Music

Nonstandard Contrapuntal Imitation

Imitation in music is when one voice announces a melody or phrase and another voice repeats it. There are lots of ways to “do” imitation. Imitation can be strict, or loose, and it can involve some kind of transformation, like slowing down the material, or speeding it up, or turning it upside down, or playing it backwards. The transformations that I’ve just mentioned all have names — like augmentation, diminution, inversion, and retrograde — and they’ve been used by composers for long enough that they can be considered “standard.” To be sure, their possibilities are boundless, but the concepts themselves have been known to composers for hundreds of years and can be found in any basic counterpoint text. One of the things I’m always looking out for in my own composing efforts is whether there might be uncommon, or perhaps unexplored kinds of imitation that could lead to new musical possibilities. In this post, I want to summarize the non-standard varieties of imitations that I’ve worked with in my canons so far. I don’t know whether any of these are new in the sense that no one’s done them before, but I can say they were all new to me when they first came to mind.

Imitation with Interval Expansion or Contraction

The idea here is that follower expands or contracts the melodic intervals of the leader; not occasionally, but repeatedly and systematically. For example, the leader might employ the whole-tone scale, where every interval has an even number of semitones, and the follower might cut all the leader’s intervals in half. I wrote about this concept in my post on Interval Compression and explored it in Canon 72 “Rhyolite” and Canon 73 “Tellurium.”

Imitation with Insertion or Deletion

The idea here is that follower periodically inserts a new measure that’s not found in the leader’s material — the follower might be extra “talkative.” Alternatively, the follower might skip or ignore one of the leader’s measures — the follower might be “forgetful.” In this way, the lag between leader and follower can change throughout the piece, even though the tempo of the parts is not changing. The important point here is that the insertion or deletion is not a one-time or exceptional occurrence, but a device that is used regularly and systematically throughout the piece. I did this in Canon 67 “Feldspar” and the two pieces that came before it, Canon 65 “Galaxite” and Canon 66 “Rhodonite.”

Imitation with Reordering

The idea here is that follower is allowed to reorder the leader’s material in some way, whether it’s individual measures or entire phrases that get moved around. I wrote about this in a previous post on Reordering Canons and explored it in Canon 94 “Cinnabar.”

Imitation in One Dimension

The idea here is that the follower is required to strictly imitate one dimension or aspect of the leader’s material while being free to alter another aspect. (The freedom to alter another aspect can also be turned into a requirement to do so.) For example, the follower might preserve the directions of the leader’s melodic intervals while freely varying their sizes: if the leader plays A and then goes up to an E, the follower would imitate this by moving upward, but perhaps landing at a D, or perhaps an F. The composer might impose a requirement that the follower must not play the exact same interval as the leader, so E becomes off limits as a landing point if the follower is also starting on A; still, the landing point must be higher than the follower’s initial note, not lower. Another variant of this idea is to say that the follower must copy the leader’s rhythms exactly, but is free to vary (or is required to vary) the leader’s melodic material. In Canon 88 “Carminite,” the follower preserves the leader’s rhythms and melodic directions while varying the leader’s interval sizes in an unpredictable way.

Monophonic Imitation

This idea is not about a transformation that the follower applies to the leader’s material; rather, it’s about the way the two voices are positioned in relationship to each other, and how the result is perceived. If the rhythmic gaps in the two voices are positioned appropriately, it may be possible to superimpose one voice on another so that they never have a shared hit. Furthermore, the voices may be placed in the same melodic range and played using the same timbre, so that it becomes impossible to distinguish them. In this case, there are still two logical voices, and one voice may still be seen on the page as imitating the other, but the listener hears a single line. In this way, a polyphonic process can give rise to a monophonic outcome. You can hear this at the beginnings and ends of Canon 92 “Ammolite” and Canon 93 “Meteorite”. In both cases, the imitation is retrograde.

Imitation with Embellishment

In “traditional” counterpoint, there’s nothing unusual about the idea that one voice might ornament or embellish a phrase as part of the imitation process. But the way we could extend this concept into “nonstandard” territory is if we make the embellishments so numerous or so significant that the phrase seems to become something new altogether. The listener might be prompted to ask whether one voice is really imitating the other, or loosely interpreting the other, or creating new material that’s inspired by the other. Another possibility is that the leader and follower could present two differing developments of the same primary line, the same common ancestor. Perhaps the two lines have a similar contour and hit the same melodic targets, but have different details, different connective tissue. In Canon 42, “Amethyst” I began to explore this idea by writing a bare canon and then repeating it in a way where the two lines were embellished in very different ways.

