A video summarizing the themes from Canons 1 through 38 (single part only):
A video illustrating Canon 2:
A video illustrating Canon 33 (note that since this is a crab canon, the video starts rewinding after it reaches its midpoint):
A video illustrating "Escher's Drum":
A video illustrating the component rhythms from "Escher's Drum":
A video illustrating Fifteen Beats:
A playlist consisting of many canons from all throughout music history:
---Q&A with Rudi---
Q: What is a canon?
A: You could say it's a piece of music built on the idea of an echo. In a two-part canon you have one part that leads and another part that lags a little bit behind, echoing everything the leader does.
Q: Isn't that how the children's tune Row, Row, Row Your Boat is often sung?
A: Yes, Row Row Row Your Boat is a simple canon. You could start singing Row, Row, Row Your Boat at one moment, and I could start singing the same tune from the top, a few moments later, entering right when you get to the word “gently.” In that case, you're the leader (or “dux” in Latin) and I'm the follower (or “comes”).
Q: Can any tune be sung that way?
A: Not really. Try this kind of staggered, overlapping presentation with a tune like Happy Birthday and it would sound horrible – our voices would clash – but if we do it with Row, Row, Row Your Boat we harmonize because the tune is constructed to fit on top of itself. We could do it with Frère Jacques, but it probably wouldn't work with Itsy Bitsy Spider. Of course, canons can range in style and scope from children's pieces to the marvels of Bach's work in The Art of the Fugue and The Goldberg Variations.
Q: What were your goals in composing a whole album of canons?
A: I love listening to canons and I wanted to write music that I'd enjoy hearing.
Q: What makes canons interesting to hear?
A: When you listen to a canon you're always hearing something new mixed with something old – the next idea or gesture is superimposed on the idea you just heard a moment ago. The old material may sound familiar when it resurfaces, but it may also sound different in its new context, with its new accompaniment. When we listen to any kind of music we tend to remember what we heard in the preceding moments, and that memory still rings in our mind as we absorb the current sounds. You could say all listening is canonic, in that there's always some kind of echo transpiring in our mind's ear. But in a canon, the echo becomes explicit, and we get to hear a running fusion of the present with the immediate past.
Q: That sounds intriguing, but in reality aren't canons sometimes confusing to hear?
A: They demand active listening. It helps to be aware that you're listening to a canon – not just any kind of piece – and to try following the relationship between the parts. Sometimes that's easier to do once you've listened to the piece several times and become familiar with the theme. I find some music, even very good music, gets boring if you listen too often, while other music only grows in interest the more you listen. Canons often fall in that second category – they can sometimes be hard to approach, but they are also hard to exhaust.
Q: The canon is an old form, tracing back to the Middle Ages – the famous piece 'Sumer Is Icumen In' comes to mind – would you say the canon is outdated?
A: As a form, the canon is really quite simple: you've got a leader and a follower – that's it. There's nothing like the structural complexity of the classical sonata where you have sections – introduction, exposition, recapitulation – each with its own characteristics and substructures. The simplicity of the canon concept makes it timeless, in my opinion, and that's borne out by the fact that we have canons by Machaut, canons by Stravinsky. And canons have lots of technical parameters that can be varied – the lag between the parts, the interval that separates them vertically, the style of counterpoint used, the manner of imitation – creating an endless variety of challenges and possible solutions.
Q: You said the canon is simple as a form – does that mean canons are easy to write?
A: No, I mean the concept of a canon is simple to describe, but the execution can be excruciatingly difficult, because the follower places so many restrictions on what the leader can do. Unlike fugues where imitative or canonic writing can be interspersed with free counterpoint, a canon has to sustain its own “echo” from beginning to end. Whatever the leader does in one section must be crafted so it will blend with the previous section that's being repeated by the follower, and that requirement never lets up.
Q: Why bother with the restrictions of the canon form? If you can incorporate short imitative passages into freer pieces, why confine yourself to writing in canonic fashion from start to finish?
A: From a composer's perspective, the challenge of a tighter, more restrictive form can be inspiring – to discover how much is possible under the severest constraints. From a listener's perspective, the more restrictive form lets you relax in a sense, because you know exactly what you're getting – when the form is clear, you can focus on the content. And it can be exciting to hear how the composer pulls it off; even if you haven't tried writing a canon yourself, you might still be aware that you're witnessing a difficult and uncommon feat, something like a puzzle being stated and cleverly solved in front of you.
Q: So would you say canons are all about technique and cleverness?
A: No. Writing a canon sometimes feels like working through a puzzle, but I think a good canon must transcend its puzzle aspect and communicate something as a piece of music. And that's why I'm not entirely opposed to “cheating” in some cases.
Q: What do you mean by cheating? Isn't the canon a strict form?
A: There are many canons where the leader and follower indeed play identical music. Strict canons are great from the standpoint of a music copyist: you don't need to write the tune out twice, you can just write it once and indicate when the players should enter. But sometimes in writing a canon you find you've explored all the “legal” options – perhaps you've arrived at something satisfactory – but you realize you can make the music come alive if you allow a few subtle details to vary between the leader and the follower: perhaps one voice plays a certain note sharp while the other plays it natural, perhaps one voice uses an ornament while the other remains plain. There's also a choice to be made at the end: you can let each voice finish on its own schedule, or you can bring them to a unified cadence, in which case you often have to stop the canonic imitation and write some free counterpoint for the conclusion.
Q: I understand the point about freedom at the cadence, but what's so difficult in canon writing that it would push you to bend the rules in the middle of the piece?
A: Maybe the crux of the challenge is that solid bass lines don't always make the most interesting melodies, and interesting melodies are sometimes not harmonically forceful enough to make strong bass lines, but anything you write in a canon must function both as a bass line and as a melody line. Sustaining a sense of harmonic clarity and forward motion along with melodic interest can be very challenging. In freer forms of music, you can move from one harmony to another by having the bass line “announce” the change while the upper parts flesh it out. But in a canon, if the bass line is the leader, any gesture it makes is going to be parroted later in the top voice. This creates situations where the harmonic movement sounds redundant, or else you're locked into one harmony for longer than you want to be there. Most of the time you can work around this by relying on musical double meanings – on the fact that notes, even entire gestures, have multiple interpretations and can belong to more than one harmony. But in some cases you can get great musical value by allowing one or two subtle alterations in the line depending on whether it's serving in a bass or melodic capacity.
