[Smt-talk] Headlam on Orbifolds

[Dmitri Tymoczko]

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.

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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