Zero-Sum Games in Dvořák

A year ago, I had something to say about Dvořák's Ninth Symphony in E Minor. With the return of March, I thought I would do so again. The end of the symphony is a little unusual. Not the final nine bars, but the three bars right before it, shown in reduction on the left side of the example below. The musical rhetoric—the entire orchestra loudly alternating between the major tonic chord and some other chord—would seem to suggest that tonic is being affirmed, as we would expect in a tonal symphony’s concluding measures. However, the “other chord” is enharmonically a half-diminished seventh chord built upon D, far from a chord rooted on B (or maybe A) one would expect. If anything, this chord, coupled with an E-major triad, destabilizes E as tonic: the notes in these chords fully constitute an A harmonic-minor scale, except the A, as shown on the right side of the example below. Play an A-minor chord after all of this and it might justify this oddity, but the symphony’s imminent end will probably feel farther away, not closer.

But, from another point of view, this harmonic alternation is consistent with other concluding aspects of the symphony. In the traditions of the per aspera ad astra symphony and the cyclic symphony, Dvořák lifts many of the work’s minor-mode themes into the major mode during the work’s final minutes. However, one theme goes the opposite way: the major-mode chorale at the beginning of the second movement, of which the first four chords are provided on the left side of the example below, is brazenly put into the minor mode toward the end of the fourth movement. Never after does the final movement restore this chorale to major, but one could hear the D half-diminished seventh chord as offering such a restoration in the abstract.

If one considers the E-major triad with a doubled root, the idealized voice leading among adjacent chords at the beginning of the chorale is always balanced; that is, the amount of total semitones voices each that go down and go up are the same, as shown in the example below with the left-hand-side columns of numbers that each sums to zero. (This is also true for the six-voice realization discussed a year ago.) This balance also occurs in the music with the D half-diminished seventh chord, as shown in the example below with the right-hand-side columns of numbers that each sums to zero. This would also be true for any of the four half-diminished seventh chords rooted on D, F, Ab, or B. But only the D half-diminished seventh chord brings back two of the three notes in each of the two balanced non-E-major chorale chords (Bb major and Db major), as shown with the four slurs below. In fact, it is the only chord—regardless of quality—that 1) brings back two of three pitch classes in these two triads, 2) can achieve balanced idealized voice leading with the root-doubled E-major triad, and 3) does not contain a half step between any of its notes.