I’ll follow up with a separate post about non-standard canonic constraints — ways of restricting what’s considered “permissible” in a particular canon — ways that lead to interesting musical outcomes, but ways that don’t involve the style of imitation per se.

Music

Canon #94, Cinnabar

I’m pleased to announce Canon #94, Cinnabar, written this past December and completed on Jan 1, 2022.

When I started working on Cinnabar, my aim was to explore the idea of a reordering canon. I wrote an outline and spent many days developing it, only to find that I had arrived at a dead end. My efforts at development — so diligent, so persistent — had only made the outline worse! This creative disappointment transpired in a span of several days when I was already in low spirits, for no particular reason, just generally feeling crumby. In an effort to lift my own mood, I went back to an earlier version of my outline and thought “Let me try doing something silly with this.” The idea was to decorate my carefully-planned outline in a totally preposterous way, entirely for my own amusement, writing nonsensical themes and absurdly discordant counterpoint with no plan anymore to turn it into a finished piece. But once I got going, the “silliness” began to acquire its own internal grammar, and the composition process turned quite serious again, but now it was unstuck.

I chose the name Cinnabar for two reasons. First of all, the interval palette and melodic style of the piece reminds me somewhat of my Canon 74, Ruby, and both substances are bright red. Second, I wanted a name that contained some element of humor. The names of gems and minerals aren’t all that funny in general, but cinnabar stood out to me because when I say cinnabar I think of “cinnamon bar” or “cinnamon bun” or “Cinnabon,” and this thought leaves me mildly amused, what can I say?

Let’s talk about the structure of the piece. The soprano plays a series of six-measure phrases, with some space left at the end of each phrase. We can label those phrases as 1, 2, 3, 4, 5, up to 14. The bass plays the same phrases as the soprano, but in a different order. That’s what I mean when I call Cinnabar a “reordering canon.” In this piece, the bass order happens to be 3, 1, 5, 2, 7, 4, 9, 6, 11, 8, 13, 10, X, 12. (“X” represents a bass-only phrase that’s not stated by the soprano). When the two parts are executed simultaneously, we hear a series of phrasal pairings or contrapuntal episodes separated by brief pauses. At the beginning, we hear phrase 1 over phrase 3; next, we hear phrase 2 over phrase 1; then phrase 3 over phrase 5, and so on. For the most part, each phrase is restated two episodes after we first hear it, which is to say the restatements are close together but usually not consecutive. Some phrases are heard for the first time in the soprano, while others are heard for the first time in the bass (i.e. if we think of the bass as reordering the soprano’s phrases, it might place some phrases in an earlier position). The piece ends with a mirror inversion of the opening episode, where we hear phrase 3’s inversion over phrase 1’s inversion.

Most phrases undergo a chromatic transposition when they are restated; so phrase 1 begins on a C# when we first hear it in the soprano, but it gains a pickup gesture leading to an F# when we hear it in the bass. One of the challenges of writing the piece was finding the right transpositions for each episode (I say episode here because each phrase has a planned intervalic relationship with its companion phrase, so the two must be transposed together). The goal is to maintain a sense of variety throughout. Since the episodes are tonally ambiguous and the piece has no specific key, it is possible to transpose the episodes arbitrarily, but experimentation reveals that the options are not equivalent. Care is taken to avoid starting too often on the same note or in the same range, and also to avoid having the phrases start or end too squarely on the beat, and also to avoid reusing the same pickup and ending gestures, so that each episode begins and ends in its own distinctive way.

The relationship between the two voices in this piece is what I might describe as “choreographed chaos.” The voices may sound at times like they are contradicting each other, or tripping over each other, or going off in their own directions obliviously, but they still give each other enough space that their own individual features can shine through the discord. Whatever “destructive interference” occurs between the parts is meant to be occasional and not constant — that’s to say, the clamor is full of deliberately positioned gaps that allow us to peek inside and see each protagonist clearly. As we hear the parts interact, we might think “This can’t go on much longer — it’s unsustainable!” But they always reach a discernible cadence or stopping point after each six-measure span, as if they had been rescued just in time — saved by the bell. These evenly timed cadences are one giveaway that the seemingly chaotic relationship between the parts has been carefully planned.

A dissonant interval palette is used throughout, and great attention is paid to maintaining this palette, so that if a prominent consonance were to occur, it might sound “wrong” in contrast to all of the staged clashes and pratfalls that come to sound “right.” Tonal centers are never maintained for long, but they are allowed to develop just long enough that we can feel pulled towards certain notes and can experience others as unexpected or out of place.

