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?

## Replies

As usual, I only understood half of what was said, but it brought back memories of an arrangement I once created for a small barrel organ. I was quite young at the time, barely in my twenties. The organ's owner had asked for a classical arrangement for his instrument, as he was tired of all the popular music available for it. I cautioned him that it might be costly, as I needed to purchase the orchestral score and then adapt it for an instrument with only 27 pipes. To my surprise, he said money was not an issue, even though these kinds of organs were typically used at fairs and by traveling entertainers.

To cut a long story short, I completed the arrangement. Using my basic computer and one of the earliest printers, I printed the score on a roll of paper filled with a multitude of zeros, each of which needed to be punched out. He was delighted with the result until I presented him with the bill, which amounted to about 600 guilders, equivalent to 600 euros today. It was a bit steep, but I was struggling financially at the time.

Once the roll was ready, he let me listen to the music, and it sounded wonderful. We even had some fun with the paper – at my request, he played the music in reverse.

One year later I sold that score to a company that used to deal in organ rolls. That company paid me handsomely too. As I said, I needed the money.

I don't think you're mad, just playfull.

I suppose it's hard to grasp what I'm trying to say in the abstract. So I decided to (painstakingly) present a simple example (among hopefully more to come) to illustrate what I mean.

To keep things simple, I will use a 12-beat time fragment, divided into 4 bars of 3/4, and notes within one octave, which has 12 pitches. (Yes, the correspondence of 12 beats to 12 pitches is intentional.) Here's a 12x12 chart that we will fill in as we go along:

First, let's start off with the following rhythm that I pulled out of the top of my head:

Suppose we beat out this rhythm on the note C. The corresponding piano roll snippet would look like this:

If we rotate this chart counter-clockwise by 90°, we get this:

This gives us the pitches C, D, D#, E, F, F#, G#, A. This set of pitches suggests a harmonic A minor, with some modifications. So we can arrange the pitches in a way that somewhat makes melodic sense, and fit it onto the original rhythm. Here's what I came up with:

If we plot this melody onto our chart, we get this:

[Argh, I seem to have hit the max character limit. To be continued in the next post, where we will do a 90° rotation!]

... continuing from above, if we now rotate the chart counterclockwise by 90°, we get this:

If we now translate this back to traditional notation, this is what we get:

So we can say that this melody is the 90° rotation of the one posted in the previous comment.

Some interesting points to note: notice that the rhythm here is the retrograde of the original rhythm. If we rotate this again by 90°, we would get the retrograde inverse of the original melody. If we rotate

thatby 90°, we would get the retrograde inverse of this melody. And finally if we rotate that by 90°, we get the original melody. In other words, there is a 4-fold rotational symmetry between these 4 versions of the melody.Also, since the pitches we selected for the melody is derived from the 90° rotation of the original rhythm, each of these 4 variants will sport the same rhythm or its retrograde. This doesn't have to be the case; I've come up with another example that sports two distinct rhythms, and the pitches of one melody is determined by the rhythm of a second melody, and vice versa. To be posted later!

Alright guys and gals, are you ready for more? ;-) We're going to look at a more interesting example involving two distinct rhythms. As in the last example, we'll stick with 4 bars of 3/4 and 12 pitches of an octave, just for simplicity. Here are two rhythms I randomly picked:

Let's plot these rhythms on our 12x12 chart from last time. This time, due to how the pitches came out, it suggests a key of Cm or Eb, so I decided to spell the pitches with flats instead of sharps. But otherwise it's the exact same 12 pitches. Here's what we get:

I used B for the first rhythm and C for the second just for spatial separation; since this is just the pure rhythm the actual pitches don't matter. Now, let's rotate this chart by 90° to get the pitches corresponding to each rhythm:

The time axis here isn't important; we're just looking at the pitches. For the first rhythm, we get Db, D, Eb, F, G, Ab, A; and for the second rhythm we get C, D, Eb, F, Ab, A, Bb. As mentioned this strongly suggests Cm or Eb as the base key, with A possibly interpreted as the leading tone of the dominant key.

Now we'll try to arrange the pitches in a way that makes melodic sense. The second set of pitches seems easier to work with, so let's start with that. Here's the melody I came up with:

Notice that I took the pitches corresponding to the

secondrhythm, and fitted them onto thefirstrhythm. This is intentional; this is how we will relate the two rhythms together under a 90° rotation. Note that rotation by 90° interchanges pitch with rhythm, so by using the second set of pitches we will obtain the 2nd rhythm once we perform the rotation. And conversely, by using the first rhythm, once we rotate by 90° we will obtain the first set of pitches.To avoid running afoul the per-post character limit, we'll chart this melody fragment in the next post. Stay tuned!

