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I’m happy to announce a new offshoot of my canon writing efforts: a collection of polyrhythmic etudes in canon form.

For some time I’ve wanted to bring more polyrhythmic interaction into my canons but I’ve held back because my primary goals are clarity, coherence, continuity, and my earlier efforts to work with polyrhythms led to a greater sense of disorder than I was seeking. The 3:2 polyrhythm occurs in some pieces, like #52 “Pyrite” and #55 “Magnetite,” and there’s a bit of 5:2 in #51 “Serpentine,” but I haven’t gone much further than that.

On another front, I had recently been interested in exploring the potential of “forbidden” parallel octaves in counterpoint and when this new polyrhythm project began I was looking at ways to build a canon outline from a series of alternating simple and compound octaves. These experiments with parallel octaves morphed into framework that — I soon realized — could be varied to create a series of canons showcasing all polyrhythms with divisions of 11 or fewer. I’ve included a more detailed description of this framework in the album notes on Bandcamp. The pieces feature simple melodic gestures using mostly chromatic motion, so the lines remain easy to track as distinct even as their rhythmic interactions get very complex.

For me these pieces occupy an interesting place at the border between the purely mechanical and the aesthetically provocative. As study pieces, they are schematic and repetitive, but the effect of the repetition differs from piece to piece: in some cases it may seem tiring while in other cases it is reassuring as the ear tries to grasp a very complex rhythmic relationship. Some of the pieces seem to reveal most of what they contain in one or two listenings, while others are intriguingly elusive and draw one back for repeated encounters. While I acclimate quickly to the 7:2 study, for example, and find it goes on a bit longer than I’d like, the 7:5 study, which is actually the same length as the 7:2, challenges my ear in a way that makes me want to hear it again right after it stops. 11:2 seems straightforward, but 6:5 is disorienting in a way where it seems to be simultaneously speeding up and slowing down at every moment — I’ve never heard anything quite like it. I hope other listeners might find some of the same variety in the collection and come across a few examples that are genuinely perception-expanding.

I wrote these exercises as part of my own study as a composer, as a way to expand my understanding of various rhythmic juxtapositions and I hope they may also be valuable for keyboardists who could use them for hands-on practice, and also for listeners who want to sample some exciting rhythmic phenomena that are not commonly showcased.

I remember standing in the living room when I was ten or eleven, trying to listen to a certain record I had placed on the turntable. I didn’t know anything about Bach at the time, but I had the sense that my parents’ record collection was full of surprises, and at an early age I had a craving to surprise my own ear. Although Bach would eventually become an obsession for me, I did not experience love at first hearing. It seemed to my uninitiated ear that this music was purely mechanical. I had sometimes heard older people criticize works of art for being technically perfect yet lacking in some special quality like feeling or spirit, and I wondered what the adults had meant by this; in listening to a Bach organ fugue that very first time, I thought I might have found an example of such a well-executed flop. The notes seemed to have logic to them – they moved in discernible patterns – and there were lots of things happening at all times. The sound of the organ was huge and made the speakers shake as they worked to reproduce it. The music was busy and complex, but without ever becoming disordered. Knowing very little about music composition at the time, I could nevertheless sense that what Bach had done was difficult. The question that eluded me was “Why do it?” He had put all of that craft into making something which exhibited clear logic from one moment to the next, but which came off in the end as a lifeless jumble, a well-built tangle of notes, coherent, loud, but carrying no meaning, no message beyond evidence of technical facility, as if the composer had merely been bragging, “I know how to do this.” As the music continued interminably it conjured in my mind an image of its weary composer writing note after note under some grave and joyless obligation.

Years later, considering that many of the best moments of my life have been spent listening to Bach, and considering how his music has swayed me through its passion and profundity, revealing itself as deeper in every successive year of my acquaintance with it, I am fascinated by the fact that it could ever have sounded superficial or joyless. What changed? What was I lacking back then that I could not recognize its greatness? I knew nothing of Bach’s biography and little of music history overall – I didn’t have a broad listening background and wasn’t familiar with many other composers to compare with Bach – but these were not my salient shortcomings. Without knowing much about music then, I still found sense in Mozart and Beethoven when I first put their discs on the turntable, Vivaldi too, but not Bach. If good music requires concentration I was certainly willing to concentrate, to put all else aside and try my best to make sense of what emanated from the speakers. Bach’s music didn’t “click” simply because I lacked awareness of one specific concept: counterpoint.

