<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div>Dear list,</div><div><br></div><div>I would like to make some points that I find crucial for the discussion about recursion in prolongational (Schenkerian) structures and about pertinent empirical evidence.</div><div><br></div><div>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:</div><div><br></div><div><blockquote type="cite"><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">It is typically possible to interpret a sequence of symbols recursively. For instance, we could take the sequence</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">ABABABABAB...</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">And group it as follows:</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">(ABA)(BAB)(ABA)(BAB) ...</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">Then, by asserting that (ABA) "stands for" (or "represents" or "prolongs") A, while (BAB) "represents" B, we could come up with a "higher level" version of the same pattern</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">ABABABAB ...</font></div></blockquote><div><br></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><br></div><div>Naturally, meter and grouping in this example also enhance the probability that the recursive model pertains to the listener's perception in this example. What I would stress, however, is the way in which features in the score may testify or fail to testify to the compositional significance of prolongational models. Therefore, I would like to comment on Dmitri's following suggestion:</div><div><br></div><blockquote type="cite"><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">The big complication is that it is possible to perceive intrinsically nonrecursive sequences in a recursive fashion. I think music theorists could probably stand to pay more attention to this possibility. For instance, I think you can make a good case that the grammar of elementary tonal harmony is, intrinsically, (largely) non-recursive. Theorists such as Schenker, Lerdahl, etc., propose that we nevertheless *perceive* music recursively -- imposing a recursive organization on an intrinsically not-recursive structure, roughly as we did with the ABAB sequence above. This is potentially true, but it is quite hard to test empirically. I think discussions of the issue of recursion could stand to pay more attention to the distinction between intrinsic structure and our psychological organization of a structure.</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; "><br></div></blockquote><br></div><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. 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. While it may be hard to test empirically whether people perceive music recursively (and different people have, of course, different perceptions), it is possible to approach empirically the question whether composers did (do) so, by assessing the extent to which recursive (prolongational) models are supported by such parameters in their music. </div><div><br></div><div>Of course, the question is much more complicated in real music than in the above example, but for those interested I might mention that that I have attempted to approach this issue in my recent (and ongoing) work on Bach.</div><div><br></div><div>Olli Väisälä</div><div>Sibelius Academy</div><div><a href="mailto:ovaisala@siba.fi">ovaisala@siba.fi</a></div><div><br></div></body></html>