The penultimate passage is marked lacrimoso and may be the closest I’ve come in any of my canons to writing a passage that sounds vaguely Chopinesque. The sorrowful ethos of this one passage is an unanticipated break from the scherzando quality of the piece, and an intensifier of the return to scherzando in the final passage.

I mentioned that Cinnabar reminds me of Canon 74, Ruby, but Ruby has a severe demeanor, not a comical one; the resemblance is in the musical materials used, not the overall tone. If I try to think of another canon that has a comical element, #49, Gallium comes to mind. In that piece, the two voices start out with a harmonious, tonal relationship, then progress towards increasing discord and tonal ambiguity, and then suddenly spring back to clarity and alignment. I think that many of my other canons have bits of humor here and there, if you know where to look — but the current #94 and the earlier #49 are two I’ve written so far that actually strive to maintain it across the span of the piece.

Music

Reordering Canon

My upcoming composition, Canon 94, has an unusual structure. I call it a “reordering canon.” There are two ways to frame this concept. To start out, we need to subdivide our melodic material somehow. We might focus on individual measures, or sections containing some fixed number of measures, or phrases occupying a variable number of measures. Once we’ve chosen a way to subdivide our melodic material, we can describe a reordering canon in either of the following ways:

  1. It’s a canon that has a leader and a follower, like any typical canon has, except the follower is allowed to rearrange the leader’s material — playing the leader’s melodic subdivisions (measures, or sections, or phrases) in a new order. In this case, the leader’s order is considered to be the original or primary one. We would label the leader’s melodic subdivisions as 1, 2, 3, 4, etc. The follower’s order would then appear as a scrambling of that sequence, like 3, 1, 4, 2.
  2. It’s a canon where we dispense with the roles of leader and follower. Instead we begin with an unordered set of melodic fragments. Each of the voices is responsible for playing some permutation of the set. That’s to say, each voice must play all of the fragments in the set, but each voice may play them in its own unique order. So one voice might play 4, 1, 3, 2 while another plays 3, 4, 2 1.

The only difference between these two descriptions is that in the first case, we assume that the material has an “original” order (maybe we are working with a pre-written tune). This lets us identify a leader that plays the original tune and a follower that rearranges it. In the second case, the material does not come with any specific order so it’s not meaningful to assign the roles of leader and follower.

To see what possibilities this idea opens up, and what new challenges it creates, let’s first review how a standard canon works. In a standard canon, the leader and follower play the same material in the same order, but the follower begins after a delay. We hear the follower constantly echoing the leader, always a few measures behind, never catching up.

Here’s a diagram of a standard canon consisting of four sections. I’m using the term “section” to refer to some number of measures — the same number of measures in the canon’s lag or delay. In this case, the bottom line is the leader; the top is the follower.

Look at the arrows. The blue arrows show transitions: section 1 flows into section 2, which flows into section 3, and so on. The red arrows show contrapuntal pairings: section 1 is heard above section 2, section 2 is heard above section 3, and so on. Together, the arrows represent the constraints that make canons interesting, and that make canons hard to design.

Examining the arrows in this diagram, we can see that the inner sections (all but the first and last) have four distinct responsibilities:

  1. Each section has to make sense coming after the previous section — it functions as a successor.
  2. Each section has to make sense coming before the next section — it functions as a predecessor.
  3. Each section has to sound good below the previous section — it functions as a bass line that accompanies its predecessor.
  4. Each section has to sound good above the next section — it functions as a soprano line that accompanies its successor.

If you understand this, you understand the basic challenge of canon writing.

But not all canons work like this.

It might seem that an echo — the unmistakable experience of hearing the follower repeat what the leader announced a few moments earlier — is the defining aspect of a canon, but in fact the term “canon” has long included pieces that have no discernible echo. In a retrograde canon, the two lines may start at the same time, with one line playing a reversed or backwards version of the other line’s material. Here’s how that might look:

As we listen to such a piece, we don’t hear a follower echoing the material that the leader had played a moment ago. Instead we hear two contrapuntal parts that may seem to be doing totally different things. In the diagram above, we see that soprano begins with section 1 while the bass begins at the same time, with a reversed version of section 4. We have to wait till the end of the piece to hear section 1 restated in the bass, but it occurs there in a backwards form, which may sound nothing like the forwards form.

If retrograde canons lack an echo, why do we call them canons at all? We call them canons because one voice can still be seen as imitating the other voice. Imitation is key. Imitation is what makes it a canon. The follower’s material is wholly and systematically derived from the leader’s material. It’s just that the imitation here is a complex kind — the imitation involves a transformation, namely reversal.