Alright, we're continuing from the previous post, to chart this melody fragment:

Here's the corresponding chart:

And here's the chart rotated by 90°:

In traditional notation, this corresponds with this melody fragment:

This fragment sounds like it's starting a new idea but doesn't quite finish; the first melody fragment sounds like it ends on a cadence. So let's join these two together to get:

And there you go, now we have a phrase that can serve as a fugue subject or theme of a waltz or something. :-D Due to the way it's constructed, it has a very deep internal consistency: the pitches of its first half determine the rhythm of its second half, and likewise the rhythm of its first half determines the pitches of its second half. Together, they form a happy couple. ;-)

This is great Teoh. A potential goldmine for finding latent material in an idea.

I found a thematic fragment that exhibits invariance under 90° rotation! Check it out:

Here's the corresponding chart that shows 4-fold rotational symmetry:

This theme fragment is in fact determined by only 2 notes: due to the 4-fold symmetry each note automatically implies 3 others, so choosing two already implies all 8. It's rather tricky business to choose the exact 2 notes that result in a sensible-sounding result. 😅 I haven't tried more notes yet; probably I'll have to expand the chart in order to accommodate the resulting 12 notes without getting too cramped.

The sort of precise mathematical operations, such as rotations on subjects, that you suggest aren't feasible in the diatonic/modal system because it's a discrete set of pitches which lack the fine, granular resolution to support them. You have to migrate to a continuous domain of frequencies wherein such fluid transformations can be achieved.

This continuous Pantonality is in no way a (necessarily) a departure from the fundamental harmony that undergirds the traditional diatonic system. The simpler the ratio of simultaneous frequencies, the more sonorous; the more complex that ratio, the greater the dissonance. These principles are deeply rooted in our physical reality, and immutable.

One of the profound distinctions between a pantonal (in the sense I use it and mean it, ie, continuous) and diatonic system is finitude vs infinitude. Given two such systems, bound above and below, let two chords of a finite number of voices be stated in succession. The possible configurations in the diatonic system, while probably immense, are still finite, whereas in the pantonal system, no matter how narrowly bound, they are infinite. Diatonicism is combinatorily infinite if unbounded in range or duration, but countably so, whereas the possible pantonal intervals that can be enumerated are uncountably infinite, no matter how narrowly bound.

If you experiment long enough with this notion of mathematically precise transformation of subjects, you will naturally be led to the same conclusions, like a single point of immense gravitational pull, we all somehow end up in the same place, when toying with things founded in universal natural principles.

I offer the following example for your consideration, although it is in first movement sonata - symphonic form, rather than fugue. The principles of motivic tranformation across a continuous frequency spectrum are the same that you proffer here, and can not be realized within the boudaries of diatonicism, nor any finite, scalar set of pitches:

If we allow truly arbitrary matrix transformations on themes, then yes, we would need to use a continuous pitch/time domain, and that would be difficult to reconcile with our present discrete pitch/time system. However, there are several mitigating fronts here.

First, and perhaps most surprisingly, is the fact that although the set of real numbers is continuous and cannot be rationalized into any discrete sequence, the subset of numbers which we can name is actually discrete, and countable. (If you're interested in pursuing this topic further, look up countable models of set theory -- there exist countable models of theories that prove the existence of uncountable sets. It's paradoxical but ultimately the consequence of the fact that we can only name a countable subset of anything, and so we can never check the model against an uncountable number of distinct objects to verify that it has faithfully represented all of them. The model only has to explicitly represent a countable subset of any uncountable object, because the rest are not nameable by us and therefore are indistinguishable. Then it just "pretends" that they are there by assigning certain truth values to propositions that assert the uncountability of a set.) Therefore one can always find a discrete representation even of something that's ostensibly continuous. (Of course, this does not guarantee such a representation will be practical, or be in any form that can be computed with.)

Secondly, we can restrict ourselves to a subset of matrix operations which would produce results from our starting thematic material that can easily be rationalized into a discrete system. Rotations by certain special angles, for example, have algebraic (rather than transcendental) matrix entries, and it is a theorem of algebraic numbers that any algebraic extension field can be represented exactly by a vector field over the rationals. For example, if we fix a square-free integer r, then the field generated by adjoining √r to the rationals can be represented as 2D vectors with rational coordinates, closed under field operations. So rotations by multiples of 30°, for example, can be exactly represented as vectors of the form (a+b√3), and retain the same form under field operations. We can therefore map these rational coordinates back to our discrete pitch/time system for at least a discrete subset of cases. Themes that are amenable to this treatment can then be part of our considerations when we wish to apply this operation to our work.

Of course, this mapping may be more complex than the simple charts I've shown so far. But simplifying assumptions can be made in our applications, as long as we preserve the desired relational properties (I.e., that applying said operation N times reproduces the original theme, if we're talking about rotation by 360/N°).

It seems Ning and Soundcloud don't like to cooperate in discussion comments, so, a workaround. As odious as the object may be, it took considerable time and effort. After months of toil, the monstrosity emerged: The Dream (or Nightmare, if you wish), realized