Here is an updated version of Canon #18, Petrified Wood, a piece I first wrote in January 2015. I’ve added a repeat and a new ending.Continue reading
Here is a new version of Canon #22, Abalone, a piece I first wrote in February 2015.Continue reading
My latest canon takes inspiration from birdsong. It’s my first canon that refers to a sound-world outside itself, and at over six minutes, it’s also my longest canon so far. It’s able to be long because it’s not a single continuous canon but rather a montage of over twenty-five canonic episodes, each one attempting to capture something that I’ve heard in the universe of bird vocalizations. Each episode is a strict inversion canon, where the follower moves in the opposite direction from leader, and most episodes are what I might call “rapid-fire” canons where the follower enters very quickly after the leader. The piece doesn’t have an plot per se, but among the canons I’ve written, it’s the one that might be the most suggestive or accommodating of a plot. It was particularly fun to write, and I’m particularly excited to share it:Continue reading
Announcing Canon #90, “Thorite.”
This piece is a sequel to Canon #89, “Thulite” — I chose the name “Thorite” simply because it’s another mineral beginning with “th.”
Like its predecessor, this piece uses the octatonic or diminished scale. The articulation is staccato, and the voices almost never coincide. Each voice speaks in the pauses and gaps left by the other voice. Sometimes these gaps are between phrases (creating a sense that the voices are “taking turns”) and other times they’re within a phrase (creating a sense that the voices are speaking over one another).
This framework (octatonic scale, staccato articulation, no shared hits) allows for a good amount of freedom in composition. I found that I didn’t need to micromanage the intervallic relationships between voices in the same way that’s necessary with other canons. There’s one exception. I tried to keep the voices from playing the same pitch in direct succession, because the sound of an octave (even if staggered) is distinct enough that the listener might expect it to have some special meaning; if no special meaning is found, the experience is slightly disappointing. Other than avoiding octaves, I could pretty much let the voices flow according to their own whims without having to plan their vertical relationship on a note-by-note basis. This meant that I felt comfortable writing the piece from beginning to end without going through my typical iterative outlining process.
One thing I loved about writing this piece is the way it all grew from the material in the first twelve bars. In writing those first twelve bars, I kept facing the challenge of inventing new material — pulling something out of thin air, so to speak — but once those few bars had been written, the rest of the work was about restating and recombining the opening material — finding new meaning in it — all the way to the end.
On a technical front, the leader is in the bass most (but not all) of the time, and there’s a one-bar lag between leader and follower. The piece is notated in 4/4. It stays mostly in the octatonic scale containing C#-D but occasionally visits the other two octatonic scales (the one containing B-C and the one containing C-C#). It starts with imitation at the major sixth above, but the interval of imitation shifts around throughout the piece.
A rough outline of the piece might go like this. First there’s about ten bars of “conversation” where various key phrases are introduced. The conversation gives way to a passage that I call the “drunken zigzag.” Here, the voices ascend in an up-down-up-down fashion that is highly patterned, but is made to sound chaotic through a jagged, unpredictable rhythm. After the drunken zigzag, there’s a bit more conversation that then leads into the first “stretto,” a passage where the same theme is layered on top of itself multiple times. It’s not exactly akin to a fugal stretto and one could argue that stretto isn’t the right word to use here, but it’s the best one I’ve got at the moment so I’m using it. Next we have a bit more conversation with echoes of themes from earlier conversations, and then we have a restatement of the drunken zigzag, this time with some doubling and a different interval of imitation between the voices. This second drunken zigzag then leads into a new stretto that’s based on the themes from the first two bars of the piece, rearranged so a phrase of the form AB now becomes BA. After this middle stretto we hear more conversation (again, with echoes of the previous ones) and this leads into another drunken zigzag that’s special because it’s at half the speed (e.g. the rhythmic values are doubled). Next there’s some connective material that has the purpose of launching us into the third and final stretto, which uses the same rhythmic structure as the first stretto, but now the theme is descending instead of ascending. This third stretto visits all three transpositions of the octatonic scale, and leads us back to the C#-D scale where the piece began. The piece concludes with a repeated statement of a now-familiar phrase, first introduced in the second bar of the piece, with the voices finally, conspicuously imitating each other at the octave.