Q: How often do you break the rules in the album?
A: Roughly half of the pieces are strict and the other half include some exceptions here and there. In the pieces where I bend the rules, I always try to do it subtly – you want to stay true enough to the canon form that the listener can maintain their bearings, can be confident that their effort in “matching” the lines in their ear will not be frustrated by lots of superfluous deviations. The ideal exception is one that makes the music sound better without even being apparent as an exception.
Q: If rule-bending is musically advantageous in some places, why did you keep any of the canons strict? Why not bend the rules in every place where doing so might be fruitful?
A: One motivator was laziness, actually. You spend thirty hours working on a canon and in the end you're left with 30 seconds of music – I was always looking for ways to get a little more length out of these pieces, and when you make a canon “invertible” you can basically double its length with no extra work. But for a canon to be easily invertible you've got stay away from alterations that need to be adjusted after you perform the inversion – strictness is your friend in this case.
Q: Let's back up – what's an invertible canon?
A: It's a canon where the positions of the voices can be swapped. That's to say, if the bass voice is the leader and the soprano is the follower, you can repeat the canon with soprano as the leader and the bass as the follower and it still sounds good. This shouldn't be confused with writing a “canon in inversion” which means that the follower doesn't directly copy the leader, but instead presents an upside-down version of what the leader does.
Q: But what makes some canons invertible – where the leader and follower can be swapped – while others canons don't allow for this?
A: To talk about inversion means we're assuming a systematic treatment of dissonance and consonance where certain intervals are considered acceptable on strong beats while other intervals must be avoided or properly prepared and resolved. In many, but not all of the canons in the album I follow a traditional approach that treats a perfect fourth between the bottom and top voice as a dissonance while the perfect fifth is a consonance. When you take the bottom voice and transpose it up by some number of octaves to the top position, you find that perfect fifths become perfect fourths, meaning that an interval that was legal on strong beats has now become illegal. However, if you avoid perfect fifths on strong beats and allow only thirds, sixths, and octaves, you never run into this problem. Thirds become sixths through the process of inversion, and vice versa; and octaves remain as octaves. You write the canon once following these special restrictions, and then almost as if by magic, you've got a second section of the piece that arises by swapping the parts – it's clearly related to the original canon, and it's “legal,” but it sounds different enough to engage the ear as new material.
Q: You said you used traditional contrapuntal practices in many places, but not all – where did you break with tradition, so to speak?
A: In some cases I used melodic resources that would be considered modern – like the whole tone scale (#6) and the octatonic or diminished scale (#19) – while still adhering to a traditional treatment of dissonance. Canons #15 and #31 use lots of chromatic motion on top of a consonant whole-tone framework. In Canons #14 and #21 I flipped the rules, emphasizing dissonances instead of consonances on strong beats. In Canon #32 I experimented with an alternate tuning system, dividing the octave into 11 instead of 12 pieces. That said, I try to not think of musical resources “traditional” and “modern” since it's not the most telling distinction; sometimes “modern” music can sound stale and “traditional” music can sound fresh, and it's really freshness that matters.
Q: Some of the canons are labeled as being “at the fifth” or “at the third” – what does that mean?
A: The most common kind of canon is where the follower enters an octave above or below the leader, playing the same notes as the leader, but transposed up or down in register. There are also canons where the follower might enter at another interval, like a third, or a fifth. In these cases, when the interval of imitation is anything other than an octave multiple, the composer has a choice to make. If the follower is to stay in the same key as the leader, the follower will need to adjust some of the leader's melodic intervals (for example, in C major with imitation at the third, the interval of a major third from C to E would be imitated as a minor third from E to G). If the follower is to proceed in its own key, then it may copy the leader's intervals exactly (a major third from C to E could be imitated as a major third from E to G#). I wrote some canons of the first kind, where the leader and follower remain in the same key, including Canon #3 and Canon #22 at the third, Canon #5 at the fifth, Canon #8 at the second, and Canon #21 at the fourth, and Canon #43 at unison. I also wrote a set of six bitonal canons, #23 through #29, where the follower strictly imitates the leader in a different key: each canon in that group is invertible, and together they explore all 12 possible key distances between leader and follower.
Q: How did you go about composing those bitonal canons?
A: You have a choice of whether to emphasize or conceal the clash between keys; I pursued both objectives in different ways. One the hand, I tried to write lines with clear cadences and a very strong sense of key. When such a firmly tonal line is played in canon, there's an inevitable tug-of-war between the two key centers. But I wanted this contest to be graceful instead of messy, so I used only consonances on strong beats, crafting the lines so they would mesh smoothly from the standpoint of vertical sonority. Even when the keys are as far apart as C major and F# major, it's always possible to build a consonant framework using intervals that are enharmonically equivalent to thirds and sixths. The combination of each line pulling towards its own key center while at the same time blending smoothly with the other line makes the six bitonal pieces my favorite works in the album.
Q: What other special types of canons did you explore in the album?
A: In a canon in inversion (also called a canon in contrary motion), the follower plays an upside down version of the leader's melody (#12, #13, #37). In a retrograde canon, also called a crab canon or a canon cancrizans, the follower starts at the same time as the leader and plays leader's line backwards; if the canon is invertible, the entire thing may be “rewound” when it reaches its end (#31, #33). In a prolation canon (also called a mensuration canon, or a canon by augmentation and diminution) the follower copies the leader at a different speed (#35). A spiral canon ends higher or lower than it started and can be repeated in a continuously rising or descending fashion (#39). An accompanied canon includes one or more lines – often a bass line – that move freely and don't participate in the canonic imitation (#36). A variation canon allows one line to be ornamented and developed differently from the other (#42). Many canons are invertible at the octave but less commonly a canon may be invertible at the tenth (#40) or the twelfth (#41). A rhythmic tiling canon is a composition in which each player repeats the same pattern, with each player beginning at a different time. The pattern and entry points are crafted so that once all the players have begun to play, there will be exactly one player striking every beat. That’s to say, every beat is covered by one of the players but no two players ever coincide on the same beat. The last two pieces in the album, “Escher's Drum” and “Fifteen Beats” are explorations of this rhythmic concept.
