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<TITLE>Re: [Smt-talk] Headlam on Orbifolds</TITLE>
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<FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>Hi all: A bit of wheel-reinventing is going on here — think of distance on the harmonic series to help with harmonic thinking, and motion within these intervallic spaces for melodic distances. In C major, C to C# is a vast harmonic distance, a small melodic distance. C-G is a small harmonic distance, a larger melodic distance. C-F# is the wild card — big melodic and harmonic — and non-directly - functional in tonal contexts — its always points away from tonic. When harmonic function begins to break down, the harmonic/melodic distinction correspondingly breaks down in favor of melodic intervals, which begin to pervade both spaces (bII instead of V or both in Berg’s Piano Sonata, half-diminished 7ths to major-minor sevenths everywhere — and in general Mark DeVoto’s “creeping chromaticism”) -- this is old news, although worth thinking about. <BR>
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I’m certainly not saying that we can’t model incomplete chords (“geometry is useless in this instance”, etc.), merely that one of the original DT postulates was (is?) that staying in one area of the orbifold was “good” -- as in staying in the triad area or seventh chord area, associated with Chopin op. 28,no. 4. I like the orbifold model precisely because it keeps multi-sets ( = incomplete chords). But in it’s PE form, it doesn’t elegantly show progressions with ICs staying in one area. If this is no big deal, then there’s no concern. But imagine reshaping it to have 000 – eee in the “middle” and 00x to eex “around” the middle and “complete chords” on the “outside”, aligned so that 000 040 047, etc. are as close as possible — now the types of progressions Brown and I referred to in our reply are more elegantly displayed.<BR>
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Or perhaps something like this: tightly wound circles of 000 to 060, etc. then with 012 to 01e etc., as another circular layer. Then connected end to end. Then we could “buzz” around 000 040 047 if the occasion called for it.<BR>
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0t0<BR>
0e0 1e1 eee<BR>
000 101 202 303 404 505 606 707 808 909 t0t e0e <BR>
010 111 e1e<BR>
020<BR>
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Even more fun in the quad model, where F- F- Ab – Eb resides.<BR>
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I like incomplete chords. I don’t throw up my hands when I hear them. I think the orbifold can model them — just not elegantly in the current format. <BR>
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Dave Headlam<BR>
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On 3/13/09 6:10 PM, "Dmitri Tymoczko" <dmitri@Princeton.EDU> wrote:<BR>
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</SPAN></FONT><BLOCKQUOTE><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>I associate part-writing with counterpoint in the sense that it deals with the surface elements of a composition and the threads of succession. Your earlier exchange with Ildar, however, implied, on his part at least, that incomplete harmonies were some pivotal issue. Let us add a definition of "chord" to be an element of counterpoint, which is to say, on the real surface of a composition, and further say that it is a selection of elements of the surface–without regard to rhythm (including simulataneity)–isolated for evaluation with respect to the referent harmonies (in tonal music, triads and maximally even tetrads). If we wish to give a chord a name, then we are including all of its elements in the concept of voice leading, while counterpoint has no such obligation. Thus, I can read bars 9-11 of the Waldstein as built upon GBF, three of the four notes of a G7 harmony, without their representing that harmony; there need be no D. Yet bar 13 has but one element of a G triad and I do read that as a full G harmony and accomodate the voice leading accordingly. You and I are with Rameau in this perception of voice leading, but most Schenkerians assume that voice leading is what they show; sometimes it is, sometimes it isn't. <I>Züge</I> don't need support for all their displacements.<BR>
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I think I follow what you're saying. The example is definitely useful.<BR>
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What I was saying to David (not Ildar) was that you can use orbifolds to model surface harmonies if you want: for instance, you could model the relevant passage of the Waldstein as the three-voice (Ab, C, F)->(G, B, F) in three-note chord space. That's a perfectly reasonable thing to do. I think you might say that we're modeling the actual "part writing" here.<BR>
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On the other hand, you can also model voice leadings that are only hinted at by the surface. For instance, in the recap of the Waldstein, measure 172, you find (F, F, Ab, Eb)->(Ab, F, B, D), which I take to be an embellishment of the common semitonal pattern (F, Ab, C, Eb)->(F, Ab, B, D). This sort of semitonal play between seventh chords is ubiquitous in nineteenth-century music, and is the engine of an awful lot of chromaticism. I think it's useful to identify it as lying in the background, obscured to some extent by the part writing.<BR>
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What I don't quite understand is Dave's suggestion that geometry is useless whenever we confront incomplete chords -- as if we have to throw up our hands and give up the moment we see a progression like (F, F, Ab, Eb)->(Ab, F, B, D). I disagree: we're still free to use all the standard techniques of analysis, including postulating octave displacements, voice exchanges, additional voices, passing tones, and so forth. Part of the value of the geometrical models is that they give us powerful tools for understanding the ubiquitous background schemata that are (sometimes) only hinted at by the musical surface.<BR>
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DT<BR>
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</SPAN><FONT SIZE="2"><SPAN STYLE='font-size:12.0px'>Dmitri Tymoczko<BR>
Associate Professor of Music<BR>
310 Woolworth Center<BR>
Princeton, NJ 08544-1007<BR>
(609) 258-4255 (ph), (609) 258-6793 (fax)<BR>
<a href="http://music.princeton.edu/~dmitri">http://music.princeton.edu/~dmitri</a><BR>
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Dave Headlam<BR>
Professor of Music Theory<BR>
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Eastman School of Music<BR>
26 Gibbs St<BR>
Rochester, NY 14604<BR>
(585) 274-1568 office<BR>
dheadlam@esm.rochester.edu<BR>
<a href="http://theory.esm.rochester.edu/dave_headlam">http://theory.esm.rochester.edu/dave_headlam</a><BR>
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