[Smt-talk] Headlam on Orbifolds

Dmitri Tymoczko dmitri at Princeton.EDU
Sun Mar 22 10:37:27 PDT 2009

Hi Dave,

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.

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 ...


First, we should distinguish two questions:

	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?

	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?

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.

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.

About these I have two things to say:

	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.

	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.

> 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.

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.

> 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 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.

About permutation equivalence, it's very important to remember two  

	1. "Equivalence" in my sense doesn't mean "equivalent for all  
purposes" only "equivalent for some reasonable purpose."

	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.

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.

> Coherence (IMO) comes not from these elements (except in a local  
> sense) but from large-scale returns / prolongations, etc. of keys  
> and motivic ideas.

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.

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.

> 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)

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.

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.

> 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

These are discussed extensively in "Scale Theory, Serial Theory, and  
Voice Leading" (Music Analysis 2008).


Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)

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