Counterpoint is a special thing that happens in music when two or more melodies are played at the same time, and as if by magic (but in fact by craft) they fit together perfectly, maintaining their independence without seeming to clash, remaining distinct but not indifferent to each other; in fact, supporting, even enlivening each other through their dialogue. When I first tried listening to Bach, I knew there was lots of “stuff” going on, but it didn’t dawn on me that within that seemingly indecipherable mass of sound, there were distinct threads that I could follow, and that I could learn through lots of practice to follow several of them at once. It didn’t dawn on me that all of those notes were not just slithering and writhing against each other in some aimless, endless way but that the “tangle” of sound was in fact the conjunction, the overlap of multiple melodic arcs each of which had its own thrust and direction and personality. It didn’t dawn on me that the composer might not decide for me which primary “tune” I should focus on at any given time, but might present a tapestry of multiple interlocking tunes and leave it to me to explore the whole fabric with my own ear, to decide where my focus should go, to experiment with different patterns of placing my attention and see for myself which were the most rewarding.

At some point after my first, bewildering experience with Bach, I read about counterpoint in a music history book – the idea sounded interesting enough – and I thought I’d search for some examples of it. I had read that Bach’s music was full of this special thing but I certainly hadn’t noticed it before. I went back to my parents’ record collection and found a simpler Bach piece than the five-voice fugue I must have stumbled on the first time. My Rosetta Stone for understanding counterpoint, and by extension, most of Bach’s oeuvre, was the choral prelude Wachet auf, ruft uns die Stimme. It begins with a flowing melody accompanied by a simple, almost plodding bass line, but back then the very distinction between melody and bass was new to me, and listening to this piece was perhaps the first time I realized I could focus on “what was going on up high” as a separate thing from “what was going on down low” in the organ’s register. After a few moments, a third tune enters the mix, the “Sleepers, Awake!” melody itself, which organists usually assign a distinct register, meaning that a different set of organ pipes is used to play the tune, so it stands apart from everything else that’s going on, as an oboe would stand apart from a violin and double bass. The chorale tune is further distinguishable from the elaborate opening melody because it has a very simple contour and a steady, even rhythm. I found myself drawn to this tune when it entered, placing most of my focus on it, but I was astonished to realize that while this tune was sounding, the bass and opening melody never stopped or even scaled down their activities, but continued on their own merry ways with everything interlocking seamlessly. So this must be counterpoint, I thought. I continued to experiment with placing less attention on the chorale tune, and trying to follow just the bass, or the other melody, even as the chorale tune called for attention. I found lots of pleasure in hearing how the texture switched between two and three voices as the chorale tune would enter, then pause, then enter again.  Finding the interplay of these three voices so exciting, I was surprised to remember that I had heard the same piece a few months earlier, but it had washed over me with little effect: I hadn’t even noticed that chorale tune as something distinct – how could it have escaped my attention? In my earlier listening, before I knew what counterpoint was, I had only sensed the sound becoming thicker at some point, as if more of the same indecipherable muck had been piled on – more notes going every which way – not the introduction of a new, connected voice that I could isolate and follow.

After this first experience of grokking counterpoint, I returned and found bits melody I could follow in other Bach pieces, the ones that had mystified me before. I began what would be a lifelong routine of challenging myself to see how much I could hear in Bach, how many lines I could follow, how well I could could understand their relationships, how “wide” I could make my ear. This is not to say that all of Bach was immediately intelligible now, but that I had found a way of approaching it, a way of probing its riches. This change in my listening style – listening explicitly and actively for counterpoint – made the music seem clearer and more sensible, yes, but its effect was bigger: it helped me connect with the music’s inner thrust, so that now I could be excited, I could be surprised, I could be moved, and I did move, waving my arms, tapping, swaying, even jumping at moments of great tension, when no one was looking. Knowing about counterpoint not only helped me understand the structure of Bach’s music but it helped me glimpse at the soul that had seemed absent before. Now that I understood the idea behind Bach’s technique, his music would never again stand out in my mind for its technique, now it would stand out for the passion it expressed through that technique.

