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<div>Dave,</div><div><br></div><div>I separated out your comments about orbifolds since they aren't relevant to sequences, and since they continue an old discussion between you, Brown, and myself.</div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size: 14px; "></span></font></blockquote><div><br></div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size: 14px; ">The orbifold approach will work for complete chords, but rather than the permutational equivalence that orbifolds are based on, something like 000 040 004 047 (CCC CEC CCE CEG) equivalence more closely matches tonal practice, to avoid constant careening from the middle to the edges and back to pick up incomplete chords (see Headlam and Brown reply to Tymoczko in Science). We also have the Chopin situation of combined triads and seventh chords that require coordination between orbifolds of different cardinalities.</span></font></blockquote><div><br></div><div>I have several things to say about this.</div><div><br></div><div>First, the point of using orbifolds is to faithfully model voice leading distances. In this context, incomplete chords (or chords with doublings) are very different from permutations and octave transpositions. Octave displacement and permutation preserve voice leading distance (at least when applied uniformly): hence, for any three-note chord (C, E, G) in any octave and any order, there is an E major chord two semitones away from it (using the taxicab metric). This is not true of an incomplete C major chord. For instance, there is no E major chord, either complete or incomplete, that is *two* semitones from (C, E) or (C, C, C). A tripled unison and a complete major triad simply have different voice leading potentialities.</div><div><span class="Apple-tab-span" style="white-space:pre"> </span>Consequently, if you try to glue together (C, C, E) and (C, E, G) you destroy your space's connection to voice leading, and this is a large sacrifice indeed. This is why -- as discussed in my reply to you guys -- I think one typically wants to use orbifolds to represent background or conceptual structures (often involving complete triads) which may be only incompletely represented by the musical surface. I really don't see any difficulties here -- this is not very different from what we do when we engage in Roman numeral analysis ... we say an incomplete C-E "represents" a conceptual C-major triad.</div><div><br></div><div>Second, if you want to faithfully model the musical surface, then it's not a problem that your representation of musical objects moves back and forth from the center of an orbifold to the edges. Why should it be?</div><div><br></div><div>Third, voice leadings between chords of different cardinalities are discussed in Cliff, Ian, and my "Generalized Voice Leading Spaces" (under the rubric of "C-equivalence"). While there are some technical issues, there's no fundamental problem: you can say, for example, that the C7 chord is closer to E major than Eb7 is, in (infinite-dimensional) OPC space. (An efficient algorithm for calculating minimal voice leadings in that infinite-dimensional space is provided in "Geometry of Musical Chords.") The real problem is that these spaces are difficult to visualize. </div><div><br></div><div>Fourth, there's a small problem of ambiguity -- CE could represent A minor in the right context, so you don't simply want to glue CEG and CE together. Again, I think this suggests the strategy of using your musical intuition to translate CE into a complete triad, and then model the result geometrically. Since this strategy produces analytical insights that I find useful, I'm quite happy with it.</div><div><br></div><div>DT</div><br><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>