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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'>Dmitri: <BR>
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There’s certainly nothing wrong with asking basic questions and looking for elemental issues — but within a context . . . . <BR>
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</SPAN></FONT><OL><LI><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>I said C and C# are separated harmonically. If we add D, then we are talking about C and D which are not as far removed, and C# can mediate. But these are two vastly different scenarios. Indeed “a semitone is a vast distance harmonically” needs a context, precisely the “direct / indirect distinction” that is basic to #IV and other such approaches. (C-F#, versus C-G(F#), C-G (D-F#) etc. see Lewin GMIT)<BR>
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2. IV and ii6/5 (long history here, Rameau, the double emploi, etc.) indeed have a voice-leading connection (Rothstein dissertation on special nature of 5-6), as often one is of contrapuntal origin.<BR>
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3. In tonal music, indeed there is a small repertoire of chords with similar intervallic characteristics and SOME voices tend to move stepwise — other voices jump around — depending on harmonic progression and function , etc. Coherence (IMO) comes not from these elements (except in a local sense) but from large-scale returns / prolongations, etc. of keys and motivic ideas. <BR>
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4. no doubt that C-C-C sounds different than C-E-G -- I don’t have stats at the ready on how much complete chords appear and when — but it is at least arguable whether permutational equivalence (PE) like C-E-G_, E-G-C, G-C-E applies to tonal music, and perhaps we can also argue about C-C_C_, C-C-E, and C-E-G being in some senses equvalent. But there is something to debate here. (unmusical is too strong here). In this context please see Howard Cinammon’s dissertation for voice-leading in 19th c. music — in tonal and symmetrical contexts — way ahead of it’s time and should be required reading for all 19th c. aficionados.<BR>
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5. oribifolds: PE as above, IC = incomplete chords, which don’t stay in one area — I’ll give up that old one about the “stay in the same area of the orbifold” (although that was a big part of your media splash) <BR>
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5a. Stravinsky rotational arrays: from single note unisons to complex chords, same with Perle in the Sonata a quatro — Ravel in le gibet (Bb), in C (in the recorded version with high Cs) -- some examples of “unisons” in complex contexts — and of course, Berg’s Wozzeck at Marie’s death (B s), but it’s true (I think) that harmonic “consistency” is more often the case -- ( look at Wallace Berry, Struc. Functions chapter 2 — a real gem on texture).<BR>
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6. The last one is worth a dissertation — what is voice-leading? If you want to turn voice-leading into “voice-leading” (like Lewin’s interval into “interval”, Forte’s set into “set”, Perle’s sum into “sum”, etc.) as a generalized notion then you have to lay out the premises<BR>
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</SPAN></FONT><OL><LI><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>an actual physical “small” connection between two notes, or an abstract connection between two possible implied entities<BR>
</SPAN></FONT></OL><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>2. governed by group theory / Euclidean solid angles / etc. or more topological / stringy<BR>
3. a function, operation, etc. (not a thing but a motion, etc.)<BR>
4. explanatory or defining? (see Roeder on Schoenberg) What is it supposed to reflect faithfully without becoming circular?<BR>
5. part of some chronological , progressive syntax or recreated constantly<BR>
6. a subset / superset of Lewin? Explains harmony or explained by harmony? Prolongational or associational? <BR>
7. kinetic, evolutionary (the voice and the hand), cultural (childrens nursery rhymes)<BR>
8. Schenkerian, Perlian, Fortean, neo-R ? Is the search for step-wise motion part of the ten commandments of music? <BR>
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What’s the VL in Schoenberg’s opus 11? Blasse Washerin? (see James Baker on opus 19/ I) In Berg’s opus 5? In Webern’s opus 5/III? (see Mark Sallmen’s fine work here). Same principles as in WTC I — or all new, or some transformation? Rothstein has an article on stepwise motion in Martino’s music — same as in WTC I? What about in Webern’s op 5/II -- along Lewin’s network model of 026s — harmonic space or voice-leading within the space of the trichord? (what’s the diff?)<BR>
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Certainly an interesting topic — some wheels are worth reinventing, but don’t forget to mention Goodyear.<BR>
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Finally, In this and other contexts, please read Walter O’Connell’s article on Tone Spaces in Die Reihe 8. One of the great articles — thanks to Andy Mead for pointing it out to me (I’ve written on it, in the Western Ontario Studies in Music Series) -- it deserves a wide audience — if anybody knows who O’Connell was, I’d appreciate hearing about him.<BR>
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Dave<BR>
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</SPAN></FONT><BLOCKQUOTE><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>Let me start by saying that I appreciate the opportunity to talk about these issues ... I know my ideas are confusing to some people, and I'd rather discuss things openly than have people think I'm off in some incomprehensible mathematical never-never land. Let me also reiterate that my book, which should be out pretty soon, goes into detail about all of this, and tries to provide accessible explanations of how one can use geometry sensitively in analysis.<BR>
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OK, so you wrote:<BR>
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</SPAN></FONT><BLOCKQUOTE><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. <BR>
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It's true that I'm asking really elementary questions -- revisiting stuff we all learned when we were eleven. In my defense, I'd say it's possible we know less than we think we do. If we're using square wheels, a little reinventing might not be so bad!<BR>
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Let's take your example: "in C major C to C# is a vast harmonic distance." I suspect we both think that the keys of C major and D minor are close. I suspect we agree that this is a harmonic fact. One potential explanation for the "closeness" is that the C diatonic scale can be turned into D melodic minor ascending by way of the (small) semitonal shift C->C#. In other words, key distance (a "harmonic" notion) may have something to do with efficient voice leading between scales (a "contrapuntal" fact). I believe that one can provide some good empirical evidence for this hypothesis. <BR>
