[Smt-talk] Headlam on Orbifolds

Ildar Khannanov solfeggio7 at yahoo.com
Sat Mar 14 09:51:02 PDT 2009

Dear Dmitri,
thank you very much for your elaborate clarification. I am especially excited to hear the distinction between NR theory and your own. I remember Richard Cohn discussing the two-step paradox at one of the West Coast Chapter conferences. Indeed, the two half-step moves have to be interpreted in some way and placed in a context of a theory. I think both Lewin's transformational idea and RN parsimony are not considering topology yet, while your approach is all about topology.
To your taxi-cab metric I would add "the 7 bridges of Koenigsberg," or, my favorite, a comparison of a bagel with the handbell. The ultimate fun of topology is that "size does not matter," or that size is conditional upon the system of measurement or underlying space. Two parallel straight lines do not cross. Two parallel straight lines necessarily cross if they are located on a sphere (Lobachevsky). In general, the novelty of topology can be seen in the difference between the blueprint and the map, or between the topographical map and topological map, in which the latter suggests spatial logic without imposing a single measurement system.
I do not see any problems with your approach, but I have some questions to basic assumptions about adjacency made by Schenkerian and Fortean theories. 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? 
 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. 
If you need a mathematical explanation, look no further than Nicomachus. Greeks were also preoccupied with the relationships of tones and applied math in their analyses. They have found that a superparticular relationship rules harmony. Duplex (1 ands 2), sesquialtera (2 and 3), sesquitertium (3 to 4), sesquiquartum (4 and 5) and sesquiquintum (5 to 6). And nowadays, the superparticular numbers are called “smooth numbers.” Is not this what Richard and you are looking for in voice leading? In this calculation, the semitone represents the relationship of unrelated numbers. As for the Schenker’s graphic representation of voice leading, topology can provide an interesting turn. For centuries, the stars in constellation Centaurus were considered to be adjacent to each other. It is known today that alpha Centauri is just 1.4 parsecs from the Earth while beta Centauri is whopping 107 parsecs. There is, however, a proxima Centairi, the adjacent to alpha,
 but nobody for centuries has seen it. 
Best wishes,
Ildar Khannanov
Peabody Conservatory
Johns Hopkins University
Baltimore, MD.
solfeggio7 at yahoo.com

--- On Fri, 3/13/09, 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>, "smt-talk Talk" <smt-talk at societymusictheory.org>
Date: Friday, March 13, 2009, 5:12 PM

Ildar wrote:

I am sorry for playing a devil's advocate for a moment. Of course, I appreciate your great work with geometry. It is just sometimes the topoloical aspect is overshadowed in your presentations by a simpler  principle of measurement by half-steps (which probably come from Neo-Riemannian understanding).

No need to apologize, and thanks for the kind words!

If I understand you, the "simpler principle" you refer to is just the "taxicab" (or "smoothness") metric, which provides one of the simplest ways to measure distance on orbifolds.  From this point of view, (C, E, G)->(C, F, A) is has size three, since one voice moves by one semitone, and another moves by two semitones.  By contrast, (C, E, G)->(C, F, Ab) is size two, since two voices move by semitone.  There's no contradiction between this way of thinking about distance and the geometrical perspective; this is just one of the many possible metrics you can choose.

It would be a mistake, however, to associate this method of measuring distance too closely with either the Tonnetz or neo-Riemannian theory.  From a NR-perspective, F major is *closer* to C major than F minor is -- the progresson F->C is LR, whereas f->C is PLR.  (Using the "taxicab" metric, the opposite is the case.)  Consequently, NR-theory doesn't seem to explain why F minor so often appears as a passing chord between F major and C major -- from an NR perspective the progression F->f->C moves away from C major and before moving back toward it.  The moral is that neo-Riemannian distances (measured in "units" of LPR or in edge-preserving Tonnetz-flips) are *not* voice leading distances.  This is a complicated and subtle issue about which much more could be said.

As far as I understand, the first geometrical models in which all distances represent taxicab distance are those provided by Douthett and Steinbach in 1998, particularly "Cube Dance."  These models are all embedded naturally in the relevant orbifolds representing n-note chords.

And I am ready to choose certain  metric from a number of others offered by topology. For example, I will decline your proposition that the E and Ab triads are the closest to C triad. In the metric of tonal-functional harmony, the G triad is the closest to C triad. You simply cannot fit any other triads in between in a meaningful progression. That is what I meant in my rather awkward critique,  that the geometry of note heads on the staff presents the metric which is drastically different from the one we use when playing and singing most of tonal music.

Oh, OK, I should've clarified the following.  Rachel Hall and I have worked to describe acceptable *voice-leading* metrics.  There are other notions of musical distance (including tonal distance, or that offered by NR-theory) that can be perfectly reasonable in some circumstances, but that don't measure voice leading.  Acoustically, you might think that G4 is very close to C3, since it's the second overtone of the lower note.  But you're not measuring voice leading if you think that C3->G4 is a "small" motion.  Again, a lot more could be said about this.

BTW, I would say that you can fit other triads between C and G.  In C major, the progression C->a->d->G is perfectly reasonable.  And you sometimes find G->e->C, both in the baroque and in 19th-century music.  The question of whether tonal harmony is fifths-based or thirds-based is a complex one; and you can make a good case for thirds, rather than fifths.  Interestingly, then you're back to voice-leading, since third-related diatonic triads are linked by single (diatonic) step voice leading.


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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