I’ve tried to blog about each canon I write, but there’s always much more to say about the piece than I can hope to write down. Reaching my ninetieth canon feels like something of a milestone so I’m planning to try something new. I’ll make a video where I walk through the piece and try to share how it works. Stay tuned for that.
Announcing Canon #89, “Thulite.”
Many of my canons pursue an orderly relationship between leader and follower. As much as I want a canon to be interesting and provocative – a goal which sometimes results in musical complexity – I still want the canon’s structure to be clear. It’s not sufficient that the follower echoes the leader; this echoing must also be apparent to the ear. There’s a certain classic sound of contrapuntal imitation that I’m usually going for, regardless of whether the musical language is antique or modern. The leader announces a melody and we hear the follower reproduce it in a way that seems preordained, inexorable, “right” – like the voice of God. The leader sets an expectation and the follower fulfills it.
In writing my latest canon, “Thulite,” I wanted to try for a different aesthetic. I wondered if I could write a canon where the follower didn’t seem to be imitating the leader – where the two voices actually seemed to be antagonizing or contradicting each other – even if in fact, the follower was copying the leader exactly. Perhaps if the leader behaved in a more erratic, jagged, and unpredictable way than I usually seek, the follower too would sound unpredictable, as if it were doing its own thing altogether, even if that “thing” had been heard once before. Perhaps if the voices occasionally became entangled, the leader/follower relationship itself would be obscured.
To achieve this goal of a canon that doesn’t sound canonic, I put aside my typical writing process where I start with a skeletal outline and add detail in a series of iterations. Instead I composed the piece “on the fly,” one measure after another, without knowing anything about what might come next. The piece became a series of brief episodes, some of them having the jerky, erratic quality that I had been going for, and others having more of the classic imitative sound. The piece goes back and forth between chaotic and structured. But even in the latter case, the voices still seem to tug against each other with accents that don’t line up nicely.
The piece reminds me of jerky, sudden, shifting movements of a cat-and-mouse game.
When I sat down to write, I intended to generate new material from beginning to end, but then I found a way to reuse some of what I had already written. One section from the early in the piece is repeated near the end with the same rhythmic structure but different melodic contours. And then, at the conclusion, we hear the material from the very beginning of the piece repeated in a different “key.” I put the word “key” in quotes because the piece is not conventionally tonal. Most of the material is in the B-C half-whole diminished scale (or octatonic scale). We hear enough material in this scale that its sound becomes familiar – a kind of tonal comfort zone – even though there is no specific center. In the middle of the piece, the leader remains in the established B-C half-whole diminished scale while the follower shakes things up, transposing material to the two other half-whole diminished scales (the A-sharp-B one and the C-D-flat one). As the piece approaches the conclusion it settles back into the pure B-C diminished scale, but then, in a surprise, the final passage is stated with both voices in the foreign C-D-flat diminished scale.
Importantly, the voices never play on the same beat, though they come as close as a 32nd-note away. I wanted the rhythms to seem whimsical and less ordered than my typical style, but I tried to achieve this without anything fancy like changing meters, complex polyrhythms, metric modulation, or the like. The score is notated in 4/4 and looks quite conventional. The imitation is strict: the follower copies the leaders intervals and rhythms exactly.
Some of the challenges in this piece were 1) keeping a sense of variety within the selected tonal material of the half-whole diminished scale, 2) maintaining a sense of gestural discord between the voices and avoiding the classic sound of imitation, 3) disrupting the beat while maintaining enough of sense of meter that the music can be heard as tugging against something, 4) creating episode-to-episode transitions that sound natural on the one hand but unexpected on the other.