Q: Do you have a tune in mind when you start writing a canon?
A: Usually not. The theme emerges in many little steps as I write. In fact, many of the melodic ideas in the album would probably never have occurred to me if I had tried to “write a tune” from scratch. The tune only comes together through the continuous cobbling and tinkering that the canon form requires. It's good not to have too many preconceptions about the theme when you start because you'll inevitably have to change and adjust your ideas again and again to get them to fit in the canon form.
Q: Do you have a specific process for writing canons?
A: Well, one of the most perilous approaches is to write a really interesting, florid melody lasting a couple of bars, then copy it to the other voice and try to write a counterpoint above it, then copy and repeat. That process can easily lead to canons that drift. Measure by measure they sound good but on a larger scale they fall flat, and they're hard to revise since you've already elaborated each line and committed to so many little details up front.
Q: So how do you proceed, if not linearly?
A: The most valuable practice I follow is to start with an outline, then elaborate it in stages. The initial outline is usually in whole notes, like a first-species counterpoint exercise. I spend a lot of time on the outline, just listening to really simple arrangements of whole notes – I might spend hours and hours on that. I try to apply the “beer test,” which is to ask whether you'd want to hear that particular whole-note sequence while kicking back with a beer. Is that simple sequence of notes compelling enough that you'd listen to it on your own time, when you're not trying to “write a canon,” but when you're just relaxing and looking for a bit of enjoyment? Once you have a good sketch, only then do you begin developing it.
Q: What happens as you develop the sketch?
A: You try various ways of connecting pillars in the outline, adding passing tones, runs, suspensions, experimenting with different rhythms, and so on. You listen to each line separately – you sing it over and over – and then you play the lines together. You try adding detail, then stripping it away, adding, then stripping again. This process is nothing new; in fact, it's akin to what Thomas Morley described in his Plaine and Easie Introduction to Practicall Musicke from 1597, where he shows how a “divided” passage may be developed from a “plaine” outline.
Q: How do you know when you've got it right?
A: I'm always looking for a balance of clarity and interest. Sometimes you can add detail in a way that makes one line more beautiful on its own, but the detail blots out or overwhelms the other line: one part of the canon becomes more “interesting” but the whole becomes more “confusing.” I'm always looking for those details that add to the beauty of each line and also improve the balance and separation between lines. I know a canon is done when it holds my interest from start to finish (there are no “dead” spaces where my attention fades) and when I can hear the lines clearly (there are no “blurry” spots where the parts seem to interfere with each other) and when I can sing one line on its own and enjoy it as a tune.
Q: Do you try to keep the lines equally active at all times, or do let one line recede as the other comes to the fore?
A: For the most part, I strive to maintain simultaneous interest in the parts, so that the ear's focal points are not prescribed, but rather the ear will discover something engaging wherever it chooses to focus. I don't think there should often be a single best or most rewarding place to put your attention: good counterpoint is immersive – something to bathe in. That said, a piece has to breathe – phrases must begin and end, not continue indefinitely, and there should be occasions when one line gives way to the other.
Q: How long does it take to finish a canon?
A: Some of them come together in a day or so; many of them take several days. Usually I'll spend a day on an outline, then another day or two on elaboration, then another day on corrections and refinements. A couple of the pieces took considerably longer, like the rhythmic tiling canons at the end of the album: each of those was a multi-week effort.
Q: Why do some of the canons use lots of ornaments like trills?
A: In some cases I used ornaments just because I liked how they sounded, but in other cases I had an additional motivation. I think ornaments can serve as “markers” that help you follow the canon. When you hear an ornament in one line, it stands out, it calls your attention. Then when you hear that ornament come up again in the second line, you recognize a correspondence between those two points in time.
Q: In a few of the canons, like #1 and #11b, you used different instruments for each part. Why didn't you do this more often? Why didn't you always vary the instruments to help the listener hear the parts separately?
A: It's true that when two timbres are very different – as are flute and harpsichord, for example – it becomes possible to “eavesdrop” on the music at any time, even if you haven't followed it closely since the beginning, and quickly discern which voice is which. But my aim was to write lines that would stand apart on their own, without requiring the aid of timbral differentiation.
Q: But even the most well-written canon can be challenging for the listener to parse. Why not give the listener a little extra help?
A: In fact I did try varying the instruments in many cases and found that it opened a can of worms. Yes, I do think it's possible to use multiple instruments in more places in album than I did, but the prospect is not as straightforward as it seems.
Q: What challenges did you encounter in trying to use multiple instruments?
A: A line that's written with one sonority in mind might not sound good when the sonority is changed. You've got to find two instruments that each work well for the given melody, and that combine in a way that preserves the clarity and resonance of the intervals between the parts. I find certain intervals lose their substance when the timbres are too far apart: consider the ringing sound of perfect fifth announced by two trumpets, or even by two mandolins, versus the same interval announced by a trumpet and a mandolin together. The "mixed fifth" will still sound like a fifth but it might not "ring" in the same way when spread across such different voices. I want to be able to hear the intervallic relationships between parts as clearly as possible – as if the two parts were two gears directly interlocking – and not as if they were slippery wheels, loosely touching. Yet another consideration is that when the timbres are too distinct I find the process of identifying analogous gestures is slowed down. I can still recognize that a melodic fragment played by the flute is identical to one played moments ago by the harpsichord, but the identity may be less apparent or may take a moment longer to perceive. And so I have sometimes found myself listening to canons with disparate timbres and having an easier time separating the voices but a harder time perceiving their relationships moment-by-moment.
Q: What kind of canon did you find to be the hardest to write?
