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."

The Art of Parsimonious Orchestration

"It's not the arrival, it's how you get there."

– Unidentified passenger on The Orient Express.

I left off the previous post with the question "What do 'we' really hear [in music]?" The reason the 19 chords of Beethoven I used as an illustration have become well-known to many (well, at least a few) music theorists is because they represent one of the longest strings of neo-Riemannian transformations ("transformations" here meaning chord progressions) identified in the music literature.

En passant:

• As in many fields, in contemporary musicology an analytical theory gains greater credibility every time a confirming instance is found in the real world (meaning, generally, music composed/performed before the theory was systematically formalized). 
• Music-analytic theories are radically retrodictive.
• When a sufficiently large number of peer-approved confirmations has been collected, the theory is recognized by a cohesive, but not necessarily universal, peer group as "true." But given the overlap of theories that are incontestably true, the turf battles fought within the musical academic community (giant egos to one side) appear, to this outsider, to be fought almost entirely over applicability, relevance, importance, significance, generalizability, etc., and rarely over facticity. 

But back to those 19 chords. Following is a reduction of the score that demonstrates a form feature that arises from Beethoven's orchestration of the passage.

While it is incontestable that the entire passage is the string of neo-Riemannian transformations[1] R,L,R,L,R,L,R,L,R,L,R,L,R,L,R,L,R,L (summarily, (RL)⁹ or, using the ad hoc computer pseudo-code notation from the previous blog entry,  X¹⁸) the orchestration of that passage indicates something more is going on – at least it was in (flagrantly flouting the intentional fallacy) Beethoven's head.

Some might say the following is an alternative reading, but I see it as a concomitant. The score reduction above shows that Beethoven was doing a game of major-minor hopscotch between the strings & horns (lower staves) and the woodwinds (upper staves). Red boxes indicate major triads and blue indicate minor triads. If you were to eliminate the blue box material, the result would be a perfectly logical sequence of major triads whose roots follow a succession of perfect fourths. The same will happen if you eliminate the red box material resulting in a sequence of fourth-related minor triads. The result of alternating the two certainly is the RL sequence. But, once again, Beethoven doesn't do a "straight" alternation. His orchestration keeps them separate creating a tension not present in a straight RL.

After beginning with the strings-horns doing the major triads with the woodwinds alternating minor triads taking us from C major to D minor, there is a pause, after which the woodwinds do the major and the strings-horns the minor, getting us from D minor to Eb major. Then, starting at Eb major after another pause, the strings-horns take up the major while the woodwinds alternate the minor sequence again, and this lasts until we arrive finally at A major, ready to make a final quick chromatic hop to E minor. All in all, a little like driving from Denver to Las Vegas by way of Atlanta. Not the shortest route, but a most interesting one. For those who like graphs and circles, here is what happens (string-horn moves are inside the circle, woodwind moves are on the outside):

So which is it? What's the right way to hear this passage? Should I follow the straight RL sequence, or should I follow the hopscotch counterpoint of major and minor triads? Can I hear both at once? Or if I'm the conductor, which should I bring out? And we haven't even brought Schenker into this wonderful mess!

Enough of Beethoven for now.


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[1] In neo-Riemannian lingo, R is the "relative" transformation and L is the "Leittonwechsel." There are several ways of defining these two basic transformations (the third basic transformation is P for "parallel"), but simply here, since we begin with a major triad, R indicates "raise the fifth of the triad a whole tone" resulting in a minor triad (CEG→CEA). Then, from the minor triad, L indicates "raise the fifth a semitone" yielding a major triad (ACE→ACF).