My typical writing process usually leaves me with a result that seems final and unchangeable, but Thulite, written in an atypical way for me, opened so many possibilities, and could have gone in so many directions, that I feel the piece is only one of many possible manifestations of the intentions at play. Usually it feels right to naming a piece after a gemstone or mineral, but with Thulite, for the first time, I wondered if I should change my naming scheme and use a more descriptive title. To keep things simple, I decided to stick with stones for now. My next steps will be to write a few more canons that explore the stylistic ideas that Thulite brought into focus for me.
How is it possible that all of music… uh… make that all of Western music… uh… make that much of Western music is formed from only twelve notes? How is it possible that some of the greatest music takes flight with fewer than twelve notes — only seven, or only five? This is a question that often occurs to people taking their very first steps in learning about music theory. If you knew absolutely nothing about the technical side of music, but you only knew of the infinite variety of musical experience – the way music can make you feel ecstatic and depressed and every shade in between, the way music can keep you entertained for hours on end – you might be shocked to learn that all of these diverse and brilliant riches are constructed from at most twelve elements. It seems unbelievable that you’d never get bored of those same twelve elements repeated over and over. But then you get used to the idea. As you learn more about music theory, you may feel that you understand how it all works, but the question “Only twelve?” might still cross your mind from time to time. I myself was thinking about it the other day and imagining how I might respond to a beginning student who insisted that there simply must be more than twelve notes. I imagined several responses I’d give the student, and I’ll record them here.
The first response is a cheeky one: asking more questions. So you’re surprised that there are only twelve notes? Then how is it possible that everything we taste is built from five basic flavor sensations: bitter, salty, sweet, sour, and umami? How is it possible that everything we see is based on three primary colors: red, blue, yellow? How is it possible that all of Shakespeare, no, all of English literature – everything we’ve ever read or written, and everything we will ever read or write – is formed from 26 letters and some punctuation marks? How is it possible that all of the content we’ve ever seen on the Internet – every video, every news article, every comment, every social media post – is represented using only two elements: ones and zeroes?
The second response is to dispute the premise. Yes, Western music has twelve named note but each named note may occur in many different ranges. The note A0 at the bottom of the piano’s range is a very different sonic experience from A7 near the top of its range, but they are both called A. A piano actually has 88 keys, not 12, and each of those 88 sounds different. If someone endeavors to sing the note A at 440hz, their voice may wobble slightly, or they may deliberately employ vibrato, in which case we hear a range of frequencies hovering around the 440hz mark. And in fact, a piano can be tuned in different ways – equal temperament is only one option. The use of twelve named notes is a simplification that conceals a much wider variety of sonic material that music can and does exploit. Music has cymbal crashes and washboard scratches and whispered words and other sounds that don’t have a specific pitch. And there is some Western microtonal music that uses 19 named notes, or 22, or 48….
A third response is to point out that twelve notes actually give rise to a very wide variety of permutations and combinations, which each have their own distinctive qualities. Lets say I want to make a sound combining four of the twelve named notes, and I want to pick one note to be the lowest, another (possibly the same) note to go above it, a third note to go above that, and finally a fourth note to go on top. I can do this in 12^4 = 20736 different ways. Now what if I want to create sequences of note combinations? The possibilities explode.
A fourth response is that notes can be delivered in an infinite variety of ways. A note can be loud or soft. It can be long or short. It can be played by a piccolo or a tuba or a guitar… or an entire ensemble. You can attack a note directly or you can slide up to it, or down to it. On each instrument there are countless articulations. There may be only one named note A, but that A can take countless forms.
A fifth response, perhaps the most interesting to me, is that notes can give different meanings to each other and can renew themselves in our perception. A sequence of notes can put a listener in a certain state of mind, and that state of mind then determines how the listener hears further notes in the sequence. If I play the note C a few times, it may come to sound familiar to you. If I play a C chord, followed by a G7 chord, followed by a C chord again, then the note C will sound like “home.” Even though you may have heard the F# chord millions of times in your life before, the F# chord would sound strange and unexpected if you heard it at this particular moment. Your perception of a note or a chord is not governed as much by your history of hearing it over your lifetime as it is by the context that’s been created by the notes and chords you’ve heard just moments ago. If I had played a different chord sequence – F# followed by C#7 and then F# again – it would be the note F# that you’d experience as “home” while the note C and its chord would sound alien and unexpected.