As I kept practicing as a counterpoint listener, I found that not only did I like counterpoint, I loved it, I came to need it. It became my nectar, my elixir. I spent years not only exploring Bach’s music but searching for great counterpoint throughout the Western classical canon and also wherever I could find it in traditional musics of the world, and sometimes in jazz. I committed to learning how write it, and my goal in life became to someday produce a sample, an offering of beautiful, rapturous counterpoint, even just thirty seconds or a minute of the kind of stuff that had given me such indescribable pleasure in Bach. I told myself that as long as I could someday understand the craft of counterpoint well enough to write one little thing, no matter how modest in scope, just one miniature piece that reflected the essence of what I loved so much in Bach, I would be fulfilled.

But why counterpoint? What is so much better about having multiple melodies play simultaneously than to just present one pretty tune the listener can enjoy without distractions, without having to follow it alongside other tunes competing for its attention? Why make the audience struggle to disentangle many things that could be offered sequentially, as if three beautiful poems were read aloud at once, instead of each being allotted their rightful moment to shine? Yes, acquaintance with the concept of counterpoint can turn what seems like indecipherable complexity into sensible, intelligible complexity, but why should music favor complexity at all? What does counterpoint bring that cannot be expressed through simpler means? Looking at a counterpoint textbook, one finds a preponderance of rules and restrictions and gets the sense that writing counterpoint is one of the most painfully difficult things to do music, so what justifies all this effort?

My answer is that counterpoint is uniquely stimulating, uniquely engrossing, and that when the writing is good and the listener is experienced, counterpoint induces a state of flow, a state of total engagement, where the work of following multiple lines does not feel effortful, it feels blissful. In effective counterpoint, the multiple lines are not merely juxtaposed without clashing, they are matched so they become mutually reinforcing: they don’t coexist, they frolic, they dance. One melody does not “steal the show” from another but echoes it here, contrasts against it there, seems to converse with it in a way that makes the two more interesting and dynamic than any one might be. Counterpoint is something a listener can get lost in, delightfully lost, and when music is full of counterpoint it presents so many pathways for the listener’s attention that each hearing will be different, the music can never be exhausted: you constantly switch from following one line closely, to dividing your focus between two or three lines, to stepping back and observing the whole fabric, and no matter how or where you “move” as a listener, you notice something new and beautiful, some detail you hadn’t noticed before, or some new way of listening to a familiar detail that makes it shimmer like never before. Since counterpoint consumes so much of your awareness, it makes you forget, it helps you forget everything else, so that in listening closely to just a short contrapuntal passage, you may feel like time has stopped. Because of its challenges for the listener and the composer, counterpoint is sometimes seen as a way of demonstrating learnedness – it has the reputation of being dry and academic – but when it’s good it’s the opposite: fresh, propulsive, almost primally compelling. But as with sex, or spiritual devotion, or any transformative experience, you need to try it before you really know what it’s like, and your first time is not guaranteed to be all that great.

Parallel octaves are forbidden in traditional counterpoint because – so the textbook explanation goes – they destroy the independence of voices. They create the impression that two simultaneous melodic lines – entities which should be perceived as separate at all times – have become indistinguishable, temporarily fusing into one. Of course, parallel octaves occur when instruments or voices explicitly double each other, but in that context the listener is not meant to hear separate melodies: the composer has assigned the same line to multiple players to achieve a certain sonority, not to suggest polyphony. The problem occurs when contrapuntal independence is the composer’s goal, which is to say that the parts are not supposed to be heard as doubling each other, and yet they seem to be doing just that, crossing the chasm from polyphony to monophony and leaving the number of voices – even if for just a moment – indeterminate.

The detection of parallel octaves could be called an industry within the larger field of music theory instruction. It is a rite of passage for every beginning student of harmony and counterpoint to submit an exercise and receive it back with the parallels circled in red (and points deducted accordingly), to perhaps question the sensibility or fairness of this demerit, but to eventually realize that if he or she wants to pass the class it will be necessary to get good – very good – at identifying and avoiding parallel perfect consonances. The teacher may say “First you need to learn the rules before you can break them” but the question of when and how the rules should be broken is prone to deferment. Stepping outside of an academic context, there are two broad scenarios where a composer need not think too deeply about parallel octaves but can simply adopt an appropriate blanket policy. First, the composer can say, “I aim to look back and write counterpoint that is idiomatic to some stretch of the Renaissance or the common practice period, and since the composers of those times avoided parallel octaves, I must do so too.” Second, the composer can say “I am a 21st century person, writing in a fully contemporary style, and there’s no need for me adopt the rules of the past, particularly given that composers before me have already broken those rules to such great effect.” In each case, any detailed inquiry into whether the parallel octave restriction serves a valuable musical purpose can be cut short, and the composer may simply embrace the attitude of whatever style he or she wishes to adopt.

