"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.
- 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 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):
Enough of Beethoven for now.
 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).