A: Canons with special restrictions like crab canons and canons in contrary motion present special problems: you have to write lines that sound good in reverse, or upside down, and that's not a very “natural” thing to do. There's usually a way to pull it off, though, and the reduced set of options means that these canons sometimes seem to write themselves. For me, the hardest form turned out to be the prolation canon, where one voice goes at a different speed than the other. As such a canon progresses, the lines don't maintain a fixed relationship – the distance between the two parts is constantly narrowing or expanding. It makes the whole thing harder to work with. When I worked on Canon #35 I assigned each note a number and wrote the numbers in the two staves as a way of keeping track of the changing leader/follower relationship. And then there's a situation where the two lines might come together at one point and begin to diverge, and it's nearly impossible to avoid parallel octaves at that point.
Q: What is the easiest kind of canon to write?
A: I wouldn't say it's “easy,” but the accompanied canon introduces flexibility that doesn't exist otherwise: the free bass line can be used to clarify the harmony and diversify the texture.
Q: How did you first learn about canons?
A: I first got interested in canons when I was a teenager reading Douglas Hofstadter's book, Gödel, Escher, Bach; if not for Hofstadter's inspiration I don't know if I'd be writing canons today. I was also inspired to undertake a lifelong study of counterpoint by a composer I studied with at that same formative time, Stephen Siegel.
Q: What canons by other composers do you like listening to the most?
A: My favorite canons are Contrapuntus XII through XV from Bach's Art of the Fugue. Those are the works I think of when I think “greatest examples of the canon form.” But there have been a number of pieces I've gravitated to over the years without considering at first that they were canons. I had probably listened to Bach's Goldberg Variations a few hundred times, letting the music wash over me, before really focusing on the nine canonic variations and trying to follow their structure as such. And I fell in love with Ockeghem's Missa Prolationem before I understood what a prolation canon is, or how Ockeghem's piece is constructed entirely from such canons. On a lighter note, I should say life would incomplete without Henry Purcell's boisterous rounds on “secular” themes.
Q: The canons in the album have nicknames referring to gems and minerals – how did those names come about?
A: I wanted to make the album easier to navigate. When I got past 25 canons or so I couldn't really remember which number belonged with which canon. I thought about using bugs as nicknames but I felt bugs had a bit too much personality to impose on these largely abstract pieces; gems and minerals seemed more neutral. I like the image of sound vibrating the stones or reflecting off the stones. In some cases there's a very loose connection between the nicknames I chose and the content of the pieces (darker stones for some of the darker or more mysterious sounding pieces) but elsewhere it's arbitrary.
Q: How is the audio rendered?
A: The current album includes digital renderings that I made with MIDI playback software and sound libraries. Although there's no live performer here, I feel the digital renderings represent the music well enough that some joy can be had in hearing them. That said, my goal is to create a sequel to this album with live performances of these pieces, so I'm looking out for performers to collaborate with.
Here are the most basic details about each canon including the interval relationships between the voices and any special techniques used, along with some personal comments about the composition process. To indicate interval relationships, I write a phrase like “at the octave above; at the third below,” which means that in the first section of the piece, the follower enters an octave above the leader, and in the second section of the piece, the follower enters a third below the leader. Intervals are listed in simple form, though in many cases they are actually compound (I might write “at the fifth above” when really the follower is a fifth plus an octave above the leader). Note that the canon numbers do not correspond directly to the track numbers in the album.
Canon 1 - “Quartz”
at the octave above
The piece opens with a simple motive and an overall sense of stasis; it then follows a path of gradually increasing tension. The core material ends an octave higher than it begins and is stated three times in ascending ranges, reaching towards the highest notes on the instrument. The tension built at the end of one statement gives way to repose at the beginning of the next statement. Successive statements include some small rhythmic decorations.
Canon 2 - “Emerald”
at the octave below
The core of the canon is a set of four simple but jaunty phrases, each with a distinct rhythmic pattern. The four-phrase set is itself stated four times, passing through multiple modes, in such a way that the beginning of one statement overlaps the end of the previous statement.
Canon 3 - “Pearl”
at the third below
Somewhat similar in structure to “Emerald.” A brief but sturdy theme and its follower are repeated eight times, in an overlapping fashion, passing through a cycle of keys that begins and ends in C major. Some variety is introduced by altering the last note of the theme so that the ending is heard either as an upward-moving or a downward-moving gesture.
Canon 4 – “Topaz”
at the octave below
The heavily ornamented theme and its follower are repeated three times in progressively lower ranges. The theme consists of two phrases, one with fast descending scalar runs and a slower descending chromatic sequence, and the other with a more even orientation. There is an echo of energetic Baroque style.
Canon 5 - “Coral”
at the fifth above
A gently rising line with a simple ternary rhythm. The canon is repeated three times in progressively higher ranges. A mood of warmth and gratitude.
Canon 6 - “Obsidian”
at the octave above
The theme is somber, spare, mysterious. Exclusive use of the whole-tone scale gives the piece a somewhat floating, anchorless quality, but the writing aims for the same strict approach to preparing and resolving dissonances that is employed in many of the canons here with more conventional tonalities. The canon material is restated four times with transpositions that keep it within one whole-tone set, except for the third statement which is transposed chromatically into the other whole-tone set, creating the highest point of tension in the piece.
Canon 7 - “Bloodstone”
at the octave above
The piece explores what's possible within the tight rhythmic constraint of uninterrupted quarter notes. The steadily marching theme includes several sections of repeated pitches that function like a pedal (and in the follower, an inverse pedal). The theme and its follower are repeated four times in sequence of different keys and ranges, creating a feel of shifting “terrains.” The overall mood is one of "searching."
Canon 8 - “Tourmaline”
at the second above
The theme is energetic but, heard on its own, it may come off as oversimple or repetitive; it gains interest here by interacting with its follower, which tracks it closely, always responding a step higher and keeping the dialogue in motion.
Canon 9 - “Jade”
at the octave above; at the octave below; at the octave above
A buoyant, playful theme with some chromatic inflections.
Canon 10 - “Sapphire”
at the octave above
A fabric of vocal phrases in Renaissance style.