There is cause for deeper questioning in a third scenario, when the composer aims to look forward and back at the same time, writing in a style that combines elements of the past with new ideas. Specifically which elements of the past should be preserved, and is the parallel octave restriction among them?  There’s room for questioning too if the composer says, “I want to write the best counterpoint I can, no matter what style it happens to fall in, and I’ll adopt any rule, or break any rule, to make the counterpoint as good as possible.” Of course this begs the question of what goodness means as applied to counterpoint. Certainly it is a valid goal to write counterpoint that is deliberately thick and tangled, immersing the listener in a jungle of melodic ideas where independence is less important a value than denseness or maximality of texture; however, for the purpose of this essay I want to consider only two-voice counterpoint where clarity and perceptual independence of the lines is treated as the fundamental aesthetic goal. Our metric is the ease or difficulty a listener has in understanding the two lines as separate entities, in keeping track of them as they proceed, in following their courses and comprehending those courses as distinct. (This is a subjective metric as it depends on the nature of the listener, and yet there’s enough in common between listeners that generalizations are not altogether futile; in referring “the listener” I am in fact referring to myself, and yet I believe that what’s true of me has some likelihood of applying to you as well, so for simplicity I will gloss over the distinction.)

The question is: do parallel octaves always degrade the perceptual independence of melodies in two-voice counterpoint, or is it possible to use parallel octaves in a way that preserves or even enhances this independence? Can parallel octaves be used in a way that does not confuse a listener who is trying to follow two separate lines, perhaps in a way that even pleases or excites the listener? Is such a “virtuous” use of parallel octaves possible only in a contemporary style, or can it be achieved in a traditional style where the rules and conventions of historical counterpoint are left in place except for this one major departure?

I believe that while it’s tempting to speculate and theorize about such questions the best way to address them is through experiment, that is to say, to try writing counterpoint with parallel octaves, to give parallel octaves the benefit of the doubt and search for the most pleasing ways of employing them, and then to listen to the outcome and reflect on it. With the composer doing everything in his or her ability to make parallel octaves work, how does it actually sound? Perhaps one reason parallel octaves have such a bad reputation is because they are often discovered as oversights in a contrapuntal effort (“Oh, I didn’t notice them before; now I wonder, can I get away with them?”) rather than as features the composer deliberately included.

Much has been written on parallel octaves and many composers have assessed them before me, but my aim here is not to survey the vast literature but just to share my own experience: no matter how much has already been said and done, I think every composer should perform the basic experiments anew in his or her own “laboratory.” I have experimented with parallel octaves in just a few of my first sixty canons, but I wrote Canon 62 and Canon 63 with the goal of studying them. I will describe a few of my observations from the composition process and then share the pieces themselves so you can draw your own conclusions.

First I should say I think parallel octaves must be examined separately from parallel fifths. While octaves have a transparent quality that could be likened to water, parallel fifths have such a distinctive signature that, even if they can be used without sacrificing contrapuntal independence, they may simply be the wrong “flavor” to introduce into a certain musical dish, so to speak. In my own experience, I have found that parallel fifths are easiest to incorporate into a texture that emphasis octaves, fifths, and fourths (the so-called perfect intervals), whereas parallel octaves are more flexible and can be blended into a thirds-based texture without triggering a sense of “that doesn’t belong here.” In these notes, I’m only considering octaves.

A first observation is that, like any element in art, parallel octaves can seem intentional, or they can seem accidental, depending on how they are employed. While there is no formula for making any artistic element convey intent, here is one specific way I’ve tried make parallel octaves recognizable as deliberate in my work: employ parallel octaves in short runs, followed by octave-free passages, in a discernible alternating pattern. In sections where parallel octaves occur, make them the only thing that occurs in that confined space.

