[Smt-talk] Headlam on Orbifolds

Ildar Khannanov solfeggio7 at yahoo.com
Sun Mar 15 21:03:27 PDT 2009

Dear Dmitri,
thank you again for your extensive comments and arguments. The title of the book sounds very intriguing, alluding to ancient Greek topics. I wish to see it translated to many languages one day.
I have adopted the idea of heteregeneous character of music, strangely enough, from reading Deleuze. Now, I see that it is a fruitful idea. 
Best wishes,
Peabody Conservatory
solfeggio7 at yahoo.com

--- On Sun, 3/15/09, Dmitri Tymoczko <dmitri at Princeton.EDU> wrote:

From: Dmitri Tymoczko <dmitri at Princeton.EDU>
Subject: Re: [Smt-talk] Headlam on Orbifolds
To: "Ildar Khannanov" <solfeggio7 at yahoo.com>
Cc: "smt-talk Talk" <smt-talk at societymusictheory.org>
Date: Sunday, March 15, 2009, 2:00 PM

Ildar wrote:

Yes, there is a phenomenon of melodic stepwise motion and it plays an important role in music. It was known well before Schenker and voice leading has been the goal of many national pedagogic systems of the past.  But this motion does not override other factors such as tonal functions. They work together making harmony, or part-writing (as mentioned in one of latest postings), a heterogeneous system. Harmonic progression cannot be reduced to melodic stepwise motion by debunking the tonal function, but it needs melodic stepwise component for the coherence of the finished product. In fact, the requirement of stepwise motion is mandatory for inner voices, it is applied to melodic voice together with the demand for strong functional relationship, but it is not mandatory for the bass.  In general, it is nice to see voices in a progression moving stepwise, but what is the engine that moves them, if not the tonal function? What makes the G “wanting to go to
 C?” What does the word “resolution” mean to you?

I agree with you that music is heterogeneous.  As you say, different rules apply to the bass and to upper voices; efficient voice leadings often occur in the upper voices while the bass moves by leap.  This is why I typically model upper voices separately from the bass.

I also agree that functional tonal music involves genuine harmonic laws that cannot be reduced to voice leading.  In fact, I would say that the voice-leading principles governing 18th-century music are quite similar to those governing 16th century music -- avoidance of parallel fifths and octaves, efficient voice leading in upper parts, similar cadential formulae, etc..  What is new is a set of genuinely harmonic laws, of the form "ii goes to V but not vice versa." 

Let me also say that a lot of these issues are clarified in my book ("A Geometry of Consonance"), which is basically done.  (The manuscript is being torn apart by readers and editors as we speak.)  I try to explain how to use all these new geometrical ideas in a practical analytical context, addressing all these points and providing lots of analytical examples.

In this respect, I see no problems feeding a different data into your system. Let us imagine that a semitone does not represent the smallest distance in all cases. Ernst Kurth wrote so much about B to C relationship, that this distance is not the closest and it takes much energy to cover it. Schenkerians also talk about “neighbor leap,” meaning that a leap can function as an adjacency. Motion from C to G seems to present a larger distance than that from C to C#. For a mathematician this is the fact. For a person who has spent 11 years in Ear Training class studying tonal structure this is not exactly so. From a certain point of view, the distance between C and G is small. The distance from G to C is even smaller (our musical space is warped!). The distance from C4 to C5 is almost indistinguishable.

I believe that there are many different distances relevant to music.  For an even simpler example, consider a scale -- in C major, the distance from C to D, like the distance from E to F, is "one step."  A scale provides a contextual distance measure, the scale step.  This in turn provides a metric on the pitch-class circle, and hence the orbifolds.


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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