Music uses gesture and pattern to make certain notes and chords sound familiar while others sound foreign. Once a perceptual frame is established, music can shift it around, making the now-foreign sound familiar and the now-familiar sound foreign. In this way, you can listen to a four-hour concert that uses only twelve named notes, never getting bored with any of those twelve, because the context in which you’re hearing those notes – and thus the meaning those notes acquire – is constantly changing. A C that you hear at the beginning of the concert may not sound the same to you, may not mean the same thing to you, as a C that you hear in the middle or at the end. That C and in fact all of the twelve named notes are only vessels that assume different meanings and affects according to the infinite variety of contexts that they create for each other and the infinite variety of moods or perceptual frames that they put us in. Music uses notes to give meaning to other notes, and as the music continues, the meanings change. While the meaning of C can change over the course of a concert, it can also change from one moment to the next: C might serve as the root or anchor of the chord you’re hearing now — it might be a stable note in this instant — but it might become the tension-giving seventh of the chord you’ll hear next.
A sixth response is that we don’t only hear notes — and the melodies and harmonies they create — when we listen to music. We hear rhythm too, of course. (Another question: How is it possible that most of the rhythms in Western music are made from whole notes, half notes, quarter notes, sixteenth notes, thirty-second notes and their dotted and triplet varieties?) Beyond that, we hear texture. We hear performers, we hear composers, we hear the people the music is about, or the people it’s for; we hear something of the history of when it was written or performed; we hear the memory of when we heard it first or heard it last; we hear the way we feel right now as it relates to the sounds that confront us. We may be something of a different person each time we listen, and listen again, to the same piece. There’s that.
What are all the possible ways to voice a seventh chord?
This post is a quick note on that question, with a focus on how a guitarist might think of things. I’m posting this so I can refer to it later.Continue reading
Announcing Canon #88, “Carminite.”
The essence of a canon is imitation: the follower copies the leader. In some canons, imitation is exact: the follower plays the same notes that the leader played moments before. In other canons, imitation involves some kind of transformation. There might be a chromatic or diatonic transposition. There might be an inversion, where an upward movement is imitated as a downward movement and vice versa. There might be a stretching or shrinking of note durations. There might even be a reversal or retrograde statement of the material. In all of these cases, imitation is deterministic: once the leader plays a note, it’s clear what note the follower must play – there’s no choice. Of course, during the composition process, liberties may be taken. The composer might decide that even though the follower is supposed to play a certain note to stay faithful to the leader’s example, the required note doesn’t sound good and a different note will be chosen. These exceptions to the imitation scheme are precisely that: exceptions and not the norm. When I write canons I often agonize over these exceptions: is it better to keep the canon strict or to make alterations that diminish the formal purity of the piece while making it more musically engaging?
For a long time I’ve wanted to write a canon with a different concept of imitation: one that would be approximate or flexible instead of prescriptive or fully determined. There are lots of ways you could flesh out what “approximate imitation” means. The idea I had in mind was that interval direction would be preserved while interval size could be changed arbitrarily; meanwhile, rhythm would be preserved exactly. That’s to say, if the leader plays two quarter notes that move up by a major third, the follower would have to play two quarter notes as well, and they would have to move up as well, but they could move up by a minor second or a perfect fifth or any other interval.
Writing a piece like this, I hoped, would help me address a question I’ve been thinking about for years: what makes a canon sound like a canon? What formal elements are essential to creating the experience that we seek when we listen to a canon? Is the strictness of the imitation important, or is it dispensable? Is rhythmic imitation more important, or is the replication of melodic contour more important? To shed any light on this, a piece employing approximate imitation would have to be more than a canon that takes liberties here and there — many canons already do this. It would have to be a canon where the leader’s intervals are very frequently changed by the follower, not in a predictable way, like always cutting them in half (see Interval Compression), but in a free and unpredictable way.
