Let’s be candid: there are few things more infuriating than other people’s stupid beliefs. What is an example of a stupid belief? Take the idea that the coronavirus is a hoax. I call this idea stupid because the virus has caused 640,601 worldwide deaths as I type, and I call this idea infuriating because it leads to behavior that increases transmission, making the situation worse for all of us. When I hear someone fiercely advocate for such a stupid belief, even saying that masks are bad for our health, I feel a sense of shock. How could a person possibly believe a thing so wrong? To answer the how question, I must examine why I am shocked in the first place.

My working theory of belief, I confess, is a simplistic one. I tend to think that every person is on a search for truth. Beliefs are the outcome of a search for truth. We all search for truth in our own ways, and it turns out that some people are quite bad at conducting this search, so they end up with beliefs that don’t make any sense. When I am confronted with such ludicrous beliefs, I tend to give the believer the benefit of the doubt. They are on a quest, just like I am on a quest, but they took a wrong turn in this particular case, just like I have certainly done in other cases. The outcome of their failed search for truth may be an outlandish and toxic belief, but nevertheless it arose from a desire that’s noble and universal: the desire to understand the world.

When I am shocked by a ludicrous belief, my shock comes from not understanding how anyone could be so ignorant of the facts in front of them, so incapable of simple reasoning, so inept at the most basic things we do when we search for truth. You would think that a person who believes the coronavirus is a hoax wouldn’t survive a day on this planet. Forget about their catching the virus. Surely they’d burn themselves on a stove, thinking the flame was a hoax, and promptly get hit by a bus, thinking physics is a hoax. But they don’t. It turns out that lots of people who believe horribly stupid things are actually great at surviving in the world. Not only do they not get hit by buses especially often, but they sometimes achieve great popularity and wealth despite their believing – or perhaps because of their believing the equivalent of 2+2=5. The fact that this surprises me tells me that my working theory of belief needs revision.

After years of watching American politics, observing the American response to climate change, and now the American response to the coronavirus, I’m forced into a sad conclusion: people, for the most part, aren’t on a search for truth, and beliefs are not best conceived as the outcome of a search for truth. Beliefs are simply stories that people adopt because there is a payoff of some sort. When we are shocked that someone holds a particularly irrational belief, our shock arises because the payoff is not apparent to us. If we were to identify the payoff, then we would find it clear why the person holds the belief.

A belief is an investment. If you are to make a financial investment, like purchasing a mutual fund or a piece of real estate, you first need to encounter it. You might find it on your own, or someone might actively pitch it to you. You might resist at first, but a skilled salesperson might overcome your objections. When you buy it, there might be a transaction cost. As you hold it, there will be ongoing maintenance costs, but you will accept these because there is a hopefully larger return. The investment might have some “intrinsic” value, but this intrinsic value might be totally disconnected from the returns you experience. If market conditions work in your favor, you might experience a great return on something that is intrinsically worthless.

When you adopt a belief, there is a transaction cost too. You might have to abandon something else you currently believe, and there might be something unpleasant in the new belief that you have to come to terms with. You might have to admit you were wrong in the past. As you hold the new belief, there will be an ongoing maintenance cost. Your own mind might periodically challenge the belief, finding gaps and contradictions within it. Your friends and family might challenge the belief, arguing with you, and even ceasing to associate with you. If you keep holding the belief in light of these costs, that’s likely because the returns are greater. If the belief happens to be true, then perhaps it helps you solve problems or navigate situations that require an accurate model of reality. But even if the belief is false, it may have other, more attractive returns. The belief might make you feel good. The belief might place you in a community of other believers whose company you enjoy, or distinguish you from those whose company you hate. The belief might make you feel superior to those who make you afraid. The belief might win you the attention you want. The belief might get you a job. The belief might serve as a social bonding mechanism, a calling card, or a salve.

The cost of maintaining a belief may depend somewhat on its veracity, but only somewhat. One would hope that it is harder to maintain a false belief than a true one, since the false belief would conflict with observed reality or contain logical inconsistencies that trouble the mind. But this is only a problem if the believer constantly searches for logical inconsistencies among his or her many beliefs and constantly tests those beliefs against observed reality. If the believer isn’t in the habit of doing these things, the maintenance costs go down considerably.

I used to imagine that a person’s many beliefs existed in some vast common space in their mind – mingling with each other as if in a great ballroom – and that if two beliefs contradicted each other, the host would inevitably notice this and feel disturbed enough to seek a resolution. I don’t see it this way anymore. The space of belief is more like a forest at night, filled with hiding places. Denizens of the forest who might antagonize or kill each other if they met in daylight hide quietly in their own nests.

A person might hold two contradictory beliefs without those beliefs ever meeting and doing battle. And a person who valiantly endeavors to rid their beliefs of contradictions will have a hard time of it, because that person has no way of acquiring a complete inventory of their beliefs, no better chance than a naturalist might have of cataloguing every living being in a vast and dark forest. The mind harbors no unified list of beliefs that can be printed out. You can try to write all your beliefs on paper but I would wager that there are dozens of beliefs you will not know about and dozens that you may be too uncomfortable to write. There may be some that you begin to write and promptly deny. There are beliefs that only become apparent through one’s actions, in situations where one is too distracted to articulate them as beliefs.