A second observation is that, brief periods of octave-doubling in two-voice counterpoint do not immediately trigger a shift from a polyphonic to a monophonic mode of listening. If the two voices first firmly establish their independence in the listener’s ear, and then align in a brief episode of octave-doubling, the listener can still rely on register, on the distinction between high and low, to track the lines separately until the doubling ends and the independent motion resumes. This doubling shouldn’t go on too long, or the sense of polyphony will eventually erode, but it is not as though the mere occurrence of parallel octaves is some kind of polyphony-killer that will immediately annihilate the perceived distinction between lines and cause the great edifice of Counterpoint with a big C to come crumbling down. Even if the lines have not yet established themselves as independent entities, such as in a case where one line enters alone and then the second line begins to double it, the contrast between the single and doubled texture may still put the listener on alert that two entities are now in the mix: if the second line then stops doubling and goes its own way, the listener may accept the texture as polyphonic and not expect a return to monophony, no matter that the introduction may have signaled monophony.

While parallel octaves can make it difficult to discern the relationship between lines, this confusion can be offset if the lines contain short, recognizable motifs that the listener can easily remember. If one recognizable motif overlaps another recognizable motif in a way that creates sections of parallel octaves, the listener can still perceive the lines as distinct because their structures are already known. The listener can be cued to understand that two separate things have come into temporary alignment rather than permanently fusing or giving up their individuality. Where the motifs are unfamiliar or harder to recognize, parallel octaves may create more confusion.

Contrary to undermining the objective of polyphony, parallel octaves can actually be used to energize or reinvigorate a polyphonic texture. They do this by providing contrast. Where the lines are consistently independent, the listener takes this independence as a given, but if that independence is sacrificed briefly, the eventual return to independent motion can seem all the more exciting and notable, particularly as the ear has had a momentary respite from the sometimes-taxing challenge of tracking divergent lines. Withholding a feature temporarily is one way a composer can make the listener pay more attention to the feature when it occurs, thereby increasing a sense of that feature’s abundance (the feature here being polyphony itself).

If independence of voices is really the highest objective in a contrapuntal composition, the composer should know that many factors are bigger threats to independence than parallel octaves may be. For example, I find it really hard to keep two lines separate in my ear when the lines are simply boring. That is because part of what it means to “hear two lines as distinct” is to remember the lines – to remember something different about each one – and boring lines can be hard to remember because, by definition, they don’t carry our attention or display any remarkable features; one may perceive their independence note-by-note, but after listening across a longer span of time, the entire episode of these boring-but-independent lines becomes a blur in the listener’s mind, and it didn’t matter that they were entirely free of parallel octaves. Similarly, lines that are overly complex can be hard to remember, and while the listener might maintain a sense that there are two different things going on, he or she might not actually follow them, in which case the experience of polyphony is compromised, even if the fact of it isn’t. If the composer says “I avoid parallel octaves in service to the objective of contrapuntal independence” then it behooves the composer to place higher weight on the objective than on rule-following itself, and if contrapuntal lines can be made more interesting and therefore more easily heard as independent by suspending a rule, this is the time to do it. Some composers believe that parallel octaves can always be “corrected” without sacrificing melodic interest, but while mandated revision is certainly a good way of discovering new ideas, I don’t think it always leads to something better.

I feel, after many years of exploring counterpoint, that the early, basic lesson about noticing and avoiding parallel octaves is worthwhile since it builds a composer’s awareness of interval relationships and contours, it builds a habit of careful scrutiny of the score, it builds respect for the objective of contrapuntal independence, and in many cases it does represent the best musical choice. But I have also come to feel that parallel octaves are not always antithetical to the objective of contrapuntal independence, and that in fact they can sometimes support it, not only in contemporary styles but even when writing counterpoint that is otherwise traditional.  I am excited to continue experimenting with them.

In Canon #62 – “Spinel,” you will hear full, overt runs of parallel octaves lasting four beats, constantly alternating with four-beat sections where independence is regained. When the parallel octaves occur, you might think the canonic imitation has stopped and the voices have suddenly come together without a lag, but that’s not what’s happening. In fact, this is a true canon, where one voice is always imitating the other with a four-beat lag forever maintained; the parallel octaves arise because the line has a self-similarity which allows it to perfectly coincide with its delayed version in certain places. The top voice is the leader, with the bass imitating it a third below. There are three sections: the outer two are essentially the same, with the bass in the closing section transposed an octave lower. The middle section is interesting because the bass there is transposed an octave higher, causing what were parallel octaves to become unisons, creating an even stronger sense of fusion. In each section, the music gets higher and higher, passing through many different modal centers, and through a series of rhythmic and melodic variations, as the same basic pattern is energetically and ceaselessly repeated.