Canon 11 and 11b - “Hematite”
at the octave above and below, with multiple switches in leader/follower position
The canon pursues an effect reminiscent of Escher's never-ending staircases where one might climb up and yet find oneself moving down. The canon motif consists of many small steps that suggest movement in one direction, while the motif actually progresses in the opposite direction over time because occasional large leaps in the opposite direction cancel out the frequent stepwise motion in the original direction. The canon was conceived around the whole tone scale with strong beats restricted to one whole-tone set and outlying notes used as passing tones. The ethos is busy, mathematical, and strange. Two versions are presented, a slow but brief original version and an extended version at a faster tempo.
Canon 12 - “Amber”
in contrary motion – at the second above; at the seventh below
This piece is a canon in contrary motion, which is to say that one line is more-or-less an upside-down version of the other. The mood is peaceful and contemplative.
Canon 13 - “Carnelian”
in contrary motion – at the octave above; at the octave below
Another canon in contrary motion – a companion piece to “Amber,” with a similar mood.
Canon 14 - “Fluorite”
at the octave above; at the octave below
The mood is mysterious, dark, perhaps with an aura of impending change. I envision an icy terrain at dusk. With fourths, seconds, and sevenths used liberally on strong beats, the sonority is largely dissonant – though the dissonance strives to be of the open, ringing variety rather than the closed, crowded, or murky variety.
Canon 15 - “Flint”
at the octave above; at the octave below
Similar to “Hematite” in technique as well as in ethos. Conceived on a whole-tone framework but employing the remaining pitches as passing tones. Multiple sections with different leader/follower relationships bleed into each other without a clear break. There is a sense of dizzy motion towards some unknown objective; with a constantly changing perspective it's unclear whether progress is taking place.
Canon 16 - “Moonstone”
at the octave above; at the octave below
Unlike the other canons based on the whole tone scale, this piece uses dissonances liberally on strong beats; it also avoids chromatic passing tones that are used in other canons to “smooth out” the whole-tone line. The theme is jagged but aiming for coherence; the mood is slightly restless.
Canon 17 - “Onyx”
at the major second above; at the minor seventh below
A playful theme with moments of seriousness. Bright, with some dark under the surface. The imitation is largely chromatic as opposed to diatonic; there is a hint of bitonality here.
Canon 18 - “Petrified Wood”
at the octave above
The theme begins with an earnest, probing quality, becoming more light-hearted towards the end, which is in free counterpoint. A Lydian modality emerges by the conclusion.
Canon 19 - “Jet”
at the octave above; at the octave below
This dark and pensive canon employs the octatonic or diminished scale, which consists of alternating half and whole steps. While I've always been drawn to the other symmetric scale mentioned in these notes – the whole-tone scale – I find the octatonic scale challenging to work with. I don't hear the same "built-in" character in the octatonic scale that I do in the whole tone scale; rather, it feels to me like a chameleon entity that can become many things depending on how it's used, and so the question in working with this scale is how to achieve a sense of consistency and coherence without falling back on undue repetition. My aim in this piece was not to use the resources of the octatonic scale to create jagged, wild lines, but rather to see if I could preserve a sense of melodic continuity and tonal groundedness, even while invoking all eight pitches periodically. The sonorities here are mostly restricted to thirds, sixths, and tritones; notes outside the octatonic collection are used occasionally in an unaccented, passing capacity. The canon begins with the leader in the bass; later, the material is repeated a minor third higher with the voices swapped. Instead of cadencing on the octave, each section concludes with the sonority of the major third, which, although it's used throughout the piece, only reveals its brightness when brought into focus in those ending positions.
Canon 20 - “Turquoise”
at the sixth above
Somewhat reminiscent of “Sapphire” – phrases inspired by Renaissance vocal style.
Canon 21 - “Mica”
at the fourth above
This canon uses dissonances (mostly seconds and sevenths) on strong beats except at phrase endings where thirds and other consonances appear. I find my ear gets used to the emphasized dissonances very quickly and they almost come to sound consonant in their own way. The voices are a fourth apart, with the exact interval switching between perfect and augmented. I hear the main modality here as Lydian (the bottom voice begins with a phrase in F Lydian and the top voice imitates it starting at B, but I don't hear the top voice as being in the separate mode, B Locrian; rather I hear it in the modal context established by the bottom voice). There's a mood of curiosity and playful exploration.
Canon 22 - “Abalone”
at the third above
This canon explores modal pairing and contrast. The leader begins in E Phyrgian and the follower proceeds a third higher in G Mixolydian. The material is then diatonically transposed down a third (with an upward octave displacement in the top voice), placing the leader in C Ionian and follower in E Phrygian. The material remains essentially the same after the transposition but the last passage is modified slightly to avoid tritone issues that appear in the second pair of modes. The mood is contemplative, with subtle differences in character between voices and between sections.
Canon 23 – Platinum
at the tritone above; at the tritone below
A strict bitonal canon at the tritone: the bottom voice leads in C major; the top voice follows in F# major, then they swap. The clash in tonalities made it challenging to work on the piece: many times when I tried to listen to the C major voice in isolation, I had to give my ear a long rest before I could really hear it, because the conflicting F# major tonality was still in mind, still preventing me from hearing C as a tonal center. However, when the two tonalities are brought together in the full composition they are prevented from colliding directly: the piece situates consonances – thirds, sixths, and octaves – on strong beats, so the bitter dissonances one might expect to hear in such a bitonal context are never realized. Instead of seeming to battle with each other, the lines appear as identical twins separated by glass – something keeps them apart, preventing direct communication, but they are nevertheless inextricably linked.
Canon 24 - “Gold”
at the minor third above; at the major sixth below
A strict bitonal canon at the minor third and major sixth: the bottom voice leads in C major; the top voice follows in Eb major, then they swap. Heard on its own, the theme is bright and joyful. The mixing of of distant tonalities has a somewhat muting or canceling effect on the palette, but the brightness of the theme is not easily hidden.
