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<div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Olli Väisälä raises a bunch of good points:</div></div><div><blockquote type="cite"><div><font class="Apple-style-span" color="#000000"></font></div></blockquote><br><blockquote type="cite"><div><font class="Apple-style-span" color="#000000">I </font>would like to make some points that I find crucial for the discussion about recursion in prolongational (Schenkerian) structures and about pertinent empirical evidence. <span class="Apple-style-span" style="-webkit-text-stroke-width: -1; ">First of all, I do not think this issue can be adequately treated merely on the basis of symbol sequences, as suggested by Dmitri Tymoczko's post:</span></div></blockquote><div><br></div><div>I wasn't trying to suggest that the issue can be comprehensively treated in this way. I was just trying to point out a distinction that is important, and that is rarely discussed. </div><div><br></div><blockquote type="cite"><div><div>Musical events are not adequately determined in terms of such sequences, since each occurrence of "A" and "B" may relate differently with aspects such as meter, design, register, emphasis, etc. , and these aspects are crucial for both the expression (by the composer) and perception (by the listener) of prolongational relationships. Consider a very simple example in 4/4 time:</div><div><br></div><div>bar 1: I–V–I–quarter rest</div><div>bar 2: V–I–V–quarter rest</div><div>bar 3: I–V–I–quarter rest</div><div><br></div><div>If this occurs in a composition, we may speak of empirical evidence for the hypothesis that the recursive interpretation, (IVI)(VIV)(IVI) = IVI, models a pattern relevant for composition. The model has considerable explanatory power for the way in which the composer has related the chords with features such as meter and grouping.</div></div></blockquote><div><br></div><div>In the one place where I've published on this (<a href="http://music.princeton.edu/~dmitri/tonaltheories.pdf">http://music.princeton.edu/~dmitri/tonaltheories.pdf</a>), I mention exactly this possibility. The point is that a supporter of recursion is going to have to rest their case on rhythm, register, etc., rather than on the Roman-numeral harmonies themselves.</div><div><br></div><div>On this view, the harmonic organization and the "recursive" organization will in general conflict. For instance, there's an extremely well established, purely harmonic grammar, that claims that tonal music organizes itself into (speaking loosely) tonic-(predominant)-dominant-tonic cycles. And indeed Olli's example does exactly this: it's a series of overlapping I-V-I cycles.</div><div><br></div><div>Presumably the supporter of recursion will agree that this organization-into-harmonic-cycles is an important feature of the music. Which means that they will claim that we need to conceive music in a disunified way: on the one hand, nonrecursively, as a series of concatenated cycles; on the other hand, recursively, as a series of recursively nested harmonies.</div><div><br></div><div>To my mind, this raises interesting questions vis-a-vis Schenkerian ideas about musical unity. Clearly, we have a disunified picture here, with the harmony and register/rhythm articulating very different organizational schemes. The issue is analogous to that discussed by Rick Cohn, with respect to motive.</div><div><br></div><blockquote type="cite"><div>As my example clarifies in a very simple way, a sequence may be "intrinsically nonrecursive," but perceiving it recursively may be supported by features that are intrinsic to the music. </div></blockquote><div><br></div><div>I would say: the sequence of harmonies is intrinsically nonrecursive, but the music as a whole may suggest recursive structure.</div><br><blockquote type="cite"><div>Whether the "recursive potential" of a sequence becomes realized does not depend only or primarily on the listener's psychology, but on the composer's acts: on the ways in which he/she supports (or fails to support) certain elements and relationships through parameters such as meter, design, register etc. </div></blockquote><div><br></div><div>Here I would probably disagree: it's still a psychological matter whether one perceives the simple example Olli has described as recursive. We, in our post-Schenkerian era, tend to take this for granted, but it's entirely possible to hear (or conceive) this sort of sequence non-recursively. Just because you have I-V-I, articulated by rhythm, doesn't mean you have to hear this as "prolonging" I; you could hear three separate-but-equal harmonies. (When I take a trip from Philadelphia to New York and back, my trip does not "prolong" the state in which I stay at Philadelphia.)</div><div><br></div><div>So I'd say there's a lot of psychology here, regardless.</div><div><br></div><div>DT</div></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>