 

Canon #63 – “Celestine” – is a second exploration of parallel octaves used deliberately in counterpoint that is otherwise traditional. This piece has a one-bar run of parallel octaves occurring every six bars. If you listen in a certain way, it sounds like certain entrances of the theme in once voice spring out of the other voice; when one voice enters, it often starts off “fused” with the other voice (i.e. moving in parallel octaves with it) and then breaks apart and does its own thing. The piece is built of a repeating motive that is never varied, though it is restated at descending scale positions, causing a gentle progression downward through the modes of C major. Unusual for my canons, it stays entirely within the notes of C major with almost no chromatic alterations, yet even with so many restatements of the same motive it stays fresh to my ear. Variety is pursued through registration: fragments are transposed up and down so we hear entrances happening in all places. Technically the top voice is the leader in this piece, with the lower voice following a fourth down, but the leader/follower relationship quickly becomes ambiguous because of the transpositions (including voice crossings), and because there are no melodic or rhythmic variations that would allow the listener to identify which line is the “source” of new material.

In searching for every musical canon I could find, I came across a fascinating series titled Thirteen Canons at Each Interval by Richard Atkinson.  I was interested to know more about a composer who had undertaken such a thorough exploration of the canon form, so I asked Richard if he’d be amenable to an interview.  Richard lives in Boston, and when he’s not composing, works as a forensic pathologist.  What follows is the text of our conversation, which we conducted over email.  Start by sampling Richard’s work itself, which deserves close, repeated listening!

 

Question (Rudi Seitz): It’s rare these days to encounter a thorough exploration of the canon form in a new musical voice, like your Thirteen Canons at Each Interval.  What inspired you to focus on the canon form and what were your goals in writing the series?

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I’ve just finished a canon — #56 “Sugilite” — which explores the possibility of making music under two special constraints.  First, the piece uses strict contrary motion, which means that the follower is an upside-down version of the leader: wherever the leader descends, the follower ascends by the exact same melodic distance, and vice versa. (This technique is cumbersome enough that I’ve only explored it occasionally in my canon series so far: I had used strict contrary motion in Canon #37 “Graphite” and non-strict contrary motion in Canons #12 “Amber” and #13 “Carnelian”.) Second, the piece uses a major third on the first beat of every measure, following in the steps of other restricted interval canons: #52 “Pyrite” (fourths), #54 “Tektite” (major seconds) and #55 “Magnetite” (minor sevenths). There are a few short connecting passages that don’t participate in the canonic imitation.

Here is a visualization of the beginning of the piece.  You can see how the top voice (red) is a mirror image of the bottom voice (green) with a skew or time lag between them:

canon56-sugilite

If we eliminate the skew between the voices, the mirror relationship becomes even clearer:

canon56-sugilite-aligned.png

The challenge of working in contrary motion is that many melodic gestures are orientation-specific: they “make sense” when played in their upright form but sound confusing or unconvincing when played in their mirrored form.  The composer has to search for those gestures that are musically persuasive in both orientations.

The further challenge of working in strict contrary motion where interval sizes are preserved exactly in the mirrored voice (a major third up translates into a major third down, never a minor third) is that it’s nearly impossible to stay within the notes of a given key.  The technique lends itself best to pieces with a free or floating kind of tonality (as in Canon #56) or to pieces that are entirely non-tonal.

Why bother with the difficulties of strict contrary motion?  I think it leads to a fascinating relationship between voices, one that can be challenging to perceive at times, but one which is definitely accessible to the focused ear — a relationship where the voices seem to be “the same” in one sense, and yet in another sense they seem to be saying different things: sort of like twins with unmistakable and deep similarities, but with their own personalities and directions.  As they move in these contrary directions it is interesting to notice how similar they remain, or how different they become.