One might expect that the idea of approximate imitation would make a canon much easier to write: more flexibility, fewer constraints. So, after months of attempting it, I was perplexed at my inability to get such a canon working. The project, in fact, seemed much harder than writing a strict canon. Why should this be so? The reason is what I alluded to above: approximate imitation introduces a new burden. To really showcase this concept, you have make the follower regularly – not just occasionally – vary the leader’s intervals. So it’s like you’re writing two melodies at once. You do all the work to make the leader’s line sound good, and then, instead of having the follower copy the leader’s already-good-sounding line, you have to make the follower do something different, but still similar, and this different-but-similar thing also has to sound good. That’s hard.
In Canon 88, I’ve finally been able to create an example of approximate imitation. What made it possible this time around? I’m not sure there was any particular trick, although I did make the early decision that this would be a dissonant canon without a clear tonal center. I spent more than a week working on possible outlines, and then another week trying to develop them: two false starts before something finally came together. The false starts were interesting in their own way. I think what went wrong with both is that my melodic developments on top of the non-tonal outline started to become too tonally suggestive. I was beginning to turn a non-tonal skeleton into a tonal piece, but this was a lost cause. Each possible tonal focus was hinted but not maintained long enough to give any satisfaction. As one possible tonal center was abandoned and another suggested, it created the disappointing, sloppy variety of chaos, rather than that very different variety I seek: the one that is intriguing, potent, and structured. In the final attempt, the one that worked, I managed to create lines that were less tonally binding, so they played better with the non-tonal outline, not constantly working up an expectation and then disappointing it.
This canon also explores a V-shaped registral contour I had been wanting to execute in a canon for some time. Both voices start very high and gradually descend until they’re both very low. At the midpoint, they turn around and begin ascending till they return to their high starting range.
In the first section, the lower voice leads. At the midpoint, the upper voice takes over as leader.
The two sections of the piece have different characters: the first is more whimsical and skittish; the second is more patterned and goal-oriented.
The second part of the piece is built from a retrograde or reversed version of the outline I used for the first part. So, the bones of the piece exhibit a kind of mirror structure around the midpoint, but the details in each half are completely different.
As the canon uses approximate imitation, of course, there’s no specific “interval of imitation” used here.
Subjectively, the piece brings to my mind a kind of cartoon dance or chase: fast, erratic, and maybe a bit comical.
The choice of name, “Carminite,” is not particularly significant in this case, though I liked its sound and I noted that Carminite is a red mineral. Sometimes pieces bring to mind certain colors. Red seemed like a good match in this case but to try to explain why, I’d have to be more inventive than I’m feeling right now.
So, does this piece answer any of my questions about the nature of canons? It leads me closer to the conclusion that strictness is non-essential. Approximate imitation sounds, or can be made to sound, very much like classic, strict imitation. Perhaps proving this was my unstated intention when writing the piece. While I set out to showcase approximate imitation as something distinct from strict imitation, my impulse was to reign it in and apply it in a very controlled way. The ear can be guided into ignoring countless discrepancies between leader and follower when more important similarities call for its attention. It’s possible that as I wrote the piece, I tended to put the discrepancies in places that didn’t draw focus.
What also may have happened is that the more glaring discrepancies in my outline got smoothed over by later melodic elaboration. For example, the outline might have had a minor third leap in one voice imitated as, say, a tritone leap in the other — a difference that would be easily noticed. But these leaps might have been filled in as a run of three semitones in one voice versus a run of three whole steps in the other voice. Of course, a semitone run is different from a whole-tone run, but if the ear is more interested in the destination, these paths may sound quite similar: they are both sequences of “small” steps. So while there are many discrepancies between leader and follower situated all throughout the piece, they’re not as glaring as they could have been made to be.
For me, the experience of listening to Canon 88 is much the same as what I get when I listen to my previous canons, barring aesthetic differences from piece to piece. It doesn’t sound like an outlier to me. But this experiment yields no final or general conclusion about approximate imitation because the idea can be realized in so many different ways; only one of them is explored here. It’s perhaps a tame application of the concept, and yet it gave rise to what I feel is one of my freer-sounding pieces. At the moment, it’s one of my favorites. My goal now is to try using approximate imitation again, in a way that makes the discrepancies even more apparent.