When I look at the news and hear that throngs of people still think the coronavirus is a hoax, or still think climate change a hoax, I am astonished because I think of how painful it would be for me to hold those same beliefs. These beliefs would clash with my confidence in science and my faith in the scientific establishment. My friends, all of whom accept the coronavirus and climate change as real, would laugh at me. I would have to reconsider all the efforts I am making in my own life to try to reduce my carbon footprint. I would have to change my news sources and my idea of who is a trusted authority. When I imagine holding these beliefs, I am so caught up in the pain I would feel that I find it difficult to conceive there could be a payoff of any sort.

To understand people who call coronavirus or climate change a hoax I must imagine that the economics are different for them. For me, the transaction cost and ongoing maintenance cost of these conspiratorial beliefs might be very high; for them it might be very low. Their friends might already agree that the virus is hoax and cheer them on for saying the same. Announcing the belief might bring them love. They may not feel any great allegiance with the project of science, no matter that they use its products (computer, cellphone, modern medicine), so they would feel no hesitation in disputing scientific authorities. As they assert these beliefs, they may feel a sense of pride and power in challenging an establishment they consider oppressive. Believing the virus is a hoax may mean, for them, that they get to share more laughs and have more beers with friends. For them, the experience of holding the belief might be entirely pleasant. Why wouldn’t they believe it?

We might wish that it couldn’t be so pleasant to believe in something so dangerous, that it couldn’t be so lucrative to invest in something so wrong, but it can. A person’s beliefs may be more a record of what rewarded them than of what brought them closer to truth. Perhaps the dynamics of belief share the same skewed outcomes as market-based capitalism: beliefs are adopted, just like financial investments are made based on the immediate payoff, without accounting for externalities or hidden social costs. One can easily believe something, or invest in something that damages the common good, without ever realizing or paying for the damage, and so the damage accumulates, compounding until there is a collapse.


Canon #88, Carminite

Announcing Canon #88, “Carminite.”

The essence of a canon is imitation: the follower copies the leader. In some canons, imitation is exact: the follower plays the same notes that the leader played moments before. In other canons, imitation involves some kind of transformation. There might be a chromatic or diatonic transposition. There might be an inversion, where an upward movement is imitated as a downward movement and vice versa. There might be a stretching or shrinking of note durations. There might even be a reversal or retrograde statement of the material. In all of these cases, imitation is deterministic: once the leader plays a note, it’s clear what note the follower must play – there’s no choice. Of course, during the composition process, liberties may be taken. The composer might decide that even though the follower is supposed to play a certain note to stay faithful to the leader’s example, the required note doesn’t sound good and a different note will be chosen. These exceptions to the imitation scheme are precisely that: exceptions and not the norm. When I write canons I often agonize over these exceptions: is it better to keep the canon strict or to make alterations that diminish the formal purity of the piece while making it more musically engaging?

For a long time I’ve wanted to write a canon with a different concept of imitation: one that would be approximate or flexible instead of prescriptive or fully determined. There are lots of ways you could flesh out what “approximate imitation” means. The idea I had in mind was that interval direction would be preserved while interval size could be changed arbitrarily; meanwhile, rhythm would be preserved exactly. That’s to say, if the leader plays two quarter notes that move up by a major third, the follower would have to play two quarter notes as well, and they would have to move up as well, but they could move up by a minor second or a perfect fifth or any other interval.

Writing a piece like this, I hoped, would help me address a question I’ve been thinking about for years: what makes a canon sound like a canon? What formal elements are essential to creating the experience that we seek when we listen to a canon? Is the strictness of the imitation important, or is it dispensable? Is rhythmic imitation more important, or is the replication of melodic contour more important? To shed any light on this, a piece employing approximate imitation would have to be more than a canon that takes liberties here and there — many canons already do this. It would have to be a canon where the leader’s intervals are very frequently changed by the follower, not in a predictable way, like always cutting them in half (see Interval Compression), but in a free and unpredictable way.

One might expect that the idea of approximate imitation would make a canon much easier to write: more flexibility, fewer constraints. So, after months of attempting it, I was perplexed at my inability to get such a canon working. The project, in fact, seemed much harder than writing a strict canon. Why should this be so? The reason is what I alluded to above: approximate imitation introduces a new burden. To really showcase this concept, you have make the follower regularly – not just occasionally – vary the leader’s intervals. So it’s like you’re writing two melodies at once. You do all the work to make the leader’s line sound good, and then, instead of having the follower copy the leader’s already-good-sounding line, you have to make the follower do something different, but still similar, and this different-but-similar thing also has to sound good. That’s hard.

In Canon 88, I’ve finally been able to create an example of approximate imitation. What made it possible this time around? I’m not sure there was any particular trick, although I did make the early decision that this would be a dissonant canon without a clear tonal center. I spent more than a week working on possible outlines, and then another week trying to develop them: two false starts before something finally came together. The false starts were interesting in their own way. I think what went wrong with both is that my melodic developments on top of the non-tonal outline started to become too tonally suggestive. I was beginning to turn a non-tonal skeleton into a tonal piece, but this was a lost cause. Each possible tonal focus was hinted but not maintained long enough to give any satisfaction. As one possible tonal center was abandoned and another suggested, it created the disappointing, sloppy variety of chaos, rather than that very different variety I seek: the one that is intriguing, potent, and structured. In the final attempt, the one that worked, I managed to create lines that were less tonally binding, so they played better with the non-tonal outline, not constantly working up an expectation and then disappointing it.

This canon also explores a V-shaped registral contour I had been wanting to execute in a canon for some time. Both voices start very high and gradually descend until they’re both very low. At the midpoint, they turn around and begin ascending till they return to their high starting range.

In the first section, the lower voice leads. At the midpoint, the upper voice takes over as leader.

The two sections of the piece have different characters: the first is more whimsical and skittish; the second is more patterned and goal-oriented.

The second part of the piece is built from a retrograde or reversed version of the outline I used for the first part. So, the bones of the piece exhibit a kind of mirror structure around the midpoint, but the details in each half are completely different.

As the canon uses approximate imitation, of course, there’s no specific “interval of imitation” used here.

Subjectively, the piece brings to my mind a kind of cartoon dance or chase: fast, erratic, and maybe a bit comical.

The choice of name, “Carminite,” is not particularly significant in this case, though I liked its sound and I noted that Carminite is a red mineral. Sometimes pieces bring to mind certain colors. Red seemed like a good match in this case but to try to explain why, I’d have to be more inventive than I’m feeling right now.

So, does this piece answer any of my questions about the nature of canons? It leads me closer to the conclusion that strictness is non-essential. Approximate imitation sounds, or can be made to sound, very much like classic, strict imitation. Perhaps proving this was my unstated intention when writing the piece. While I set out to showcase approximate imitation as something distinct from strict imitation, my impulse was to reign it in and apply it in a very controlled way. The ear can be guided into ignoring countless discrepancies between leader and follower when more important similarities call for its attention. It’s possible that as I wrote the piece, I tended to put the discrepancies in places that didn’t draw focus.

What also may have happened is that the more glaring discrepancies in my outline got smoothed over by later melodic elaboration. For example, the outline might have had a minor third leap in one voice imitated as, say, a tritone leap in the other — a difference that would be easily noticed. But these leaps might have been filled in as a run of three semitones in one voice versus a run of three whole steps in the other voice. Of course, a semitone run is different from a whole-tone run, but if the ear is more interested in the destination, these paths may sound quite similar: they are both sequences of “small” steps. So while there are many discrepancies between leader and follower situated all throughout the piece, they’re not as glaring as they could have been made to be.

For me, the experience of listening to Canon 88 is much the same as what I get when I listen to my previous canons, barring aesthetic differences from piece to piece. It doesn’t sound like an outlier to me. But this experiment yields no final or general conclusion about approximate imitation because the idea can be realized in so many different ways; only one of them is explored here. It’s perhaps a tame application of the concept, and yet it gave rise to what I feel is one of my freer-sounding pieces. At the moment, it’s one of my favorites. My goal now is to try using approximate imitation again, in a way that makes the discrepancies even more apparent.


Canon #87, Barite

Announcing Canon #87 — “Barite.”

Work on this piece started in a typical way for me. I spent a week exploring various ideas from my canon to-do list, but I couldn’t get any of my sketches to take flight. I kept working through Memorial Day weekend; still nothing. It was hard not to think the long weekend would have been better spent on something else entirely – maybe I should have given up and tried again later? – but I know that every day that seems fruitless is an investment in what’s to come. In a sense, you can’t get something done unless you’re willing to accept the feeling that you’re getting nothing done, and keep going anyway. Finally on Monday evening, I noticed a simple technical option that I hadn’t yet explored in any of my canons. I’ve written a few canons in 3/4 time where the lag is one beat, but I hadn’t written a canon in 3/4 where the lag is two beats (leader starts on the first beat, follower starts on the third beat). Why not? Although Max Reger isn’t my model for canon writing, I notice he used this construct in a good number of his 111 Kanons durch alle Dur und Molltonarten. Ready for something new to work on, I abandoned the other sketches I had been struggling with and started a canon in 3/4 with a two-beat lag.

As with most of my canons, the first step is to create an outline, not one that I like, but one that I love. Why is this step so important? After all, the quality of the outline doesn’t necessarily dictate the quality of the finished piece. It’s totally possible to transform a lackluster outline into a great piece because as you’re working, you can revise the outline or simply throw it away when it stops serving you. The problem for me is that when I don’t start with an outline I absolutely love, it’s hard to find the motivation to keep struggling to reveal its potential. If I do love the outline, then that love propels me: I feel an overwhelming resolve to do whatever it takes to transform the outline into a full piece of music. So while I could probably start with cursory outlines that take a few minutes to throw together, and maybe I’d produce more pieces that way, I’m more inclined to spend hours or days creating an outline that totally captivates me, because once I’m hooked, I’ll never abandon the piece even when the going is rough. You could say I have a kind of perfectionism about my outlines, but I take the view that perfectionism itself isn’t evil: one just needs to be realistic about what one chooses to be a perfectionist about. Outlines are the good things to be a perfectionist about because they’re simple enough that you actually can make them perfect.

So I started making an outline for Canon 87, and managed to get something I loved. During the outlining stage, I don’t really know what style the piece is going to land in. My outline for Canon 87 was full of unprepared and unresolved dissonances, suggesting it would take on a modern style, but the melodic material was firmly tonal and full of diatonic sequences, and the implied harmonies all seemed to fall within the realm of “common practice.” As I developed the piece, this duality persisted: in a melodic or “horizontal” sense, the piece started sounding like something from the 18th century but in a “vertical” sense it seemed much more modern. Towards the end of the piece, the interval palette becomes more consonant, with more thirds and sixths on measure onsets; the sound is less conflicted in spirit and style. I considered revising the latter part of the piece to keep the style more consistent with the dissonant opening, but I decided instead to embrace the piece’s progression from a dissonant to a more consonant palette, and from a severe to a lighter mood.

To bring the piece to a satisfying conclusion, I knew I’d have to break out of the canon and write some free counterpoint. I was ready for ending to be a struggle as it often is. But then I came upon the idea of having the voices move mostly in parallel at the end (after all, they had established their independence by now, right? What more did they have to prove?). I brought them closer together and had them converge into a unison at the final beat, and that worked.

A few details: the piece uses diatonic imitation at the fourth above. It opens in D minor, progresses to G minor, moves back to D minor, and finally progresses to B-flat major. The imitation is fairly strict, but the bottom line takes various ornaments that the top doesn’t repeat. I chose the name Barite because, for whatever reason, the piece brought the color yellow to mind, and Barite is a mineral that can look yellow when cut as a gemstone (all of the other more familiar yellow gemstone names are taken by now). Unlike many of my pieces, Canon 87 has only one section and doesn’t go through an inversion. (It’s probably possible to get this material to work in an inverted form, but it would take some rewriting, and although I always wish my pieces were just a little bit longer, I think this one reaches a natural stopping point and doesn’t call for an extension.) The piece is based on the simplest of melodic figures: on almost every measure onset, in the bass, you can hear a note, followed by its lower neighbor, and then the note again. I like working with simple figures such as this — I like seeing how much they can do.



Canon #86, Tiger’s Eye

Announcing Canon #86 — “Tiger’s Eye.”

I named this piece “Tiger’s Eye” because it makes me imagine a tiger prowling through different environments, some sunny and plush, some barren, some dangerous, some tranquil. The piece creates a sense of shifting terrains – for my ear, at least – even though it consists of the same six bars of core material, repeated over and over (with variations).

In Tiger’s Eye I was finally able to apply some technical ideas I had been wanting to explore for some time. The first idea is to have the leader play with a pronounced staccato articulation, while the follow plays the same material legato. I had experimented with this in earlier pieces but it never seemed to work: my melodies always seemed to demand one articulation or the other, not accepting both. Here, the situation is different: the piece only sounds good with contrasting articulations. The bass has to be played staccato or the piece starts sounding too muddy, and the upper part has to be played legato otherwise the piece starts sounding too choppy. When I started work on the piece, both parts were legato and I wasn’t happy with what I heard, thinking the material itself was a dead end. It’s quite possible I would never have continued writing this piece if I hadn’t tried making the bass staccato on something of a whim; once I did that, I immediately heard some magic happening and I felt an uncontrollable urge to develop it.

The second idea I was able to try here is to freely double the lines. The doubling happens most often at the fourth, fifth, and octave, and occasionally at the third. This gives the piece a much fuller sonority than any of my previous canons. In my earlier pieces, the option of doubling the parts at the octave always seemed superfluous, and doubling at any other interval seemed to wreak havoc on the harmonic design of the piece. It works here though, because the core material is so simple melodically, and because the rhythm is structured so that the hits are staggered between the parts.

Note that the doubling in this piece does not follow strict imitation: for example, in one passage the leader might be doubled at the fifth while the follower might be doubled at the fourth. This freedom is also allowed for transpositions that happen after each six-bar cycle of core material: when the next cycle begins the leader may enter higher or lower, as it wants, and the follower too might enter higher or lower, as it wants (or rather as I wanted when writing the piece).

The piece consists of an opening section and a repeat where some melodic variations are introduced, registers are shifted for variety, and the leader and follower are occasionally allowed to stray from each other (with the follower exploring some ideas that were not previously stated by the leader). Each section begins with the bass leading one bar ahead of the follower, but towards the end, the follower splits into two separate parts, playing a “stretto” with a two-beat lag, while the leader continues as usual. After this stretto, the roles are reversed, with the top part leading and the bass following.

The foundation of the piece is a rhythmic cycle consisting of 6 bars in 7/8. What’s special about this cycle is that, when played in a canon with a one bar lag, the parts never hit simultaneously except at the onset of each measure. It’s not quite a “rhythmic tiling canon” like the ones I explored in Escher’s Drum where the parts only come together at the beginning of each full cycle, but it’s similar.

To create this particular rhythmic cycle, I started by looking at all the ways I could fill the space of a 7/8 measure with three quarter notes and an eight note. There are only four possibilities depending on where the eight note falls with respect to the quarters. After listing these out, I looked at the complimentary rhythms for each possibility: another pattern of four beats that can be played simultaneously, such that the only shared hit is at the measure onset. This image shows the initial possibilities on the upper staff and the complementary rhythms on the lower staff.



I noticed that the first and last patterns are complements of each other. I chose to put those aside and build a cycle consisting of the items in the middle, the one I’ve labeled A above, followed by its complement A′, and then B followed by its complement B′. To separate each section I added some “glue” material that acts as a phrase end. Here’s how the complete cycle looks:


Played raw, without any interpretation, this rhythmic cycle might not seem immediately arresting. So the next challenge was to take the cycle and make some music from it, repeating it over and over, finding out what kind of melodies it can accommodate and what kind of variations it can accept. How to manifest its potential? How to make it sound good? Canon 86 was my answer.


Rip, Slam, Blast

I understand that people who write news headlines face a challenge. The headline should be compact, gripping, and easy to understand. So it’s natural that editors would prefer active, monosyllabic verbs. But this leads to an inequity of sorts. I see a ton of headlines of the form:

X rips Y!

X slams Y!

X blasts Y!

Typically, X is a loud, obnoxious individual who has done nothing more than go on Twitter and post a derogatory and unfounded comment about Y. In other words, X hasn’t really done anything aside from spouting off. And yet, for not doing anything particularly hard, X gets the benefit of having their actions described with some of the most powerful verbs in the English language.

To rip, slam, or blast something suggests an act of great force and great consequence. One assumes that that the person doing the ripping, slamming, or blasting possesses great energy and is motivated by great conviction to use that energy in service of a cause. Superheroes blast things.

If all you’ve done is type some nasty, possibly misspelled, and probably false words about someone you don’t like, you’re not a superhero, and your actions don’t merit the powerful descriptors we attach to the heroic. You haven’t done anything to deserve the strength of a word like “blast.”

Journalists, if you must report on the fact that X wrote something nasty about Y on Twitter, how about not saying “X slams Y?” Instead just say what happened:

X tweets about Y

Or do you not want to do that because a matter-of-fact description would reveal there’s no story here?



Canon #85, Tin

Announcing Canon #85 — “Tin.”

When I started looking for a name for this piece, I thought of the process by which it had come into being. I had spent a week playing with outlines that might help me explore different concepts from my list of things to try in future canons. One idea that’s come up a few times is to write a canon where the voices move primarily in similar motion. Could the voices still sound independent even if they moved in the same direction most of the time? To explore this idea, I created a simple outline with a one-bar lag where each line ascends by a major second, a perfect fourth, a major second, a perfect fourth. Quickly, the outlines spirals up from the low end of the keyboard’s range to the top. While something intrigued me about this material, I put it aside, thinking it was a dead end: too short, and too uniform, to make into a satisfying piece.

I then went on a long excursion, abandoning the idea of a similar-motion canon to try out some other ideas from my backlog, and finally arriving at a new outline that I hoped to develop. Except, I couldn’t. I heard potential in the new material, but I couldn’t transform it from its raw state into music. There’s a process I go through that I might liken to tapping on a tin can to see what sounds it can make. The tin can is the outline, the seed, the initial sketch. How resonant is it? What can it do? Where is the sweet spot to strike it? Sometimes a beautiful shiny can makes only the dullest sound. That was my experience as I tested my new outline, “striking” it in different places to find a spot where some music might start coming out, but hearing nothing, nothing, nothing.

In the course of doing this, I tried putting the new material into 7/8 meter and I came up with rhythmic pattern that caught my ear and had a useful quality: within each measure, the two voices never hit on the same beat except the first. I couldn’t get this rhythmic pattern to work with the new material so, on a whim, I decided to apply it to very first outline I had created, the one I had deemed too simple to make into a piece. To my surprise, the music quickly took shape. It was like tapping on a tin can expecting a thud and hearing a long, shimmering ring. I think it was the simplicity and “hollowness” of the initial outline that made it resonate when combined with the complex rhythm.

I found that my initial material could be molded and reshaped in different ways, leading to a piece with three sections that is happily on the longer side for my canons (two and a half minutes). Bringing a canon’s frenzied motion to a stop is sometimes the thorniest part of the composition process. There were two ways I could go here. My first thought was to end the canon in a way that would preserve the austerity of its sound. The canon moves rapidly across tonal centers but now it would need to end in one specific place. How to do that without it sounding arbitrary? The ending shouldn’t be too “clean” or it might sound contrived, but it has to be strong enough to convince the listener that the piece is really over and hasn’t just stopped at an arbitrary point. And it has to maintain the energy and complexity that the listener has gotten used to. To make a long story short, I couldn’t write a convincing ending that seemed totally in character with everything that had come before, while feeling sufficiently final, so I explored my second option: an ending that takes the piece in an unexpected direction. I took a little strand of “brightness” that surfaces occasionally throughout the darker tapestry of the piece, and gave that brightness the spotlight at the end. So while the conclusion might sounds like it’s happier or sunnier or just simpler than the earlier material, it still derives from that earlier material. It’s as though the other elements of the piece have fallen away, the commotion has exhausted itself, and now the winds that were tugging against each other have come into tune.

As for the technical details: the piece is in 7/8 and has three sections, each consisting of an ascending half and a descending half. There’s a one-bar lag throughout. The imitation is generally at the fifth, except in the third section where it’s mostly at the octave. Section 2 is an inversion of Section 1 (bass and soprano are swapped). Section 3 is a restatement of Section 1 but with parts transposed so the imitation is at the octave in the ascending section and the beginning of the descending section; ornaments are added here.


Canon #84, Chalk

Announcing Canon #84 – “Chalk.”

Writing this piece has been a week-long journey. I was about to stop working on it before the week was up, and I was going to say that it’s the sort of creation where the finish line isn’t clear. With most of my canons I reach a point where I’m confident that every note is in the right place and nothing should be changed. But with some pieces, like Canon 84, I could change a whole bunch of notes and these edits might not make the piece better or worse, just different (this apparent malleability is what led me to the name “Chalk”). With such pieces one has to declare completion without the feeling of arrival. One has to accept that the so-called “final” version of the piece is just one of many possibilities with merit.

I was thinking this way about Canon 84 until I tried to go in and make some changes. Boy, was it difficult! The canon is dissonant enough that you might think it would be possible to change a few sharps to flats without the difference being apparent, but this piece doesn’t work that way. While I kept the name “Chalk,” I discovered that the piece does not lend itself to easy modification.

One of my main constraints in writing the piece was that the relationship between the lines should be “prickly.” Dissonances are emphasized, particularly the harshest ones, minor seconds and major sevenths. Another constraint was that each phrase should avoid establishing a tonal center. I think of the piece as an experiment where atonality is pursued, but not combined as it often is with other destabilizing elements like erratic rhythms, disjoint melodic contours, and an avoidance of phrasal repetition; instead, the atonal phrases try to be smooth and easy to follow, and they all adhere to the same rhythmic pattern: they’re all variations of the same idea.

To avoid establishing a tonal center I use some basic heuristics like trying to include pitch classes evenly, avoiding note repetitions that create emphasis, avoiding scalar patterns, and being very careful about perfect fourths and fifths as melodic intervals, always trying to cancel out their tonal implications. It’s easier to do this if you allow your melody to be disjoint and chaotic; it’s harder when you seek continuity.

There were a few reasons why I wanted to change this piece after my first attempt to wrap it up a few days ago. I kept noticing little violations of my heuristics for atonality, places where a note is stated one too many times in a phrase, or where a scalar pattern seems to imply a temporary tonal center. But almost every edit I tried to make in search of a “purer” atonality seemed to cause other problems, like introducing a vertical consonance between the lines and interrupting the “prickly” sonority.

And then I spent some time listening to one line of the canon alone, and I wondered if the phrases were too repetitive. I usually consider the sound of each line alone as an important part of the creative product. Sometimes there’s a trade-off between the beauty of the single line and the beauty of the two parts together: you can make choices that hurt the single line while enhancing the two-part whole. It would seem that the two-part whole should be given precedence because it’s what the listener is actually going to hear, but I take it as a matter of pride that the canon’s leader could be played alone, without the follower, and it would still be interesting. In the case of Canon 84, I struggled to edit the phrases so that when played outside the canon, they would sound more like distinct utterances and less like variations on the same thought. And I think I achieved this. I made a beautiful single-part piece with plenty of variety. But I scrapped the effort because the added variety made the canon harder to follow when heard in two parts. I realized that the repetitiveness of the phrases is actually an asset here, serving as a welcome anchor.

So this piece which I thought I could poke and prod turned out to be highly resistant to any kind of reshaping. I caught and corrected a few errors in the strict chromatic imitation this morning, and then decided not only that I would stop working on the piece, but that piece the didn’t want me to work on it any more anyway! The finish line was clearer than I thought.

That’s how I stopped working on the piece, but how did I get started on it? I was looking through some of my earlier canon scores and I remembered that I used to do something in my outlining process which, for whatever reason, I haven’t done recently. Most of my canons begin as outlines where each part is an uninterrupted sequence of whole notes. But I used to sometimes leave gaps or whole rests in these sequences, writing the phrase structure directly into the first version of the outline, instead of waiting till later to create phrase boundaries by adding rests where there had been notes: erasing parts of the outline in the same way you might give shape to a sculpture by removing parts of the original stone block. Leaving gaps in the outline makes the writing process easier because each gap frees you from a bunch of logistical problems. You don’t have to generate new material that seamlessly connects with the preceding material while playing well with the other part.

So I set out to make an outline with gaps, like I used to, and started wondering what I could “do” with those gaps. I came upon the idea of changing the interval of imitation after each gap, so instead of having a “canon at the octave” or a “canon at the fifth” I would would have a canon where each phrase in the follower imitates the leader at a different interval. (I’ve used a shifting interval in previous pieces like Gallium and Palladium, but circumstances were different in those pieces, long story.) I wondered how the effect of such a shifting interval would differ between a tonal canon and an atonal one. I decided to try things out in an atonal context. I gradually developed the core material of the piece, a set of six phrases where the follower (soprano) imitates the leader (bass) a major sixth above, then a minor second above, then a minor sixth above, then a tritone above, then a major seventh above, and finally a minor second above. Having done this, I hoped to extend the piece by inverting the parts, but it didn’t work because I had used a lot of fourths; these turned into fifths which didn’t have the proper “prickliness.” I decided instead to have the piece repeat without an inversion. But I found I could introduce some variety into the repeat through a few transpositions. In the repeat, the follower’s first phrase is transposed so the interval of imitation is an octave, and the third phrase is transposed so the interval is a major seventh. Both transpositions came about through experiment and were selected because they preserved the requisite prickliness. The leader’s final phrase is also transposed an octave higher to increase the tension at the conclusion.

What does the piece mean to me? I find it kind of thrilling to hear an interaction between the voices that is calculated, on the one hand, but chaotic on the other. It’s a loud, sharp entanglement that’s neither an argument nor an embrace. I like how the phrases are “free” from the magnetism of a tonal center but at the same time they have a feeling of direction and drive. The soundscape and phrasal repetition reminds me of earlier canons Flint and Zebra Marble but those are much more tightly patterned than this one.



Canon #83, Fulgurite

Announcing Canon #83, Fulgurite.

This is my first new canon in roughly a year. Like all my canons, it’s abstract music that has nothing to do with current events. But the need to get my mind off the pandemic is what motivated me to start work on this piece. It worked. For a few days, finishing the piece was my all-consuming obsession and all the bad news about the virus receded into the background. Of course, those few days of “flow” were preceded by many days of struggling to get started, fiddling around with ideas that seemed to go nowhere. An analogy came to mind: starting a new piece is like trying to start a fire with a few sticks on a rainy day. You keep rubbing the sticks together and nothing happens. You go to sleep cold, having failed to create even one spark. Then you try again the next day. It rains again. The wood is all wet. A few days go by and the routine of futile struggle becomes familiar. You feel guilty about wasting time. You think of other pursuits where you could be more productive. You think of obligations you’re ignoring, messages you haven’t responded to, chores you haven’t done. You wonder if you’re missing something: is it time for a new approach? Is it time to quit? You think of the fires you’ve kindled in the past, and wonder if the magic is gone forever. These thoughts arise because you’re hoping for a shortcut. But there are no shortcuts. The only path that leads to a new fire is the path through discomfort: some boredom, some monotony, some doubt. The only way to start the fire is to show up in inclement weather, day after day, so that at some unpredictable moment, when there’s a window, an opening… when conditions are right for the fire to start, you’ll be there, rubbing those sticks together and making it possible for the first sparks to form.

The names of my canons, which I take from lists of minerals, metals, and gemstones, are somewhat arbitrary, but I do try to find a loose connection between the composition and the title where possible. Fulgurite is a material formed when lightning strikes the ground, fusing sand and soil together. Canon 83 is a dissonant piece where the parts seem to be “fused” together rather than seamlessly blended. Also, I had been hoping for something like a lightning strike, and it came. For those reasons, the title seemed appropriate.

The piece is in 5/4 and continues a series of canons I’ve been working on that explore odd meters. The subdivision pattern (3+2 vs. 2+3 vs. 4+1) switches from measure to measure. Imitation is at the octave with a two-bar lag. Dissonances like major seconds, perfect fourths, and minor sevenths are emphasized on strong beats; however, the tonality of each line is fairly centralized around A minor and doesn’t stray too far afield. So the piece explores how a dissonant sonority and an anchored tonality can happen together.

When I first started work on the piece, I thought I would experiment with extreme ranges, with the leader confined to the lowest end of the keyboard and the follower confined to the highest. That experiment will have to wait for another piece; it didn’t quite work out here. However, the piece does start with the leader quite low, entering on an A2 and returning to that note repeatedly, while the follower stays relatively high. And the piece did get me thinking about range in some new ways. A major second sounds very different from a major second plus an octave, a major ninth, but what about a major ninth versus a major sixteenth? Do we consider those compound intervals as equivalent or are they different experiences, with the intensity of the dissonance reducing as the distance between notes increases? How might this reduction of intensity with each added octave change one’s contrapuntal choices?

In working on this piece I found that the soprano and bass interplay only “worked” for my ear when those parts were separated by several octaves; move them closer and they didn’t sound right. However, when I inverted the counterpoint to create the second half of the piece, the opposite was true. The inverted voices did not make sense to my ear when separated by multiple octaves, but when I moved them closer so they almost touched, I was delighted by the result. Why? I’m not sure.

The second half of the piece is not something I could have ever written from scratch. It only came about by my writing the first half, then inverting it, then experimenting with the range of the parts, and finally making a few modifications to avoid certain jarring coincidences. The second half straddles the line between order and chaos in a way that I really like. The voices are hard for the ear to untangle, but they still assert independence, making clear, assertive gestures that the ear can latch onto. It’s chaos, but it’s controlled chaos; it’s noise but it’s purposeful noise. I wouldn’t know how to sit down with a blank page and compose that kind of purposeful noise directly, but I was able to discover it in a rearrangement of the components I had already created for the first half of the piece.


Practicing Optimism

Earlier in my life, in those occasional moments when I’ve felt really dejected for one reason or another, and a friend, listening to the litany of my troubles, has suggested I simply need to have a better attitude, that I should look on the bright side, I’ve usually muttered something to the effect of “You don’t understand.” When I need commiseration or catharsis, it may feel like an insult to be told that I should be more optimistic. Such advice overlooks the details of what’s troubling me and implies that I bear the responsibility for my condition: my real problem isn’t out there in the world, but rather in my mind. If I would just think differently, I would feel differently, and if I refuse to do that, then my suffering is in some sense my own fault.

It’s easy to chafe at such advice, and to reject it, particularly when you feel that the person giving the advice doesn’t really “get” what you’re going through: the confluence of factors outside your control, the unreasonableness of other people, the inescapable thorniness of happenstance.

But whenever I’ve bristled at the idea of looking on the bright side, there’s been a tinge of epistemic hubris in my position: I’m absolutely convinced I’m seeing the truth. The truth is ugly and that’s why I’m sad. To be cheerful at such a bad time would require willful ignorance. Being sad, in this view, is almost virtuous in that it involves a refusal to look away from reality, however grim, and an unwillingness to be duped by happy fantasies. Being sad is being honest.

My stance regarding “positive thinking” has changed in recent years. I’m more open to it, less likely to resist it as I’ve just described. That’s not to say I’m able to transform my perspective from gloomy to hopeful at will, but that I respect those who can pull off such a feat and I believe there’s something for me to learn here.

Here’s an argument I accept: if you walk into a room, you may think you’re seeing the full truth of the room, but of course you’re only seeing the few parts of the room you’re looking at. You could take a photograph that shows light coming in through the windows, and someone viewing this image would sense an airy, welcoming place. Or you could take a photograph that shows dust and cobwebs in an abandoned corner, and someone viewing this image would think the place is cramped and dirty. Is one photograph more honest than the other? No, they are both honest but partial depictions of a complex reality. It’s like this with any situation: where you point the lens makes all the difference. When we feel dejected it’s often because we are pointing the lens at those things that are most troubling to us, ignoring or discounting the possibility that we could point the lens in other directions too.

Photography requires skill that can be developed through practice, and likewise we can practice and improve at finding perspectives that emphasize the hope and promise in difficult situations as opposed to emphasizing only the pain. We can be kinder “photographers” of our circumstances. We can be more receptive to what might be beautiful or inspiring, and less obsessed with showcasing what’s ugly or upsetting. We can do this not by denying reality and embracing fantasy, but simply by changing what aspects of reality we concentrate on.

From those last paragraphs, you might conclude that I have consumed the Kool-Aid of positive thinking, and maybe I have. What made me do it?

I’ll mention one of the moments that brought about a shift in my attitude. It was when a critical care nurse who had just taken my blood pressure, reviewed my medical chart, and interviewed me about my various concerns told me that I needed to be more optimistic to be more healthy. I was not in an ICU; rather, I had signed up for a course in stress management and mindfulness at my local hospital, and I was having an intake session with this nurse, who would be teaching the class. So perhaps it was predictable that she would have said something about positivity. Nevertheless, it was shocking, in a helpful way, for me to receive optimism as a prescription, written by someone I viewed as a medical authority, the same kind of person who would tell you to take an antibiotic twice a day for the next two weeks, and you’d do it without question because you trust her to know what’s best for your physical health.

The nurse gave me an explanation of how thoughts can trigger a stress response or a relaxation response, and how those bodily responses in turn affect our fitness, immunity, and overall well-being. I’d heard such stuff before, but I was ready to be reminded. When a friend blithely tells you to be more positive you can be miffed that they’re not really commiserating with your pain, but when a healthcare practitioner tells you the same, quoting research and invoking the weight of a lifelong medical career, the advice carries a different weight. Where I had always viewed the tension between optimism and pessimism as private matter, a question of personal philosophy, something an individual could reasonably ponder throughout their life, this was the first time anyone suggested to me that one of those stances was an essential component of health, and the other was not.

Another thing that’s affected my outlook is that I’ve gathered enough years to look back on now, more than forty of them, and I can see that my bad moods never got me anywhere, even though in each case it seemed like being sullen was an act of protest, a way of sending a message to reality that I did not approve of its course. The message was never received, not once. I can only conclude that sullenness is not a great way to effect change.

During the nurse’s class, I wrote down my own summaries of points that were made. I’m reviewing them now as I look for wisdom to apply in the time of COVID. On one piece of notepaper, I wrote this:

Contentment comes from believing there is meaning in life and always working to find that meaning. The ability to find meaning is something that can be practiced: you get better at it the more you try. When you’re down, it’s because you’re overemphasizing negatives and ignoring positives – you’re turning away from sources of meaning – it’s that simple.

Is it really that simple? I’m not sure, but I do recognize that the times when my life has seemed the most suffused with meaning are the times when I’ve felt the happiest. I’m intrigued by the idea that a sense of purpose is not a static quality that a person might have or lack but that it is something we can cultivate as an ongoing practice.

I’m tempted to think that “looking on the bright side” is advice that applies to normal times, when you’re basically OK; in times of extreme stress, it might not be practical or reasonable to be optimistic. But I remember that the nurse who taught the class also works with cancer patients, some of whom are at late stages of disease: with only a few months to live, she would say, it’s still not too late to embrace a more optimistic outlook, and doing so could help you make the most of what time you’ve got left. Even if your day was spent struggling with the side effects of chemo, there might have been a moment when you laughed at something or appreciated a kind gesture from a friend. Focus on that moment, and savor it. Give the good things more airtime than the bad things. Doing that can only help you.

As COVID rages, there is a lot to practice. We are cut off from so many of our most natural and familiar sources of meaning. Where we find meaning in physical togetherness, gathering with friends, camaraderie, public celebration, we must now stay home. Where we find meaning in travel, adventure, novelty, we must now stay home. Where we find meaning in live performances, plays, concerts, sporting events, we must now stay home. Where we find meaning in shared meals, we must now eat alone. We can move some of these activities online, but it’s not quite the same. As we lose the social rituals that keep us feeling connected, we may also be losing whatever economic security we have worked to achieve, fearing for our own health and those of our loved ones, and hearing daily reports of illness, death, and systemic dysfunction.

The idea is not to ignore this or pretend it isn’t happening. The idea is not to artificially think cheery thoughts. Rather, it’s to experiment with “photographing” this situation in different ways, holding the camera at angles we might not often think to use. What beauty is still transpiring in the world even as the pandemic expands? What lessons can we learn from the experience before us? What opportunities for growth and change does it present? What meaning can we find in it? Take the time to list possible answers to these questions. Even if you’re not persuaded by those answers, see how many you can come up with.

And now I come to the challenge of practicing what I preach. I will try to list a few things that have been making me feel good or giving me hope in the past few days, and I’ll try to crank the list out in ten minutes, so it’s going to be unpolished:

I’ve had some really good phone conversations with friends and family, especially my mom, over the past two weeks.

I’m glad I developed an exercise routine before the pandemic because it’s serving me well now.

I’m getting reacquainted with my music collection, listening to some albums that I haven’t heard in years.

I’m cooking every meal at home. This is the first time in my life when I’ve sustained a practice of 100% home cooking. Now that some ingredients are hard to get, I’m appreciating each meal more than I otherwise might.

I’ve still got a job. A home. A partner. A life.

I’ve watched some good movies this past week. For whatever reason, I never developed a movie streaming habit; maybe now’s the time to partake (even though, alas, streaming has a hidden environmental cost).

I’m sleeping well again.

My musical collaborator just sent me some fantastic clavichord recordings of some of my new canons. I’m eager to keep working with him and write more canons. Also, to record some of my songs. And start some new musical experiments.

Listening to those few leaders who project competence, composure, and respect for science.

I remember that by staying home, the average person is not only protecting their own health, but the health of all of us. Our isolation is a social gesture, an act of solidarity. We’re saying inside for each other.

My ten minutes is up.