[Smt-talk] Headlam on Orbifolds

Dave Headlam dheadlam at esm.rochester.edu
Thu Mar 19 12:23:35 PDT 2009


There¹s certainly nothing wrong with asking basic questions and looking for
elemental issues ‹ but within a context  . . . .

1. 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 ³a semitone is a vast
distance harmonically² needs a context, precisely the ³direct / indirect
distinction² that is basic to #IV and other such approaches. (C-F#, versus
C-G(F#), C-G (D-F#) etc. see Lewin GMIT)

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.

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.  

4.  no doubt that C-C-C sounds different than C-E-G  -- I don¹t 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 Cinammon¹s dissertation for voice-leading in 19th c. music ‹ in tonal
and symmetrical contexts ‹ way ahead of it¹s time and should be required
reading for all 19th c. aficionados.

5.  oribifolds:  PE as above, IC  = incomplete chords, which don¹t stay in
one area ‹ I¹ll give up that old one about the ³stay in the same area of the
orbifold² (although that was a big part of your media splash)

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 ³unisons² in
complex contexts ‹ and of course, Berg¹s Wozzeck at Marie¹s death (B s), but
it¹s true (I think) that harmonic ³consistency² is more often the case -- (
look at Wallace Berry, Struc. Functions chapter 2 ‹ a real gem on texture).

6.  The last one is worth a dissertation ‹ what is voice-leading?  If you
want to turn voice-leading into ³voice-leading² (like Lewin¹s interval into
³interval², Forte¹s set into ³set², Perle¹s sum into ³sum², etc.) as a
generalized notion then you have to lay out the premises

1. an actual physical ³small² connection between two notes, or an abstract
connection between two possible implied entities
2.  governed by group theory / Euclidean solid angles / etc. or more
topological / stringy
3.  a function, operation, etc.  (not a thing but a motion, etc.)
4.  explanatory or defining? (see Roeder on Schoenberg) What is it supposed
to reflect faithfully without becoming circular?
5.  part of some chronological , progressive syntax or recreated constantly
6.  a subset / superset of Lewin?  Explains harmony or explained by harmony?
Prolongational or associational?
7.  kinetic, evolutionary (the voice and the hand), cultural (childrens
nursery rhymes)
8.  Schenkerian, Perlian, Fortean, neo-R ?  Is the search for step-wise
motion part of the ten commandments of music?

What¹s the VL in Schoenberg¹s opus 11?  Blasse Washerin?  (see James Baker
on opus 19/ I)  In Berg¹s opus 5?  In Webern¹s opus 5/III?  (see Mark
Sallmen¹s fine work here).  Same principles as in WTC I ‹ or all new, or
some transformation?  Rothstein has an article on stepwise motion in
Martino¹s music ‹ same as in WTC I?  What about in Webern¹s op 5/II --
along Lewin¹s network model of 026s ‹ harmonic space or voice-leading within
the space of the trichord?  (what¹s the diff?)

Certainly an interesting topic ‹ some wheels are worth reinventing, but
don¹t forget to mention Goodyear.

Finally, In this and other contexts, please read Walter O¹Connell¹s article
on Tone Spaces in Die Reihe 8.  One of the great articles ‹ thanks to Andy
Mead for pointing it out to me (I¹ve written on it, in the Western Ontario
Studies in Music Series) -- it deserves a wide audience ‹ if anybody knows
who O¹Connell was, I¹d appreciate hearing about him.


> 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?
>> I¹m certainly not saying that we can¹t model incomplete chords (³geometry is
>> useless in this instance², etc.), merely that one of the original DT
>> postulates was (is?)  that staying in one area of the orbifold was ³good²  --
>> 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 voices.
> 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 it¹s PE form, it doesn¹t elegantly show
>> progressions with ICs staying in one area.  If this is no big deal, then
>> there¹s no concern.  But imagine reshaping it to have 000 ­ eee in the
>> ³middle² and 00x to eex ³around² the middle and ³complete chords² on the
>> ³outside², 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.
> 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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> rg


Dave Headlam
Professor of Music Theory

Eastman School of Music
26 Gibbs St
Rochester, NY 14604
(585) 274-1568 office
dheadlam at esm.rochester.edu

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