On the Verge of Non-Riemannian Musical Geometries

Its pleasure is not the comfort of the safe harbor,

but the thrill of the reaching sail.

– Robert Grudin

In the previous two posts I used the example of a passage from the Scherzo of Beethoven's 9th Symphony purportedly to introduce an alternate nomenclature for neo-Riemannian (NR) theory's three fundamental triad progressions. Here is a complete comparison of the Riemannian moves with the same moves notated as triples (a,b,c) from what I call an extensible Riemannian (ER) system.

NR progressions

[M=major triad, m=minor triad, r=root, f=fifth, t=third, s=semitone, w=whole tone]

ER moves for SC3-11

[Numerical values indicate number of semitones in 12tET]

Given that the objects being operated on in both cases are major and minor triads, there is no real difference between the two abstract voice leading systems. They are, at this point, merely different expressions for the same thing. When two moves are combined such as R and L* to create a chain as in the Beethoven example, NR analysis must first locate the fifth of the triad for each move, alternating moving it up by whole and half steps, whereas ER, for each move, shuffles a mask by a T* permutation and multiplies it by the same formula each time. The result from ER is the same as that from NR, as I intended it to be. Obviously, in this context, there is no gain. And admittedly, the NR version of "locate a note and move it up or down by x" is musically more intuitive than "shuffle, multiply and add" – given that the objects being operated on are major and minor triads.

Let's try yet another way of viewing these three progression types by focussing on common tones rather than the tones that move:

The P [parallel] transformation exchanges triads that differ in modality but have the same root; whence the pcs common to both triads are ic5-related. The L [leading tone] transformation exchanges triads of opposite modality with common pcs related by ic3, and R [relative] exchanges triads of opposite modality that have common pcs related by ic4. (Jack Douthett. "Filtered Point-Symmetry and Dynamical Voice-Leading" in Music Theory and Mathematics: Chords, Collections, and Transformations, ed. Douthett, Hyde, Smith)

Formulations of this sort that are based on common tones (certainly Douthett was just paraphrasing older formulations) make even more clear the notion that the diatonic (usual major/minor) triad is axiomatic in Riemannian and NR-based theories. The triad – the object – is ultimately in control. The operations are valid or not as they are found to conform or not to the "shape" of the object. To wit: the assumption is that if you hold any two tones of the triad constant, there is only one place for the free tone to go to create another consonant (read "valid") chord – up or down a minor or major second.

Here is the crux of the issue. Riemann & company arrived historically at a time when at least one axiom – one unexamined, unquestioned premise – was still agreed upon by all. Riemann, like Schenker, begins with the assumption that the usual triad, however you want to tune it, is the basis for Western common-practice music. And this is true. But what this means is that all (common practice) theories, whether harmonic or contrapuntal,  revolve around the irreplaceable object and not, as "tonal" and "neo-tonal" theories seem to want us to believe, its transformations .

In geometry, Euclid takes us a long way by accepting the common sense notion that parallel lines never meet. But there are other geometries that begin by denying that axiom. There are other worlds to explore. An ever-present question for us in music ought to be: what happens if we deny axiomatic status to the earth-bound triad?

This puts us inexorably back on a path to places dreamed by poets. Stefan George, among others.

"I feel wind from other planets."