Being as I’m a mathematician and fan of logic puzzles, I had a lot of fun with how I might approach the “fugue” form and tonal structure. What follows is going to be only interesting to people, like me, who delight in building such constructs, and also know some music theory. Anyone who’s spent time here will know that “how it sounds” is really the proverbial “proof of the pudding” for me — that’s the “eating.” As always, my tonal constructs are consciously outside of the normal western European traditions. Regardless, my hope is that the composition should still be enjoyable, even if the underlying theory isn’t understood.
Like many of my works, the tonal centers, and transitions between them, relate in a non-traditional way. This piece, like “September,” is built around tonal centers a descending major 3rd apart, which means only two such transitions are required to bring one back to the original key. But there are two other things worth mentioning.
Whole-tone scale. Musical whole-tone scales ignore half-steps. This piece is in the key of D. A normal D major scale would be D, E, F♯, G, A, B, C♯, D (8 pitches, 7 before the D is repeated). The whole tone version is D, E, F♯, G♯, A♯, B♯ (C♮), D (7 pitches, 6 before the D is repeated). The scale can be considered “pure vanilla,” since there aren’t any “tendency tones” (tones that inherently want to lead somewhere), and there are only two different triads (both augmented). Whole-tone scales have a long history, with a big hit in the late 19th/early 20th Centuries when Debussy and his contemporaries used them a lot.
In the fugue, I use a whole-tone scale as the underpinning of the fugue subject — the key centers rise, step-wise. When I arrive at the C♮, I hesitate, then descend two half-steps down to B♭. Being as this is a major 3rd below the original tonal center (D – see above), I’m good to go with the second fugal statement. So the sequence is…
- Subject (1: violas carry the fugal “subject”): Starts on D. Stops on C, falls back to B♭
- Answer (2: 2nd violins): Starts on B♭. Stops on A♭, falls back to G♭ (F♯)
- Answer (3: 1st violin): Starts on F♯. Stops on E, falls back to D
After a brief four-measure interlude (“link” in fugue parlance), the above sequence repeats (#4: bassoons/low brass have the subject, #5: trumpets, #6: clarinets), before transitioning to the development with its oboe solo.
Open 5ths. All chords that are triads include 5ths. In D, this is D and A. Major/minor chords include a 3rd (F or F♯), but if you know my works, you know I don’t use 3rds very often. In this piece. open 5ths, by themselves, figure prominently — they are what goes up the whole-tone scale in the fugue, as the underpinning of the fugue subject and subsequent episodes. The cellos provide this in the first three subject statements. When we reach the development, the truncated subject (first two steps in the scale), with open 5ths (D/A, and E/B), serve as accompaniment (in the clarinets) for the solo oboe. In that section, I’ve dispensed with the complete scale, using the bare-bones of the descending major 3rd transitions.
Open 5ths appear everywhere.
The Prelude. As I mentioned, I worked backwards in writing this piece, so the opening “prelude” was completed after the first two major sections were almost finished. The prelude also uses the whole-tone scale leveraging open 5ths, but a lot of added tones disguise things. The prelude is also in two parts, reflecting the fugue and the development which follows.