Here is Canon #56:

Premier performance of Canon #46 “Palladium” with the wonderful Matthew McConnell on harpsichord.  A new exploration of the idea of vertical shifting counterpoint as first described by theorist and composer Sergei Taneyev (1856-1915).  Part 1 is an initial statement of the canon with imitation at the octave.  Part 2 is a reprise with the top voice transposed up a semitone while remaining consonant with the bass (imitation is now at the minor ninth).  Part 3 is a reprise with a shifting interval of imitation.

I’ve just taken another look at my exploration of rhythmic tiling canons from 2015.  I’ve corrected a few errors and given the piece a new name.  I’ve also updated the video to include captions that explain what’s happening in each section of the piece.

 

From the notes on YouTube:

This piece is named after the artist M. C. Escher (1898-1972) because of his fascination with tessellations: the way a carefully crafted shape can be repeated across a surface in an interlocking way, so as to fully cover the surface without any gaps or overlaps.  In music, there is a related concept called a rhythmic tiling canon, where a special rhythm is repeated by some number of musicians in a staggered fashion, so that no two players ever coincide on the same beat, but every possible beat in every measure is hit by someone.  Such rhythms are akin to “tessellations” of musical time.

This piece explores eight rhythmic tiling canons that were described in the paper “Asymmetric Rhythms and Tiling Canons” by Rachel W. Hall and Paul Klingsberg (2006).  These are the only tiling canons that exist under certain constraints — where the rhythmic cycle consists of twelve beats and the entrances of the players are equally spaced.

The objectives of this piece are, first, to present the eight canons in a musically engaging way (by carefully choosing where each rhythm should begin, which beats to accent, which instruments to use, which order the instruments should enter, how to arrange the eight sections, etc.) and second, to progress from one canon to another in a connected fashion, where rhythms from neighboring canons are temporarily combined, and each section seems to morph into the next.

Captions have been added to the video to clarify the structure of the piece.  As you watch, you will notice sections where one specific rhythmic pattern is played solo, or in duo, or in trio.  Any section marked TRIO is a very special event where the full rhythmic tiling canon emerges.  In these trio sections, every possible beat is struck without any gaps or overlaps — remember, most rhythmic patterns can’t be played this way, and it’s rare to find one that’s suitable.  You will also notice connecting sections where different rhythms are juxtaposed.  These juxtapositions are planned so that no two players ever coincide on the same beat (though some beats may be skipped) and they were chosen out of many possible combinations for their musical interest.

Ignoring the question of where accents should fall, each rhythmic pattern in the piece can be written as a sequence of ones and zeros where one represents a hit and zero represents a rest.  Using this notation, the eight patterns are as follows:

Pattern A: 101001010000
Pattern B: 001001011000
Pattern C: 001011010000
Pattern D: 111100000000
Pattern E: 000100001110
Pattern F: 110000110000
Pattern G: 110100100000
Pattern H: 100100100100

Here is a rough outline of the piece:

Pattern A: SOLO, then DUO, then TRIO, then VARIATIONS
Patterns A and B: JUXTAPOSITION
Pattern B: TRIO
Patterns B and C: JUXTAPOSITION
Pattern C: TRIO
Patterns A and C: JUXTAPOSITION
Pattern A: DUO, then TRIO (reprise)
Patterns A and D: JUXTAPOSITION
Pattern D: TRIO
Patterns D and E: JUXTAPOSITION
Pattern E: TRIO, then DUO
Patterns D and E: JUXTAPOSITION
Patterns D and F: JUXTAPOSITION
Pattern F: TRIO, then DUO
Patterns F and G: JUXTAPOSITION
Pattern G: TRIO, then DUO
Patterns G and H: JUXTAPOSITION
Pattern H: TRIO, then SOLO

The geometric pattern shown at the beginning of the video is NOT by M. C. Escher, though it’s safe to assume he would have admired it.  The pattern is the fifteenth known way of tiling the plane using convex pentagons.  It was discovered by Mann, McCloud, and Von Derau in 2015, the same year this piece was composed.

The eight individual rhythms used in this piece are illustrated here: https://www.youtube.com/watch?v=vBsCfl-MgwA

This piece is part of the album Canons by Rudi Seitz:
https://rudiseitz.bandcamp.com/album/canons

An earlier version of this video was released in Jan 2015.  The composition was given a new title in 2016, some small corrections to the music were made, and captions were added to the video.  The visualization was made with MIDITrail software by Wada Masashi.

See also my followup piece, Fifteen Beats.

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