Canon 25 - “Silver”
at the minor second above; at the major seventh below
A strict bitonal canon at the minor second and major seventh: the bottom voice leads in A minor; top voice follows in A# minor, then they swap. One of the slower and more somber of the bitonal canons in the album.
Canon 26 - “Nickel”
at the major third above; at the minor sixth below
A strict bitonal canon at the major third and minor sixth: the bottom voice leads in C major and the top voice follows in E major, then they swap. A chasm between two instances of the cheerful, energetic theme.
Canon 27 - “Copper”
at the major second above; at the minor seventh below
A strict bitonal canon at the major second and minor seventh: the bottom voice leads in C major and top voice follows in D major; later, the top voice leads in D major and the bottom voice follows in E major. A gradual shift of moods, beginning with a more somber aspect but ending brighter.
Canon 28 - “Zinc”
at the perfect fifth above; at the perfect fourth below
A strict bitonal canon at the perfect fifth and perfect fourth: the bottom voice leads in C major and the top voice follows in G major, then they swap. During the writing process, I found it interesting that this pair of closely related keys was not significantly easier (or harder) to work with than some of the more distant combinations like the major-third related keys in “Nickel.” The mood is cheerful but somewhat calmer than “Gold” and “Nickel.” Again, the juxtaposition of tonalities has a somewhat muting effect on the bright, affirmative nature of the theme, but that brightness still shows through.
Canon 29 - “Sunstone”
at the octave above; at the octave below
This invertible canon is based on a very simple outline of "Do Re Mi Fa Sol Fa Mi Re Do." That pattern can be superimposed over itself with a skew of two beats to form a rudimentary canon with motion in parallel thirds and sixths. In this piece, the notes of "Do Re Mi Fa So Fa Mi Re Do" can be found by looking at the onset of each measure; however, there is so much intervening elaboration that the pattern is not necessarily obvious to the ear. There's a mood of optimism and good cheer.
Canon 30 - “Granite”
at the octave above; at the octave below
This invertible canon is based on the same skeleton as Canon 29, except in minor (“Do Re Me Fa Sol Fa Me Re Do”). The elaboration is quite different from the earlier piece, and the aspect is sparse and somber.
Canon 31 - “Zebra Marble”
I had been avoiding the task of writing a crab canon, where two voices enter together, with one voice playing a backwards version of the other voice. I'm interested in writing canons because I enjoy listening to them, and part of that enjoyment depends on an ability to follow a canon's structure as it is heard. But crab canons have always seemed to me a bit beyond the capacity of a mortal ear to follow. The idea of playing a musical line in reverse is simple to describe, but not very natural to do. Most ears have trouble discerning the relationship between a line and its backwards version as the two often sound radically different. But with “Zebra Marble” I felt I had finally managed to write a crab canon that drew me back as a listener. I took an "easy" route of basing this canon on the whole tone scale which lends itself to melodic constructions that are easy to reverse. This canon is invertible, so the top and bottom line can be swapped – when the canon is followed by its inverse, the music appears to be “rewinding” back to the beginning. Here the canon/inverse pairing is stated three times in rising half-steps. The canon is not entirely strict: the sequence of pitches in one line is indeed an exact reverse version of the sequence of pitches in the other line, but some notes have different rhythmic values in the two versions. The canon brings to mind the interacting lines in a geometric sketch – perhaps a sketch of heavenly bodies.
Canon 32 - “Chondrite”
at the octave above; at the octave below
This canon explores an alternate tuning system in which the octave is divided into 11 equally-spaced notes instead of 12. This system is often called 11EDO, where EDO stands for “equal divisions of the octave.” Here, 11EDO is expanded to 11EDO+ by adding an "extra" note at center of the octave. This extra note subdivides pitches 6 and 7 of the 11-part division to create a scale with 12 notes that is nevertheless very different from the ubiquitous 12EDO. In exploring an alternate tuning one can emphasize its unique character or search for similarities with 12EDO. Because semitones in 11EDO and 12EDO are are fairly close in width, sequences of semitones – that is, chromatic motion – sounds similar in the two systems. Chromatic motion is the focus of this invertible canon which aims for an effect that at once foreign and familiar. The skeleton from which the piece was developed is simply a rising chromatic line, stacked on itself.
Canon 33 - “Peridot”
This piece, like “Zebra Marble,” is an invertible crab canon where the music begins rewinding when it reaches its midpoint; but, unlike the earlier piece, “Peridot” is a strict crab canon where the rhythmic values of the notes in each line are preserved in the backwards version. A single liberty is taken: there's one note instance that's played sharp when the line is moving in one direction, but natural when the line is moving in the other direction. With its use of the whole-tone scale, “Zebra Marble” presents a somewhat exotic soundscape; in contrast, the aim in “Peridot” was to write in a simple, straightforward style – a little ditty that might never be suspected of being a crab canon unless one examined it closely. The mood is cheerful.
Canon 34 - “Opal”
at the octave above
While many of the canons here were developed from melodic skeletons, this piece began with a harmonic or chordal skeleton. The piece proceeds steadily through a cycle of ascending fourths. It starts in C major and follows the cycle almost all the way back to the starting key, but it cadences early at D major. Modulation can be difficult to achieve in canon writing; this canon employs relative minor chords to help achieve transitions between major chords a fourth apart. The leader outlines C major, and then moves to A minor while the follower imitates the leader in C major. The leader then moves to F major while the follower is outlining A minor, and so on. While modulation is a source of variety in music, a long sequence of modulations of the same kind can begin sounding repetitive; a challenge in writing this piece was to make sure something slightly different happens in each of the many visited keys. The mood is playful.
Canon 35 - “Lodestone”
A sequence of four short prolation canons at the octave. Section 1: The bottom voice leads; the top voice enters later at double speed; the voices end together. Section 2: The voices begin together, with the bottom voice at double speed. The bottom voice ends first, leaving the top voice to continue. Section 3: The bottom voice leads; the top voice enters later at double speed; the voices end together. Section 4: The bottom voice leads; the top voice enters later at 1.5 times the speed; the top voice catches up with the bottom voice and continues at 1.5 times the speed, ending before the bottom voice.
