The fuguists among us are well-acquianted with the traditional fugue devices: inversion, retrograde, augmentation, diminution, etc.. Back in Sep 2018, an idea -- probably a totally insane one; y'all be the judges -- occurred to me, that suggested an expanded palette of fugue devices that one could potentially use to derive new material in a composition.
The underlying idea is that these fugue devices can be viewed as geometric operations on the fugue subject. The DAW users among us would be well-acquianted with the piano roll: each note is plotted on the roll by its pitch and the time of its sounding. For our purposes, let's ignore note lengths for the time being, and just pretend that each note is a single dot on the piano roll. So, given some musical theme X, it would correspond with a specific pattern of dots on the piano roll. Now suppose we cut out the rectangular section of the piano roll that corresponds with our theme. It would be a rectangle with a bunch of dots on it. Now we can perform various geometric operations on it to produce the various fugue devices:
- Flipping the rectangle upside-down produces the inversion of the theme.
- Looking at it in the mirror gives you its retrograde.
- Rotating it 180° gives you its retrograde inverse.
- Stretching it horizontally gives you its augmentation, and compressing it horizontally gives you its diminution.
So far so good, nothing unfamiliar here. But the above picture of performing geometric operations to our cut-out piano roll suggests that there are other possibilities:
We could stretch the theme vertically, for example. I don't know what this would be called, but it would essentially stretch all (pitch) intervals in the theme. And conversely, compressing the theme vertically compresses the intervals. Conceivably one could rationalize tonal answers as a vertical stretching/compression of a fugue subject. This is interesting, but there's something else even more interesting.
We could rotate the theme by 90°. This essentially interchanges pitch and rhythm: chords become a series of repeated notes, with the rhythm determined by the intervals between the pitches in the chord. Similarly, a series of repeated notes turns into a series of simultaneous pitches: i.e., a chord. If we rotate 90° counterclockwise, for example, higher pitches would turn into notes that sound earlier, and lower pitches into notes that sound later; notes that occur earlier become lower notes, and notes that occur later become higher notes.
We need not be limited to 90° rotations. We could, for example, assign numerical coordinates to each pitch and time, and thereby turn each note into a 2D vector, whose components represent the pitch of the note and the time when it sounds. Then we can apply a rotation matrix to this vector to produce a new event: a new note that represents the original note rotated by some arbitrary angle. So we could, for example, compute the 45° rotation of a theme, and that would produce a new theme whose pitches and rhythm are determined by an equal proportion of pitch and rhythm in the original theme.
Indeed, we need not be limited to rotations; as we already mentioned above, reflections correspond with the various familiar fugue devices. So we could in fact apply any arbitrary linear transformation to our theme to produce all manner of transformed themes that are directly related to the original theme geometrically.
At this point, y'all probably think I've completely gone off my rocker (and you're probably right). But wait, there's more! 😆
Some of you are probably familiar with Messiaen's "modes of limited transposition", i.e., scales that, when transposed by each of 1-12 semitones, result in at least one transposition that has the same pitch class as the original scale. For example, the octatonic scale on C transposed up by 3 semitones produces the exact same class of pitches. One can think of it as, the octatonic scale is invariant under transposition by 3 semitones. This invariance is what makes these scales interesting.
In a similar vein, we can postulate themes that are invariant under certain geometric transformations. To start with a familiar example, take a palindromic fugue subject. Its retrograde is the same as itself, so it is invariant under retrogradation. Similarly, a theme that consists of palindromic chords (chords that equal themselves when their constituent pitches are inverted) would be invariant under inversion.
A more interesting kind of theme would be one that's equal to its retrograde inverse (something like a table canon on a smaller scale).
These are all themes invariant under one of the well-known fugue devices. But we can do better: what about a theme that's invariant under rotation by 90°? I.e., its pitches determine its own rhythm, and its own rhythm determines its own pitches! Now that would be a very interesting theme indeed! (Unfortunately I haven't come up with such a theme yet... anybody up for the challenge? 😉)
An even more interesting (and significantly harder) challenge is a theme that's invariant under rotation by 45° (which, by definition, will also be invariant under a 90° rotation). This theme not only determines its own pitches by its own rhythm and vice versa; it also has 8-fold symmetry, where both the pitches and the rhythm are also determined by their equal combination in the pitch/time axes.
One can also imagine themes that are invariant under 60° rotations (6-fold symmetry), or 72° rotations (5-fold symmetry), etc..
Now try writing a fugue where the subject is in stretto with a 45°-rotated version of itself. 😂 Or indeed, a fugue with n entries, where each entry is rotated by 360°/n degrees until it finds itself in its original form again at the very end. A rotational fugue!
Beyond fugue writing, though, this analysis suggests very interesting avenues of exploring the relationship between pitch and rhythm.
I'm reminded, for example, of a collaborative thing we did years ago on the old forum, where each of us wrote a variation to a theme by Bach (BWV 140) and we pasted the result together to form an eclectic air with variations composition. John Driscoll's entry in this collection was especially interesting: instead of some clever variation on the original Bach melody, he instead segmented it into groups of notes that he then collapsed into chords. The result was quite interesting and atmospheric. That particular device can be understood in the above context as a projection matrix applied to the notes of Bach's theme (appropriately segmented), thus compressing it in the time axis so that the original separated pitches fell together to form chords.
This would seem to suggest, to me, that there remains a vast unexplored realm of possibilities out there as far as fugue devices are concerned. Or, should I say, the geometric generalization of fugue devices. 😄
What do y'all think of this? Is this something worth exploring, or have I just gone completely mad?