If we’re going to allow “imitation” to include a transformation as extreme as reversal, and still call the piece a canon, we might ask what other kinds of imitation could be employed to good musical effect. Along with reversal, the standard ones are inversion (turning the material upside down), augmentation or diminution (playing the material faster or slower), and of course transposition (playing the material higher or lower). Those are the ones you’ll find in a textbook, but what else is possible?

I’ve been pursuing this question in many of my compositional efforts. In a separate post, I will summarize the other nonstandard forms of imitation that I’ve explored. Here, let’s delve deeper into the reordering concept, where the follower must imitate each of the leader’s sections, but the follower is free to change their order. A reordering canon, as I’m calling it, can be visualized like this:

As you can see, where the top line plays sections 1, 2, 3, 4, the bottom line scrambles that order and plays 3, 1, 4, 2. Both parts start at the same time, similar to what happens in a retrograde canon, but this diagram does not show a retrograde canon. Even if the bottom line played 4, 3, 2, 1, it still wouldn’t be a retrograde canon. That’s because a retrograde canon does more than reverse the order of the sections; it reverses all the notes inside each section. We’re not doing that here. The contents of each section are still being played in their regular, forward-moving direction. It’s just that the order of the sections has changed.

Does a reordering canon have an echo? Yes it does, but it’s a more complex kind of echo than we hear in a standard canon. In a standard canon, the echo comes from one part: the follower echoes the leader. In a reordering canon, the echo comes from both parts. That’s to say, some of the sections will be announced first by the bass and imitated later by the soprano; while other sections will be announced first by the soprano and imitated later by the bass. And the delay — the length of time between a section’s initial statement and the onset of its echo — may vary from section to section. In the diagram above, we can see that section 1 is imitated by the bass immediately after it is stated by the soprano; section 3, on the other hand, is only imitated by the soprano after a delay of one section has passed since the bass first announces it.

Is a reordering canon easier or harder to write than a standard canon? In one sense, there are more constraints at play. In a standard canon, any section has only one horizontal context: for example, section 3 would always occur between sections 2 and 4, whether we’re considering the top line or the bottom line. But in a reordering canon, each section has two horizontal contexts. In the diagram above, we see that section 3 occurs between 2 and 4 in the top line; but in the bottom line it serves an opening role and then leads into section 1. Furthermore, each section still serves in two vertical capacities: in this case, section 3 is the bass that we hear below section 1 at the beginning of the piece, and the soprano that we hear above section 4 later on.

By the looks of it, you might think that a reordering canon should be harder to construct than a standard canon, because there are more constraints at play. But there’s one difference that allows a great deal of freedom in how a reordering canon can be composed. In a reordering canon, each section still serves in two vertical roles, but no longer has to serve as the bass to its predecessor specifically and the soprano to its successor specifically; instead, it might be paired with distant sections. For example Section 3 might not have to sound good below section 2 and above section 4; instead, its companions might be Sections 22 and 36! This opens up a lot of compositional options that just aren’t available in a standard canon.

And while it might be challenging to write a sequence of sections that can be played in two different orders (e.g. 1, 2, 3, 4 and 3, 1, 4, 2) there are ways make it work. In my first exploration of reordering canons, I’ve found it helpful to work at the phrase level rather than the measure level. In Canon 94, each section is a self-contained six-measure phrase, there are pauses between the phrases, and I allow for any phrase to be freely vertically transposed. It’s still challenging to write a set of phrases that can sound good in two different orders, but I feel there’s more leeway in how phrases can be rearranged than in how the fragments or gestures inside a phrase can be rearranged. Of course, that depends on the style, phrase structure, and musical materials in use.

We might observe that the traditional contrapuntal operations like retrograde and inversion produce one specific “output” for any given “input” whereas reordering gives the composer many permutations to choose from. Is this too much freedom to still call reordering a kind of imitation, and the resulting piece a kind of canon? The whole reason I’m interested in the reordering concept is because it’s a way to introduce more freedom into canon writing, but I don’t personally think it’s so much freedom that the piece ceases to be a canon. As we’ve seen, there are still many constraints to work with, and one line is still responsible for repeating the other line’s material. If we compare reordering with retrograde imitation, we see that retrograde imitation can completely change the sound and character of a line, including each of its sections, while reordering preserves the sound of each section. Arguably, reordering is less of a drastic transformation that note-by-note reversal, even though there are more ways to do it.