Canon 36 - “Diamond”
at the octave above
This piece, in a tranquil mood, explores the idea of an “accompanied” canon. The canon is first stated in two voices; then it is repeated with a third voice, a free bass accompaniment that does not participate in the canonic imitation. The canon was written together with the bass, and there are some some fourths between the upper voices that sound dissonant when the bass is stripped away; hence a few small adjustments were made for the two-voice version.
Canon 37 - “Graphite”
in strict contrary motion – at the octave above and below
In contrast to the other canons in contrary motion in this album, the theme here and its inverse are exact "mirror" images, meaning that intervals are not altered between the leader and follower: if the leader ascends a minor third, for example, the follower descends a minor third, not a major third. The canon consists of two short sections, each of which has two subsections of its own. In the first main section, the bottom voice leads with the ascending version of the theme and the top voice follows with the descending version; as soon as the follower reaches its last note, it begins ascending and becomes the leader. In the second main section, the top voice leads with the ascending version of the theme and the bottom voice follows with the descending version; again, when the follower reaches its last note it turns around and becomes the leader. The second section includes some ornamentation not present in the first. The mood is urgent but not troubled.
Canon 38 - “Lapus Lazuli”
A sequence of short, slow canons in Renaissance style passing through different modes and making frequent use of the technique called suspension or syncopated dissonance. Imitation is at the octave, but the leader/follower relationship changes throughout the piece. Free counterpoint is sometimes used near cadences. The suspension effect is expressed well through the sustained tones of the organ, but an alternate track rendered on harp is included as well. For some inexplicable reason, the canon brought an image of fireflies to mind as I listened to the final draft.
Canon 39 - “Alexandrite”
at the fifth above; at the octave above
Bach’s "Canon a 2, per tonos” from the Musical Offering is a spiral canon that can be repeated indefinitely. Since the canon modulates up a whole step (e.g. C minor to D minor), the music gets higher and higher with each repetition. I set out to explore this spiral concept in Canon 39, which is actually a pairing of two short canons: one that modulates up a half-step and another that modulates down a half-step. The upward-moving canon is repeated twelve times, ascending an entire octave, after which the downward-moving canon is repeated twelve times, returning to the starting point. The whole-step modulation in Bach’s “Canon a 2, per tonos” is so seamless that I can never quite tell where it’s happening, though I sense the music is gradually rising. In Canon 39 I tried to make the modulations as smooth as I could, but the constraints are quite different from the Bach canon: each section is only eight bars, and the semitone modulation that takes place in those eight bars involves a massive change in key signature. I find that in performing such a dramatic modulation in a short space, there’s always a moment of disorientation that’s bound to happen for the ear somewhere; my goal was to reduce the duration of that moment and quickly usher the ear into the new landscape so there would be some feeling of groundedness before the process repeats.
Canon 40 - “Citrine”
at the octave above; at the third below
It’s not like there are throngs of counterpoint groupies waiting in the wings, ready to cheer at the sound of a canon that’s invertible at the tenth. What does ‘invertible at the tenth’ even mean? It means that unlike the most common kind of invertible counterpoint where you can move the bottom voice up some number of octaves so it takes the top position, in this case you can move it up an octave plus a third (so an A in the bottom voice would become a C in the top voice, a C would become an E, and so on). For various technical reasons it’s hard to make this work. In fact it’s so hard and the rewards are so comparatively slim that one wonders if there’s any point. Still, I felt my life would be incomplete if I didn’t try it. Canon 40 might not sound all that different from the others, but if you closely compare the first section with the second (inverted) section you might notice how the relationships between the parts have changed in a special way: the clear open-sounding perfect consonances have turned into richer, fuller-sounding imperfect consonances and vice versa. Parallel thirds and sixths are carefully avoided because, in the inverted version, they would become undesirable parallel fifths and octaves. Since the proportion of perfect to imperfect consonances is roughly 50/50 in the first section, it remains 50/50 after they’re swapped in the second section, and there isn’t a huge change in overall sonority, but the differences in position do affect how the phrases come off.
Canon 41 - “Iolite”
at the octave above; at the fifth below
The theme begins with the grave and majestic gesture of descending octaves. Where Canon 41 is invertible at the tenth, Canon 41 is invertible at the twelfth. This means that the follower, which enters an octave above the leader, can be transposed down so it falls a fifth below the leader.
Canon 42 - “Amethyst”
at the octave above
This piece reveals something of how canons can be developed in stages. The first section represents an initial stopping point in the composition process: I worked it up to the point where it satisfied my ear as a piece, but I left it spare. The second half is an elaboration of the music from the first half. What's notable about the second half is that the top and bottom voices are ornamented separately, breaking the rule that they should closely imitate each other. The idea was to explore what might be possible if the voices could be allowed to go in their own directions mid-measure, while still following the same outline and hitting corresponding notes at the beginning of each measure. Because significant variation is allowed between the two voices, this could be called a "variation canon."
Canon 43 - “Moldavite”
Although lots of the children’s rounds we know, like “Row, Row, Row Your Boat” are canons at the unison, it can be challenging write unison canons of any complexity: at this interval, the parts are inclined to cross, and when they do cross it can be very hard for the listener to keep them separate unless the parts are sung or played in distinctive timbres. The composer can go for a sense of “structured chaos” where the voices meld into an inseparable tapestry, or try to find compositional mechanisms to keep them distinct. My approach, pursuing the latter objective, was to leave long gaps where the leader plays solo and the ear can latch onto the tune, and then to confine voice crossings to the ends of phrases. The canon is strict except that the follower stops two bars short of the end of each of the leader's phrases. The recording includes two presentations of the material, one starting at G and the other starting a fifth higher at D.
