<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><blockquote type="cite"><div>what do you mean by "harmonic terms" with respect to the orbifold approach (factorized voice-leading) you are working on?</div></blockquote><div><br></div><div>Sorry, I suppose that was a bit too brisk.</div><div><br></div><div>In general, I think a lot of nineteenth-century chromatic practice can be understood contrapuntally: composers exploit efficient voice leading to move between tonally distant chords. So, you see Schubert exploiting the efficient voice leading among major-third related triads to move from C major to E major, for example. Or Wagner exploits the efficient voice-leading among minor-third related seventh chords to move from D half-diminished to B7. Looking broadly at nineteenth-century practice, I think you can make a case that many of these unusual tonal moves are motivated by voice leading relationships.</div><div><br></div><div>One piece of evidence for this is that there's a marked asymmetry between triadic and seventh chord routines: triads are more likely to move chromatically by major third and seventh chords are more likely to move by minor third or tritone. This reflects the underlying voice-leading relations: C major is "closer" (in voice leading terms) to E major than Eb major, but C7 is closer to Eb7 than E7. You can actually see this asymmetry in the statistics -- just count up all the chromatic progressions in Mozart or Schubert.</div><div><br></div><div>In the case of our sequences, I think it's hard to make the case that these minor-third progressions are motivated contrapuntally, precisely because composers seem to avoid the most efficient (stepwise, descending) voice leading in favor of ascending parallel motion. This makes it look like they're choosing the harmonic destination, rather than it being an effect of voice-leading relationships. On the other hand, with "Hey Joe" perhaps it's the descending voice leading that produces the unusual (ascending fifth) root motion.</div><div><br></div><div>I'd never thought about trying to express this in terms of orbifolds. I suppose you could though: the contrapuntal approach, roughly, involves thoughts like "I'm here at chord X; what chords are nearby?" The harmonic approach involves thoughts like: "I'm here at X, and I want to go to Y, and distance isn't really a factor right now."</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>