<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">
<div>Ildar wrote:</div><div><blockquote type="cite"></blockquote><br><blockquote type="cite"><table cellspacing="0" cellpadding="0" border="0"><tbody><tr><td valign="top" style="font: inherit;"><div>I am sorry for playing a devil's advocate for a moment. Of course, I appreciate your great work with geometry. It is just sometimes the topoloical aspect is overshadowed in your presentations by a simpler principle of measurement by half-steps (which probably come from Neo-Riemannian understanding).</div></td></tr></tbody></table></blockquote><div><br></div><div>No need to apologize, and thanks for the kind words!</div><div><br></div><div>If I understand you, the "simpler principle" you refer to is just the "taxicab" (or "smoothness") metric, which provides one of the simplest ways to measure distance on orbifolds. From this point of view, (C, E, G)->(C, F, A) is has size three, since one voice moves by one semitone, and another moves by two semitones. By contrast, (C, E, G)->(C, F, Ab) is size two, since two voices move by semitone. There's no contradiction between this way of thinking about distance and the geometrical perspective; this is just one of the many possible metrics you can choose.</div><div><br></div><div>It would be a mistake, however, to associate this method of measuring distance too closely with either the Tonnetz or neo-Riemannian theory. From a NR-perspective, F major is *closer* to C major than F minor is -- the progresson F->C is LR, whereas f->C is PLR. (Using the "taxicab" metric, the opposite is the case.) Consequently, NR-theory doesn't seem to explain why F minor so often appears as a passing chord between F major and C major -- from an NR perspective the progression F->f->C moves away from C major and before moving back toward it. The moral is that neo-Riemannian distances (measured in "units" of LPR or in edge-preserving Tonnetz-flips) are *not* voice leading distances. This is a complicated and subtle issue about which much more could be said.</div><div><br></div><div>As far as I understand, the first geometrical models in which all distances represent taxicab distance are those provided by Douthett and Steinbach in 1998, particularly "Cube Dance." These models are all embedded naturally in the relevant orbifolds representing n-note chords.</div><div><br></div><blockquote type="cite"><table cellspacing="0" cellpadding="0" border="0"><tbody><tr><td valign="top" style="font: inherit;"><div> </div> <div><span class="Apple-style-span" style="color: rgb(0, 0, 221); -webkit-text-stroke-width: -1; ">And I am ready to choose certain metric from a number of others offered by topology. For example, I will decline your proposition that the E and Ab triads are the closest to C triad. In the metric of tonal-functional harmony, the G triad is the closest to C triad. You simply cannot fit any other triads in between in a meaningful progression. That is what I meant in my rather awkward critique, that the geometry of note heads on the staff presents the metric which is drastically different from the one we use when playing and singing most of tonal music.</span></div></td></tr></tbody></table></blockquote><div><br></div><div>Oh, OK, I should've clarified the following. Rachel Hall and I have worked to describe acceptable *voice-leading* metrics. There are other notions of musical distance (including tonal distance, or that offered by NR-theory) that can be perfectly reasonable in some circumstances, but that don't measure voice leading. Acoustically, you might think that G4 is very close to C3, since it's the second overtone of the lower note. But you're not measuring voice leading if you think that C3->G4 is a "small" motion. Again, a lot more could be said about this.</div><div><br></div><div>BTW, I would say that you can fit other triads between C and G. In C major, the progression C->a->d->G is perfectly reasonable. And you sometimes find G->e->C, both in the baroque and in 19th-century music. The question of whether tonal harmony is fifths-based or thirds-based is a complex one; and you can make a good case for thirds, rather than fifths. Interestingly, then you're back to voice-leading, since third-related diatonic triads are linked by single (diatonic) step voice leading.</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>