[Smt-talk] Headlam on Orbifolds

Dmitri Tymoczko dmitri at Princeton.EDU
Mon Mar 16 10:16:28 PDT 2009

Hi Dave,

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.

OK, so you wrote:

> 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.

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!

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.

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.

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?

> 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 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  

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.

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.

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.

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.

 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.

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.

> 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.

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.

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.

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.


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

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.societymusictheory.org/pipermail/smt-talk-societymusictheory.org/attachments/20090316/98de9a42/attachment-0004.htm>

More information about the Smt-talk mailing list