[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
points:
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).
DT
Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)
http://music.princeton.edu/~dmitri
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