[Smt-talk] Headlam on Orbifolds

Dave Headlam dheadlam at esm.rochester.edu
Sun Mar 15 09:24:17 PDT 2009

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.  C-F# is the wild card ‹ big melodic
and harmonic ‹ and non-directly - functional in tonal contexts ‹ its always
points away from tonic.  When harmonic function begins to break down, the
harmonic/melodic distinction correspondingly breaks down in favor of melodic
intervals, which begin to pervade both spaces (bII instead of V or both in
Berg¹s Piano Sonata, half-diminished 7ths to major-minor sevenths everywhere
‹  and in general Mark DeVoto¹s ³creeping chromaticism²)  -- this is old
news, although worth thinking about.

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 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.

Or perhaps something like this:  tightly wound circles of 000 to 060, etc.
then with 012 to 01e etc.,  as another circular layer.  Then connected end
to end.  Then we could ³buzz² around 000 040 047 if the occasion called for

0e0 1e1                                                         eee
000 101 202 303 404 505 606 707 808 909 t0t e0e
010 111                                                         e1e

Even more fun in the quad model, where F- F- Ab ­ Eb resides.

I like incomplete chords.    I don¹t throw up my hands when I hear them.  I
think the orbifold can model them ‹ just not elegantly in the current

Dave Headlam

On 3/13/09 6:10 PM, "Dmitri Tymoczko" <dmitri at Princeton.EDU> wrote:

>> I associate part-writing with counterpoint in the sense that it deals with
>> the surface elements of a composition and the threads of succession. Your
>> earlier exchange with Ildar, however, implied, on his part at least, that
>> incomplete harmonies were some pivotal issue. Let us add a definition of
>> "chord" to be an element of counterpoint, which is to say, on the real
>> surface of a composition, and further say that it is a selection of elements
>> of the surface­without regard to rhythm (including simulataneity)­isolated
>> for evaluation with respect to the referent harmonies (in tonal music, triads
>> and maximally even tetrads). If we wish to give a chord a name, then we are
>> including all of its elements in the concept of voice leading, while
>> counterpoint has no such obligation. Thus, I can read bars 9-11 of the
>> Waldstein as built upon GBF, three of the four notes of a G7 harmony, without
>> their representing that harmony; there need be no D. Yet bar 13 has but one
>> element of a G triad and I do read that as a full G harmony and accomodate
>> the voice leading accordingly. You and I are with Rameau in this perception
>> of voice leading, but most Schenkerians assume that voice leading is what
>> they show; sometimes it is, sometimes it isn't. Züge don't need support for
>> all their displacements.
> I think I follow what you're saying.  The example is definitely useful.
> What I was saying to David (not Ildar) was that you can use orbifolds to model
> surface harmonies if you want: for instance, you could model the relevant
> passage of the Waldstein as the three-voice (Ab, C, F)->(G, B, F) in
> three-note chord space.  That's a perfectly reasonable thing to do.  I think
> you might say that we're modeling the actual "part writing" here.
> On the other hand, you can also model voice leadings that are only hinted at
> by the surface.  For instance, in the recap of the Waldstein, measure 172, you
> find (F, F, Ab, Eb)->(Ab, F, B, D), which I take to be an embellishment of the
> common semitonal pattern (F, Ab, C, Eb)->(F, Ab, B, D).  This sort of
> semitonal play between seventh chords is ubiquitous in nineteenth-century
> music, and is the engine of an awful lot of chromaticism.  I think it's useful
> to identify it as lying in the background, obscured to some extent by the part
> writing.
> What I don't quite understand is Dave's suggestion that geometry is useless
> whenever we confront incomplete chords -- as if we have to throw up our hands
> and give up the moment we see a progression like (F, F, Ab, Eb)->(Ab, F, B,
> D).  I disagree: we're still free to use all the standard techniques of
> analysis, including postulating octave displacements, voice exchanges,
> additional voices, passing tones, and so forth.  Part of the value of the
> geometrical models is that they give us powerful tools for understanding the
> ubiquitous background schemata that are (sometimes) only hinted at by the
> musical surface.
> 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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