J.S. Bach: Air from Orchestral Suite No. 3, mm.1-2

May 31, 2008

Recall that in a previous post I challenged readers to analyze the first two measures of the Air from Bach’s Orchestral Suite No. 3 in D major (a piece, incidentally, that might be better referred to as “Air Off The G-String” than by its usual nickname). The time has come to reveal the answer.

In the Pachelbel analysis, we started from the underlying basic structure and showed how the passage was constructed via the Westergaardian operations. This time, for the sake of variety, we’ll proceed in the reverse direction, starting from the passage itself and “undoing” the operations until the basic structure is revealed.

Our passage is the following:

Stage 1

Call this Stage 12. The first thing we’ll undo are the explicit arpeggiations in the first violin and continuo lines:


Actually, I did a bit more than that, as you can see. I skipped a stage in which the first violin part looks like:

How did I know that D was the span pitch of the second half of beat 2 rather than C#? That is, why did the first violin part not reduce to:

Is it because G#-E-B (or even G#-E-B-D) is a Certified Chord, whereas G#-E-B-C# isn’t? Fat chance! As an exercise in eliminating harmony, see if you can explain the real reason. (I’ll likely explain it in a future post, but probably only after we’ve formally developed more Westergaardian theory. Hint: It has nothing to do with Certified Chords.)

Eliminating the borrowed G and B from the first violin, we obtain stage 10:


What an odd interpretation of beat 2! Instead of hearing a passing motion from E to C, I am interpreting the E as a borrowing from the viola line:


(Note also the elimination of the A borrowed from the second violin line.) Why on Earth is this interpretation to be preferred to the seemingly simpler one? The answer is that the seemingly simpler one isn’t in fact so simple. Notice that the D in the second violin line is left hanging (ITT, p. 30), and therefore not displaced, after beat 1. If the D in the first violin line were to be interpreted as a passing tone, that would leave us without a D among the sounding span pitches of beat 2. However, we know from the C# of beat 3, as well as from the fact that D was left hanging in the second violin, that D must be a span pitch for some span that includes beat 2 (deeper levels will make this clearer; see below). We would therefore be compelled to regard the second violins’ D as being temporarily displaced during beat 2; that is, it must move by step to some note borrowed from another line. (The only alternative would be to regard it as (entirely) undisplaced during beat 2, but this is made difficult because of the simultaneous E: since in this scenario we’re not considering D as a local span pitch of beat 2, we’re left with understanding an implicit dissonance, which is quite problematic indeed.) Since E is a span pitch of beat 2 and C# is not, we must therefore hear the D-line as moving up to a borrowed E during beat 2. But why should we go through the trouble of understanding such a conceptually difficult situation as the D-line effectively “merging” temporarily into the F#-E line? Given the stated step motion D-C# in the first violin, isn’t it easier to regard that D as a span pitch over the span of beat 2?

Stage 8 shows transferred pitches (ITT, sec. 7.7) reassigned to their rightful homes:

Stage 1

This stage represents the transition from instrumental lines to structural lines; I have symbolized this by switching from the alto clef to the treble clef in the third line.

Next the transferred pitches are reassigned to their rightful registers:


The suspension in the top line is removed:


Rearticulations in the bottom three lines:


Rearticulation of a suspension in the second and third lines; chromatic step motion in bass:


Suspensions eliminated:


Neighbor note removed:


Finally, then, we have the basic structure of the phrase: