I’ve been searching for art to use as an album cover for the set of canons that I’ve been working on since 2014, now performed beautifully by Matthew McConnell on harpsichord. The images I’ve considered so far reveal a lot about the album itself, so I’d like to share this story of the hunt that’s now lasted five days.

In a musical canon, there’s one part that leads and another that follows.  This second part is a copy, image, reflection, or translation of the first part. One visual theme this brings to mind is the idea of an object and its shadow… preferably a curved or wavy object that evokes the ups and downs of a flowing melodic line. Indeed, when I first began my canon project, I made a cover using my own photograph of a curvy bike rack and its shadow. This is an image I captured back when I first began exploring photography with an iPhone 3GS and it’s one of my favorites from that time:


I continue to use this cover for the collection of computer-generated recordings of my canons that I maintain on Bandcamp. I love the cover and have received many compliments on it, but since it’s already in use I need to find a different image for the new album of Matt’s harpsichord performances.

As I searched around on the theme of shadows, I came across these vintage diagrams showing how to make hand shadow puppets. They were amusing to find — I admire their variety and intricacy! — but they’re not quite what I’m after…

Moving on from the shadow concept, mirrors and reflections are another visual theme that resonates with canons. Here are two intriguing mirror illustrations I found in a nineteenth century science text, El mundo fisico gravedad, gravitación, luz, calor, electricidad, magnetismo, etc. by A. Guillemin:

I really like these images, but I don’t love them enough to use on the album cover. I’ve always enjoyed the way vintage scientific engravings seem to straddle the line between art and merely functional illustration. These images strike me as falling somewhere in that happy region, but lacking the magic that would make me return to them again and again.

A friend asked me why I’m spending so much time obsessing over the album cover when most people will care more about the music itself than the cover — wouldn’t any decent cover suffice? My first answer was that I’m doing this for myself: if I am going to be looking at the cover for years to come, it’s got to be something I really love. My second answer is that I’m still committed — rationally or not — to the idea of an album as an integrated whole, including cover art, liner notes, and audio, so… every choice is important.

Continuing with the theme of mirrors and reflections, art history includes a multitude of portraits involving mirrors. Here’s a depiction of the painter Iaia of Cyzicus (“Marcia”) working on her self-portrait using a mirror. It’s from an illustrated French translation of De Mulieribus Claris, known as the earliest collection in Western literature dedicated to biographies of women:


Again, I really like it (the three images of the painter’s head, the depiction of art in the making, replete with the painter’s tools, the tiled floor and patterned background), but I don’t love it in the way that would make me choose it as a cover. Another mirror portrait that comes to mind is by Renaissance painter Giovanni Bellini, but the Tallis Scholars already used it on the cover of their Ockeghem album which I’ve listed to a lot:


One portrait involving a mirror that I do love is En nøgen kvinde sætter sit hår foran et spejl (I can’t pronounce that, but I can cut and paste!) by Danish “Golden Age” painter C. W. Eckersberg. I’m so fond of the painting that I added text and made it into a real candidate album cover, thinking for some time that it was “the one”:


In addition to its use of the mirror, and its overall elegance, I like this image for the canon album because it has the sense of timelessness — rather, that sense of being old and new at once — that I’ve been seeking in the musical compositions themselves: each canon looks back to principles of traditional counterpoint from the Renaissance and Baroque while at the same time exploring modern possibilities. The painterly technique as well as a few details (the subject’s hair style?) tell us that this work is from an earlier time, but when specifically?  The image is spare and simple enough it does not seem dated — we could almost be looking at a contemporary scene rendered by an artist conversant with traditional techniques.

This option seems to have the “classical album cover look” and I love it, so why not use it?

The painting just doesn’t feel like a perfect fit for the music. It’s an intimate scene that makes the viewer feel he or she is almost intruding, and that’s not how the canons make me feel — they don’t create the sense of witnessing a “private moment.” The Eckersberg is a sensuous image, and while I like that in some ways — first, it would be attention-grabbing as an album cover, second it insinuates that the accompanying music is beautiful and compelling — it doesn’t capture an important aspect of the compositions: their mathematical nature, the way they are carefully crafted structures, born of calculation and artifice. We can’t know what the woman in the painting is thinking, but if she’s fixing her hair in front of a mirror we could guess that she’s considering her appearance, and perhaps thinking of who she will see later in the day, and how they will see her: the image conjures the personal, the world of human concerns and relationships, but yet the canons seem more abstract than that. I’d love to use Eckersberg’s painting on a cover someday — for a different album!

Turning away from such a promising option, what else is out there?  Here’s a thought: the bike rack in my original album cover looks something like a snake. A snake is evocative of a musical canon in the way it makes beautiful patterns by folding on top of itself, or in the way the motions of two snakes — or even different sections of the same snake — seem to copy or imitate each other.  Here are my favorite snake illustrations:

My concern here is that many people — rightly or wrongly — view snakes as menacing, and I want the album cover to be inviting. In deference to the ophidiophobes among us, I decided to look further.

All of the canons in the album are named after gemstones or minerals: “amethyst,” “malachite,” “obsidian,” “silver,” and so on. This is a naming convention I adopted to impose some uniformity on the collection and also to keep myself from getting stuck trying to find the perfect descriptive name for each piece — an endless and in some cases impossible task.  Given the way the pieces are named, one possibility is to use an illustration of gemstones as the cover, preferably a print from a vintage science text:


What I like about this option — an image showing a dozen or so different stones — is that it evokes the variety of the album: each canon is in its own mood and compositional style, and while there are similarities and some common themes, the pieces are not tightly connected — they each stand on their own.  And when you have 45 independent, miniature creations all together, they make a kind of miscellany, sort of like a collection of beautiful stones of all different colors, textures, and shapes as shown in this image.  In listening to the album as I prepare the release, my one concern has been that listeners might find it to be something of a hodgepodge, with many pieces clamoring for attention but not flowing logically from one to the next since they were not written with that aim in mind. I like the way these gemstone illustrations actually set the expectation of an assortment — a beautiful, intriguing collection that’s fun to explore and that makes no promise of tight integration.

On the other hand, I worry that gemstones don’t have an obvious connection to canons other than the fact that I’ve happened to use them as canon names. Gemstones on the album cover might be a bit perplexing until the viewer — if he or she persists — comes to understand the finer details of the album.

In searching for other old science illustrations I was delighted to come across Haeckel’s Kunstformen der Naturwhich reminds me of one of the joys of my childhood: looking through my grandfather’s old botany textbooks and the drawings he made when studying to be a botanist before he was drafted. Haeckel’s images are so entrancing that I’m not surprised some people adapt them as tattoos! Unfortunately I couldn’t find a Haeckel image that seemed right for the canon theme. Here’s one I love though:


Another theme I considered are geometric patterns like tilings, mosaics, and Escher-like tessellations:


But I don’t want something too regular and seemingly deterministic. The canons have patterns in them, but what makes them interesting is how patterns are broken or redefined: ultimately each canon tries to escape its mechanical, calculated nature and become something more.

Some of the canons are perception-bending, and I even wrote one piece (#11 “Hematite”) that is designed to imitate the effect of Penrose stairs where you think you’ve been ascending until you discover you’ve actually descended:


Using Penrose stairs or some other impossible object or illusion on the cover is a strong possibility but yet I don’t want to convey the sense that the album is all about perceptual trickery and/or disorientation.

Another possibility is to use an image of the harpsichord, but that doesn’t add as much from an interpretive standpoint as I’ve been hoping the cover art would add.  Beyond the harpsichord, I could use a fantastic or imaginary musical instrument:


The above image is an automatic organ proposed by Athanasius Kircher in Musurgia Universalis. I’m intrigued by this speculative musical automaton but I worry that it would make a confusing album cover as there could be the implication that this instrument, or something like it, is actually used in the recording.

There’s also the possibility of using music notation on the cover, like the circular score of John Bull’s puzzle canon, Sphera Mundi:


Or the score of Baude Cordier’s canon Tout Par Compas from the Chantilly Manuscript, used on the back of a wonderful album by Ensemble PAN:

A solid prospect, but I’m reluctant to use another composer’s work on the cover as it could imply that this music is recorded on the album.  Also, as I mentioned earlier, I’m hoping for an image which reveals some kind of interpretive perspective on the album — sheet music seems too straightforward from that standpoint (though the circular scores above are certainly intriguing).  A friend suggested taking a photograph of one of my own scores, and possibly showing it reflected in a mirror — a great idea! — but I’m a perfectionist about my photography and I’m trying to get this album cover done without delay.  I’d rather not embark on a photographic project unless I absolutely can’t find some existing art to use.

Turning back to the theme of mathematical constructs and searching through many old geometric illustrations, I was very excited to come across the fantastical creations of 16th century German artist Lorenz Stoer, published in Geometria et Perspectiva:


WTF? It’s a beautifully bizarre combination of perfect polyhedra in the foreground, with abstract curved structures that somewhat resemble the arches of ruined buildings in the background, beyond which we see trees, a village, mountains — where is this? What is this?  I so want this to be my album cover, but alas, the aesthetic is not the right match.  The image is too sprawling and seemingly chaotic to match the canons which I view as constrained and tightly constructed, even minimalist in that they’re very short and they only use only two voices.

Yes, I’m a tough customer, but my demands are not impossible to satisfy. When I came across the geometrical illustrations in Perspectiva Corporum Regularium (1568) by Wenzel Jamnitzer, I knew I might find my cover within:


Jamnitzer was a famous goldsmith working in Nuremberg in the 16th century. His Perspectiva Corporum Regularium explores the way myriad forms can be created from the building blocks of the five Platonic solids. The work shows his remarkable ability to visualize complex geometric constructions and project them convincingly in two dimensions. I love the way he chose to balance his structures on intricate and seemingly precarious stands. He didn’t have to draw those elaborate stands, but he did, and doing it gives the images an element of whimsicality and improbability that really speaks to me. What we see here are possible structures (these are not like the impossible Penrose stairs we considered earlier), which could possibly be balanced the way they are… but only for a split second. If we are to treat this drawing as depiction of a real scene, we must be witnessing a fleeting moment while the structures were balanced on their stands before they promptly toppled off; else we are observing a more perfect, fantastical world where things just remain in balance

Why do I like this image as an album cover?  It’s old, but yet timeless in the geometrical perfection of the objects shown, and almost modern in the surreality of their balancing act. It reminds me of the way the canons themselves are balancing acts, and how each canon seems to me, in some sense impossible, and yet somehow my process of creation has wrung them all into being, and Matt has recorded them so beautifully, and they’re available to witness!

Only problem with this Jamnitzer cover is, my mom just told me she’s not crazy about it… likes it but finds it a little lackluster. I’ll keep my eyes open for other possibilities, but for now I’m treating this as “the one.”

Here’s an updated version (still in progress) incorporating some advice and guidance by my friend AG:


“Crossings” is a study in pattern perception, chaotic interference, and magic.

The piece opens with two parts that start in distant registers (one very high, and one very low) and move closer until they intersect and eventually pass each other.

When the parts are far apart they can be heard as distinct entities, but as they get closer they begin to interfere with each other. Their crossing is moment of indecipherable “chaos” where the parts seem to lose their individuality without actually fusing together into a coherent whole.

The interference happens because the parts are skewed so they sound on alternating sixteenth notes. When the parts are far apart and you hear a sixteenth note in one part immediately followed by a sixteenth note in the other part, you are likely to perceive these notes as separate, unrelated events belonging to distinct patterns. But when the parts come close together, pairs of notes belonging to different parts begin to bind together into melodic fragments. That’s to say, the tendency to interpret two successive notes in close pitch range as a linear “melody” is stronger than our ability to parse alternating beats into their respective parts.

After the parts cross, they continue moving in their given directions, growing further apart and separating again into distinct entities. They then reverse direction and move towards each other for a second crossing.

In the second crossing a bit of “magic” is pulled off. Throughout the first crossing the parts had stuck to their given patterns, but during the second crossing, amidst the “chaos,” the parts covertly adopt a new melodic pattern. As they emerge from the chaos, moving apart from each other again, the new pattern can be heard with increasing clarity.

Eventually the parts reverse direction and move towards each other for a third crossing.  This third crossing is pure and free from magic — that’s to say, the parts stick with the second melodic pattern throughout. They move apart again, and then change direction once more, progressing towards a fourth and final crossing. During this fourth crossing, some magic is again carried out amidst the chaos, and the parts emerge playing the original melodic pattern once more.

The piece is technically a canon by inversion – the parts are mirror images of each other.  It will be included as Canon #70 in Rudi Seitz’s collection:

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 a 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 of 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 in 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 with each other. 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 to 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 work in.

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 raises 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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