<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">
<div>Hi Dave,</div><div><br></div><div>It sounds like we're starting to understand each other a bit better! That's gratifying. Certainly, the conversation is helping me clarify my ideas.</div><div><br></div><div>I'm going to try your patience by addressing a few lingering issues, and by reporting on some statistical work I did to try to get a handle on how important your concerns are. This is potentially more for my benefit than yours, as I suspect that some of this will probably end up in my book, when I revise. I'm pretty sure that, by now, everybody else on the list is spam filtering our missives anyway, so it's probably just us here anyway ...</div><div><br></div><div>===========</div><div><br></div><div>First, we should distinguish two questions:</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>1) Is it the case that Western tonal music very frequently utilizes efficient voice leadings between (approximately) transpositionally or inversionally related chords, represented in the orbifolds by short line segments between structurally similar chords, more or less as I've described?</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>2) Does this observation tell us absolutely everything we might want to know about how to combine (something like) harmonic consistency with (something like) efficient voice leading?</div><div><br></div><div>I claim that the answer to Question 1 is pretty clearly "Yes." If you understand the geometry, you can open virtually any page of any 18th or 19th century score (and many 20th and 21st century scores) and find any number of these voice leadings. Heck, you can see them in standard fingering patterns for guitar chords. They are truly ubiquitous. They occur not only between chords but also between scales (as described at headache-inducing length in "Scale Networks and Debussy"). In this sense, I claim, the geometry helps you understand an awful lot of what actually happens in music -- including some of the weirder stuff, such as Chopin's E minor Prelude or Wagner's Tristan prelude.</div><div><br></div><div>The answer to Question 2, I admit, is "no." There are some subtleties I haven't discussed. Incomplete chords, as you've mentioned. (I'll get back to these in a minute.) Another interesting case are nonbijective voice leadings like (C, C, E, G)->(A, C, F, F) and (C, C, E, G)->(B, D, F, F) where if you throw out any one voice, you end up with something that cannot be modeled as a short-distance motion between transpositionally related chords. </div><div><br></div><div>About these I have two things to say:</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>Thing 1. You can, in fact, use geometry to understand voice-leadings like (C, C, E, G)->(A, C, F, F). There's actually a whole fascinating and beautiful story about these voice leadings, which I forced myself to work through in response to your criticisms. (The two cases described in the previous paragraph account for the large majority of the nonbijective triadic VLs that appear in Bach chorales.) Remarkably, the efficiency of (C, C, E, G)->(A, C, F, F) can be explained by the fact that the major chord is pretty close to the tritone. (I'd explain further but it wouldn't fit in the margins of this email.) This all belongs in my book, so I'll have to add it in.</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>Thing 2. At the end of the day, my ideas are not going to tell you how to model 100% of the voice leadings in classical music. But suppose I can help you understand 90% or 75% ... or 50% ... or even 25% of them. That's still still pretty good, I say. In general, I'm more concerned with trying to describe things that happen frequently, rather than providing a complete system that describes everything under the sun.</div><div><br></div><div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size:14.0px">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. </span></font></blockquote><div><br></div><div>It is typically the upper voices that move stepwise -- this is very clear in the statistics. Sometimes the bass + two upper voices move stepwise while an inner voice leaps. Efficient voice leading is more or less independent of progression, being about as likely to occur in I-V as V-I, IV-I, ii-V, etc. A very large number of four-voice voice leadings (~ 88%, in the Bach chorales) "factorize" into a voice leading between complete chords in 3 voices, plus one voice that adds doublings. (Of these, about 80% have stepwise motion in three voices, with one leap; in about 70% of the total number of cases, the leaping voice is the one that doesn't participate in the 3-voice voice leading.) All of this can be investigated empirically so there's no need to rely on vague intuitions.</div><br><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size:14.0px">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). </span></font></blockquote><div><br></div><div>In Bach chorales, about 97% of the consonant four-voice chords are complete; incomplete triads (thirds, fifths or unisons doubled in various ways) account for just 3%. In Josquin's "Magnus es tu" the numbers are more like 85% vs. 15%. Given these ratios, it's hard for me to get excited by the problem of incomplete sonorities. Of course, keyboard-style music may be different; unfortunately, it's much harder to investigate using a computer. That said, the Bach chorales have often (and in my view reasonably) been taken as representing harmonic patterns that occur in the background in instrumental pieces.</div><div><br></div><div>About permutation equivalence, it's very important to remember two points: </div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>1. "Equivalence" in my sense doesn't mean "equivalent for all purposes" only "equivalent for some reasonable purpose."</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>2. If you're modeling the bass voice separately, then permutation affects the upper voices only. Now, from the standpoint of elementary local harmonic coherence (C3, G3, C4, E4), (C3, C4, E4, G4), and (C3, E4, G4, C5) are all very clearly identifiable as C major chords. Here, permutation and octave shift apply to the upper voices but in the grand scheme of things it is a small change. We clearly have C major chords in each case. </div><div><br></div><div>So to question permutational equivalence is to doubt whether (C3, G3, C4, E4), (C3, C4, E4, G4), and (C3, E4, G4, C5) should in general be considered as relatively minor variants of the same basic harmony. Somehow, I doubt you that you're prepared to question permutational equivalence in this sense.</div><div><br></div><div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size: 14px; ">Coherence (IMO) comes not from these elements (except in a local sense) but from large-scale returns / prolongations, etc. of keys and motivic ideas. </span></font></blockquote><div><br></div><div>You use a nifty trick of rhetoric here: "Coherence (IMO) comes not from these elements (except in a local sense) but from ..." In other words, you agree that there's a kind of coherence that comes from these elements, namely local coherence, but you then go on to denigrate it en passant by valorizing some other kind of coherence.</div><div><br></div><div>But why denigrate? I am primarily talking about coherence in a local sense. I'm genuinely interested in it. I think there are fascinating unexplored questions about just how local coherence is possible. I doubt you really think that local coherence is unimportant; you'd certainly miss it if it were absent.</div></div><div><br></div><div><blockquote type="cite" class=""><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Verdana" size="4" style="font: 14.0px Verdana">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) </font></div></blockquote><br></div><div>Let me reiterate what I wrote in response to Question 1: efficient voice leading is represented by short distance paths in the orbifolds; efficient bijective voice leading between transpositionally or inversionally related chords is possible only by exploiting the interesting geometry of the spaces (roughly, "staying in the same region of the orbifold"); and these voice leadings are ubiquitous in a wide range of tonal styles. I stand by all of this. The statistics I've been able to produce seem to support the claim. The fact that there are *occasional* progressions like (G, B, D)->(C, C, C) doesn't change the general picture.</div><div><br></div><div>Now admittedly, it certainly helps my case to be able to provide some actual numbers -- even if I only have data about the Bach chorales. So I'm grateful to you for forcing me to generate them.</div><div><br></div><blockquote type="cite"><font face="Verdana, Helvetica, Arial"><span style="font-size:14.0px">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</span></font></blockquote><div><br></div><div>These are discussed extensively in "Scale Theory, Serial Theory, and Voice Leading" (Music Analysis 2008). </div><div><br></div><div>DT</div></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>