Canon 44 - “Malachite”
at the octave above and below
I have generally avoided the ostensible “sin” of parallel fifths and octaves in most of the pieces here, no matter whether the style is conventional or experimental. When I discovered that an early draft of innocent and placid-sounding Canon 36 (“Diamond”) was riddled with such transgressions, I began a flurry of corrections. In Canon 44, however, I decided to explore the beauty of gestures that may be considered taboo elsewhere. Inspired by Medieval contrapuntal technique, this piece treats perfect fourths, fifths, and octaves as consonant while all other intervals are handled as dissonant. Parallel fourths, fifths, and octaves are embraced, and it is considered permissible for the voices to leap – even by similar motion – into those consonances. These freedoms are counterbalanced by tight restrictions on the use of thirds and sixths, which come to prominence only at cadences. The result is a sound-world that's different from any other in the album. The piece is a sequence of three canons, each of which is followed by its inversion. The mood is solemn but exultant.
This piece is a percussion work that visits eight different rhythmic tiling canons. A rhythmic tiling canon is a composition in which each player repeats the same pattern, with each player beginning at a different time. The pattern and entry points are crafted so that once all the players have begun to play, there will be exactly one player striking every beat. That’s to say, every beat is covered by one of the players but no two players ever coincide on the same beat. This piece is a tapestry that interweaves all eight possible rhythmic tiling canons where the cycle consists of twelve beats and the entrances of the players are equally spaced. I first learned about these component canons in the paper “Asymmetric Rhythms and Tiling Canons” by Rachel W. Hall and Paul Klingsberg. My aim in “Escher's Drum” was to cycle through the eight possibilities in a seamless way, where there would be no gaps. My idea was that in progressing from a tiling canon based on rhythm A to one based on rhythm B, there should be a connecting passage where rhythm A is mixed with rhythm B. And so the current piece consists of the eight tiling canons themselves (where all three voices play the same rhythm) together with connecting passages where different voices play different rhythms. My aim was to arrange the canons so that, if canon B follows canon A, the mixture of B and A would sound interesting. In order to do this, I tested all possible mixtures of the eight rhythms to find the ones I liked best. There were more than (8 choose 2) = 28 pairs to consider, because each rhythm consists of twelve beats (three bars in 4/4 time), and when two rhythms are superimposed there can be a skew of 0, 1, or 2 bars.
At first I tried to structure the piece so that no rhythm would ever be revisited, but I abandoned this goal after finding certain rhythms to be “dead ends,” so to speak. That’s to say, I might progress from rhythm A to rhythm B and, in the transition period, might enjoy the mixture of A and B. But then, while playing rhythm B, I might not be able to find any rhythm C that hadn’t already been played, where I could transition from B to C with a good-sounding B/C mixture. In such cases I had to return from rhythm B to another rhythm previously visited, so that I could then step from that previous rhythm to a new one.
In some cases I tried to make transitions from one rhythm to another smoother by a process of subtraction. If moving from A to B, I might gradually strip notes away from A until it began sounding like it might be a stripped-down version of B. These subtracted rhythms notwithstanding, a notable aspect of this piece is that it is based entirely on eight rhythms — no more than eight. Another notable aspect is that throughout the piece, the three players never coincide on the same beat — neither when they are playing a canon nor when they are playing transition material between canons.
In working on this piece I was constantly reminded of rhythms I had heard before, rhythms from the diverse percussion traditions of Africa. It fascinated me that these eight tiling canons and their combinations can be derived on paper, through an abstract mathematical process, but they resemble rhythms that have emerged in folk traditions through a very different process of intuition and experiment.
This piece is a follow-up to Escher's Drum, my earlier exploration of rhythmic tiling canons. I had set out to surpass Escher's Drum! when I began working on Fifteen Beats, but I came away with an appreciation for the special chemistry that had transpired in the earlier piece. Fifteen Beats is more ambitious in some ways but less ambitious in others. It’s more ambitious in that it explores sixteen different rhythmic tiling canons built from rhythmic cycles of fifteen beats. In contrast, Escher's Drum! explored only eight different canons built from shorter twelve-beat cycles. For Escher's Drum I found the individual canons in a paper by Hall and Klingsberg, whereas in Fifteen Beats I found them in an enumeration by Harald Fripertinger. In the mathematical literature on tiling canons, rhythms are presented as a series of note onsets: for example, this is the pattern that Fifteen Beats begins with: . In that representation, 0 indicates a rest and 1 indicates a hit. From a composer’s standpoint, there’s a lot of work to do in turning  into something that sounds good. First of all, you’ve got to decide where the cycle should begin, and usually it will make more sense to the ear if it begins with a hit than with a sequence of rests. So our pattern would best be played as: . Next you’ve got to decide where the accents go — that takes lots of listening and experimentation — and in the end it’s a matter of taste. I chose to accent the fifth beat: . And then of course you’ve got to decide what sonorities to use and what tempo to play at. I’ve been working with two drums and a bell. When a rhythmic cycle is played by three voices in a canon, it can come out sounding very different depending on whether the bass drum enters first, then the treble drum, then the bell, or whether the bass drum enters, then the bell, then the treble drum. That’s another choice to be made. And finally of course, you have to decide how to arrange the individual canons in sequence to form a meaningful progression. That last point is where Fifteen Beats takes a less ambitious approach than Escher's Drum. In Escher's Drum, my aim was to create interludes between each of the canons, where the rhythm from the earlier canon would be mixed with the rhythm from the upcoming one. This required testing lots of rhythmic combinations to see which ones sounded the most interesting, and then arranging the canons in an order so that each canon’s rhythm would mix well with the one coming next. And to create smooth transitions I sometimes had to do more than just mix rhythms: in some cases I transformed one rhythm into another by subtracting notes or shifting the accent pattern. In Fifteen Beats, I decided to simply juxtapose the canons, so that each canon would be played for four cycles, and then the next canon would begin immediately, starting with the bass drum announcing the new rhythm without any remnants of the earlier rhythm playing in the other voices. There was still a lot of experimentation to be done in finding a pleasing order for the canons, but there were fewer restrictions to work with since the rhythms from adjacent canons were never directly mixed. When I finished Escher's Drum I had the feeling that I’d arrived at something of an optimal solution for the constraints I was working with, whereas in finishing Fifteen Beats I feel this is really just one of many ways the rhythms here could be presented. That said, it’s a way that I enjoy hearing, and I hope you will too.