Announcing Canon #87 — “Barite.”
Work on this piece started in a typical way for me. I spent a week exploring various ideas from my canon to-do list, but I couldn’t get any of my sketches to take flight. I kept working through Memorial Day weekend; still nothing. It was hard not to think the long weekend would have been better spent on something else entirely – maybe I should have given up and tried again later? – but I know that every day that seems fruitless is an investment in what’s to come. In a sense, you can’t get something done unless you’re willing to accept the feeling that you’re getting nothing done, and keep going anyway. Finally on Monday evening, I noticed a simple technical option that I hadn’t yet explored in any of my canons. I’ve written a few canons in 3/4 time where the lag is one beat, but I hadn’t written a canon in 3/4 where the lag is two beats (leader starts on the first beat, follower starts on the third beat). Why not? Although Max Reger isn’t my model for canon writing, I notice he used this construct in a good number of his 111 Kanons durch alle Dur und Molltonarten. Ready for something new to work on, I abandoned the other sketches I had been struggling with and started a canon in 3/4 with a two-beat lag.
As with most of my canons, the first step is to create an outline, not one that I like, but one that I love. Why is this step so important? After all, the quality of the outline doesn’t necessarily dictate the quality of the finished piece. It’s totally possible to transform a lackluster outline into a great piece because as you’re working, you can revise the outline or simply throw it away when it stops serving you. The problem for me is that when I don’t start with an outline I absolutely love, it’s hard to find the motivation to keep struggling to reveal its potential. If I do love the outline, then that love propels me: I feel an overwhelming resolve to do whatever it takes to transform the outline into a full piece of music. So while I could probably start with cursory outlines that take a few minutes to throw together, and maybe I’d produce more pieces that way, I’m more inclined to spend hours or days creating an outline that totally captivates me, because once I’m hooked, I’ll never abandon the piece even when the going is rough. You could say I have a kind of perfectionism about my outlines, but I take the view that perfectionism itself isn’t evil: one just needs to be realistic about what one chooses to be a perfectionist about. Outlines are the good things to be a perfectionist about because they’re simple enough that you actually can make them perfect.
So I started making an outline for Canon 87, and managed to get something I loved. During the outlining stage, I don’t really know what style the piece is going to land in. My outline for Canon 87 was full of unprepared and unresolved dissonances, suggesting it would take on a modern style, but the melodic material was firmly tonal and full of diatonic sequences, and the implied harmonies all seemed to fall within the realm of “common practice.” As I developed the piece, this duality persisted: in a melodic or “horizontal” sense, the piece started sounding like something from the 18th century but in a “vertical” sense it seemed much more modern. Towards the end of the piece, the interval palette becomes more consonant, with more thirds and sixths on measure onsets; the sound is less conflicted in spirit and style. I considered revising the latter part of the piece to keep the style more consistent with the dissonant opening, but I decided instead to embrace the piece’s progression from a dissonant to a more consonant palette, and from a severe to a lighter mood.
To bring the piece to a satisfying conclusion, I knew I’d have to break out of the canon and write some free counterpoint. I was ready for ending to be a struggle as it often is. But then I came upon the idea of having the voices move mostly in parallel at the end (after all, they had established their independence by now, right? What more did they have to prove?). I brought them closer together and had them converge into a unison at the final beat, and that worked.
A few details: the piece uses diatonic imitation at the fourth above. It opens in D minor, progresses to G minor, moves back to D minor, and finally progresses to B-flat major. The imitation is fairly strict, but the bottom line takes various ornaments that the top doesn’t repeat. I chose the name Barite because, for whatever reason, the piece brought the color yellow to mind, and Barite is a mineral that can look yellow when cut as a gemstone (all of the other more familiar yellow gemstone names are taken by now). Unlike many of my pieces, Canon 87 has only one section and doesn’t go through an inversion. (It’s probably possible to get this material to work in an inverted form, but it would take some rewriting, and although I always wish my pieces were just a little bit longer, I think this one reaches a natural stopping point and doesn’t call for an extension.) The piece is based on the simplest of melodic figures: on almost every measure onset, in the bass, you can hear a note, followed by its lower neighbor, and then the note again. I like working with simple figures such as this — I like seeing how much they can do.
