<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><blockquote type="cite"><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></blockquote><div><br></div><div>I realize that I was being a bit coy here. In fact, I do think that there's a way in which tonal music is recursive, because efficient voice leading occurs between scales as well as chords.</div><div><br></div><div>So, I think that a modulation from C major to G major involves an efficient voice leading whereby F moves to F#. This is the same basic thing that happens when you have, say, a voice leading from the C major to G major triads, such as (C, E, G)->(B, D, F). Here, two notes of the C major triad move down by step to G major. So you have the same basic process (efficient voice leading between structurally similar objects) occurring on two different temporal levels (that of the chord, and that of the scale). I've talked about voice leading between scales in "Scale Networks and Debussy"; in my book, I claim that the same thing happens in traditional modulation.</div><div><br></div><div>Scales can be thought of as large "chords" inhabiting a seven-dimensional geometrical space. In "Scale Networks and Debussy" I show how to depict the relevant portion of the space that contains familiar scales. It actually turns out that the geometry representing seven-note scales in chromatic space is very, very similar to the geometry representing three-note diatonic triads. So it turns out the local geometry of chords-in-a-scale is almost identical to the geometry of scales.</div><div><br></div><div>All of this, I think, is musically pretty plausible, as well as being theoretically quite elegant. But of course, it's a longer story than I can really explain in a short email like this.</div><div><br></div><div>DT</div><div><br></div></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>