics. I've recently been giving a talk, to philosophers and linguists, about why I think linguistics is a bad model for music theory. Whenever I try to contrast language with music, claiming that the former is more obviously recursive than the latter, I encounter dissident linguists who say, in essence, "a lot of what you say about music is true about language too." This issue is, as they say, above my pay grade, but it's interesting that some of the same issues arise in the two disciplines.</div><div><br></div><div>DT</div><div><blockquote type="cite"><div><span class="Apple-tab-span" style="white-space:pre"></span></div></blockquote></div><br><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><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">_______________________________________________</div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Smt-talk mailing list</div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><a href="mailto:Smt-talk@societymusictheory.org">Smt-talk@societymusictheory.org</a></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><a href="http://lists.societymusictheory.org/listinfo.cgi/smt-talk-societymusictheory.org">http://lists.societymusictheory.org/listinfo.cgi/smt-talk-societymusictheory.org</a></div> </blockquote></div><br></div></body></html>