Announcing Canon #86 — “Tiger’s Eye.”
I named this piece “Tiger’s Eye” because it makes me imagine a tiger prowling through different environments, some sunny and plush, some barren, some dangerous, some tranquil. The piece creates a sense of shifting terrains – for my ear, at least – even though it consists of the same six bars of core material, repeated over and over (with variations).
In Tiger’s Eye I was finally able to apply some technical ideas I had been wanting to explore for some time. The first idea is to have the leader play with a pronounced staccato articulation, while the follow plays the same material legato. I had experimented with this in earlier pieces but it never seemed to work: my melodies always seemed to demand one articulation or the other, not accepting both. Here, the situation is different: the piece only sounds good with contrasting articulations. The bass has to be played staccato or the piece starts sounding too muddy, and the upper part has to be played legato otherwise the piece starts sounding too choppy. When I started work on the piece, both parts were legato and I wasn’t happy with what I heard, thinking the material itself was a dead end. It’s quite possible I would never have continued writing this piece if I hadn’t tried making the bass staccato on something of a whim; once I did that, I immediately heard some magic happening and I felt an uncontrollable urge to develop it.
The second idea I was able to try here is to freely double the lines. The doubling happens most often at the fourth, fifth, and octave, and occasionally at the third. This gives the piece a much fuller sonority than any of my previous canons. In my earlier pieces, the option of doubling the parts at the octave always seemed superfluous, and doubling at any other interval seemed to wreak havoc on the harmonic design of the piece. It works here though, because the core material is so simple melodically, and because the rhythm is structured so that the hits are staggered between the parts.
Note that the doubling in this piece does not follow strict imitation: for example, in one passage the leader might be doubled at the fifth while the follower might be doubled at the fourth. This freedom is also allowed for transpositions that happen after each six-bar cycle of core material: when the next cycle begins the leader may enter higher or lower, as it wants, and the follower too might enter higher or lower, as it wants (or rather as I wanted when writing the piece).
The piece consists of an opening section and a repeat where some melodic variations are introduced, registers are shifted for variety, and the leader and follower are occasionally allowed to stray from each other (with the follower exploring some ideas that were not previously stated by the leader). Each section begins with the bass leading one bar ahead of the follower, but towards the end, the follower splits into two separate parts, playing a “stretto” with a two-beat lag, while the leader continues as usual. After this stretto, the roles are reversed, with the top part leading and the bass following.
The foundation of the piece is a rhythmic cycle consisting of 6 bars in 7/8. What’s special about this cycle is that, when played in a canon with a one bar lag, the parts never hit simultaneously except at the onset of each measure. It’s not quite a “rhythmic tiling canon” like the ones I explored in Escher’s Drum where the parts only come together at the beginning of each full cycle, but it’s similar.
To create this particular rhythmic cycle, I started by looking at all the ways I could fill the space of a 7/8 measure with three quarter notes and an eight note. There are only four possibilities depending on where the eight note falls with respect to the quarters. After listing these out, I looked at the complimentary rhythms for each possibility: another pattern of four beats that can be played simultaneously, such that the only shared hit is at the measure onset. This image shows the initial possibilities on the upper staff and the complementary rhythms on the lower staff.
I noticed that the first and last patterns are complements of each other. I chose to put those aside and build a cycle consisting of the items in the middle, the one I’ve labeled A above, followed by its complement A′, and then B followed by its complement B′. To separate each section I added some “glue” material that acts as a phrase end. Here’s how the complete cycle looks:
Played raw, without any interpretation, this rhythmic cycle might not seem immediately arresting. So the next challenge was to take the cycle and make some music from it, repeating it over and over, finding out what kind of melodies it can accommodate and what kind of variations it can accept. How to manifest its potential? How to make it sound good? Canon 86 was my answer.