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You can also make similar points about chords: perhaps vi and IV6 are close in part because they're linked by single-step voice leading. Again contrapuntal relationships may help explain harmonic facts -- in this case, efficient voice leading may explain the frequent intersubstitutability of vi and IV6.<BR>
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That doesn't mean there are no independent harmonic facts. But from this point of view the bare claim that "a semitone produces a large harmonic distance" is much too simplistic. Given this possibility, I think it might be worth revisiting some basic theoretical concepts. Maybe the wheel (or at least our understanding of "harmonic distance") could actually be improved?<BR>
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</SPAN></FONT><BLOCKQUOTE><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>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. <BR>
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I don't believe I've ever written anything this simplistic in a published paper, not said it to anyone who had even a rudimentary knowledge of music theory. (It's possible that I've said something like this to journalists who knew absolutely nothing about music, and who wanted me to explain my ideas in elementary terms, but that's a whole different communicative context.) What I have written, and what I believe, is that much Western music exhibits two kinds of coherence: harmonic consistency (chords sound roughly similar) and efficient voice leading (voices move by small distances, represented in the orbifold by short distance motions). One can use orbifolds to understand how these two properties can be combined, but it is sometimes necessary to interpret the music prior to doing so. For example, the bass voice often moves by leap, so to find the short-distance motions you need to remove the bass and look at the upper voices.<BR>
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Do you agree that a large amount of Western music exhibits these two kinds of coherence, and that they're conducive to listeners' sense of musical order? If not, that would certainly be a debate worth having.<BR>
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Now let's consider your example -- (G3, B3, D4)->(C3, C4, C4). Does the progression exhibit efficient voice leading? Well, not until we remove the leaping bass, at which point the remaining voices can be represented by a short-distance motion in two-note chord space. Does the progression exhibit harmonic consistency? In an abstract sense, yes, because the final chord can be taken to represent a C triad. (As I say in my response to you and Matthew, you could express this by treating the voice-leading is an incomplete form of a more basic five-voice V7->I schema.) But in a more concrete sense, no -- intuitively, the triple unison sounds very different from a complete triad, and if you model the progression in three note chord space you see a dramatic move from the center to the edge.<BR>
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Now I would argue this latter fact actually helps *explain* why Western pieces don't move willy nilly between complete triads and triple unisons -- triple unisons are typically reserved for the ends of phrases, precisely because of their very distinctive, non-triadic sound. (I believe you can start to see a clear statistical preference for complete triads, and the association of incomplete sonorities with cadences, at least as far back as Josquin.) Their harmonic difference in turn is accurately reflected by their very different positions in the orbifold.<BR>
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This principle of harmonic consistency is very basic, and very widespread: for example, jazz musicians do not move willy-nilly from very "thick" sonorities (such as C-E-G-A-D, representing a C major chord) to bare triads like F-A-C, and even less to triple unisons like F-F-F. Why? Because of the palpable violation of harmonic consistency that results. Like classical composers, jazz improvisers strive for a consistency in sound, and this involves avoiding frequent and large changes in the number of pitch-classes between successive chords. <BR>
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>From this point of view, it seems unmusical to associate (C, C, C) too closely with (C, E, G). For centuries, composers have recognized a difference in sonic quality between these chords, and have used them very differently in actual pieces. <BR>
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Do you disagree? Do we agree that sonorities like (C, C, C) sound very different from (C, E, G) and are typically reserved for special points in musical phrases? More generally, that incomplete triads are much less common than complete triads in 4-voice textures? And that the standardization of 4-voice textures may have been driven by the desire for complete triads? (Coupled of course with the need for an extra voice to sound chord roots.) If we disagree at all here, I'd certainly be interested in hearing more about why.<BR>
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</SPAN></FONT><BLOCKQUOTE><FONT FACE="Verdana, Helvetica, Arial"><SPAN STYLE='font-size:14.0px'>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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I don't think I understand what "PE form" means -- nor "ICs staying in one area." Even without understanding this, however, I can say that the spaces you've described no longer accurately represent voice-leading distances. <BR>
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One thing that's hard for people to appreciate is that if you want to create a graph or a space in which all distances model voice-leading distances, then you really don't have many choices. Basically, the mathematics forces your hand. So you might say to yourself "let's construct something a lot like the chord-space orbifolds, but we'll glue CCC to CEG" or "let's glue together Z-related sets" or "lets just move a few chords around a bit." The problem is that in doing so, you sacrifice the ability to faithfully reflect voice leading. In some circumstances, this might not be so bad -- but if you want to understand voice leading (which I do), it's a big problem. To my mind, what's elegant about the orbifolds is that *every* line segment in the space represents a voice leading, with its length being equal to the voice leading's size. Ad hoc alterations such as those you describe destroy this feature, and hence (for me) much of the space's beauty, elegance, and interest.<BR>
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In fact, I'm pretty sure that any graph or space in which all distances represent voice leading distances is going to be embedded in one of the spaces that Cliff, Ian, and I describe in "Generalized Voice-Leading Spaces." (That's the real force of the term "generalized.") Conversely any graph or space that doesn't embed naturally in any of those spaces will not reflect voice leading distances faithfully. This is a bit of a shock -- particularly if, like me, you were brought up to think about music-theoretical constructs as arbitrary or purely conventional contrivances. In this respect, I think Ian was really onto something with his more "platonic" approach.<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>
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