On Teaching: Further thoughts about rhythm

Gavin Black is director of the Princeton Early Keyboard Center in Princeton, New Jersey.

Further thoughts about rhythm, part 2

Continuing from my September column, I offer here a few additional anecdotes and ideas relevant to rhythm and the teaching of rhythm. In my next column I will explore the question of how to teach rhythm or “counting” in a way that enables a student to connect those concepts as directly as possible with rhetoric and the student’s interpretive stance.

I begin by revisiting the anecdote from the last column about my colleague who, in the eyes of a particular third party, “couldn’t count.” Since I wrote that, I have had further relevant thoughts. I remember that as I walked into the rehearsal studio after that exchange, I was very sorry to have been told my colleague’s opinion. I did not want to approach my work with this fellow musician with any preconceptions. And while I certainly did not assume that the older colleague was right, I found it hard not to have a question layered on top of my focus during rehearsal. “Was that rhythm correct? Is this where I learn that she indeed cannot count?” That was distracting.

I wondered why he wanted me to focus my attention on her rhythmic deficiencies. Why did it matter in the overall picture? Why tell me as I headed into the rehearsal that “she can’t count” rather than that “she is an amazingly compelling performer?” This seems like a manifestation of bias in favor of that which can be quantified—that which seems more objective or safer to describe.

Since that day decades ago, I have become very aware that there is a strong pull in all sorts of areas of life to focus on things that can be measured and described in a way that is objective to some extent. It is a truism that in evaluating a competition, for example, or even just in reviewing a concert or a recording, it is relatively easier to note that a performance either does or does not get all the right notes, or that the rhythms are or are not all in strict accordance with the notation, or that the tempo is or is not the same as what the composer requested.

It is much more difficult to describe how expressive or moving a performance is. This is not a bad thing, and competition judges and reviewers certainly do not fail to grapple with the elusive side of things. But this has some bearing on teaching rhythm and counting. On paper, rhythm is one of the most objective matters about music—at least with our familiar rhythmic notation. Quarter notes are twice as long as eighth notes, etc. In a way, some of what is non-objective can even be described objectively. Certain dotted-note patterns, for example, are not meant to be what they apparently look like on the page, and we can often describe the ways that they differ from that presentation.

But how do we teach subtle variations in rhythms that look the same on paper? Is it an acceptable part of our understanding of rhythm for there to be such variations? If we think not, is that for musical reasons, or is it because of this pull toward the objective? If a quarter note is really always exactly twice as long as an eighth note, if all the eighth notes are really exactly the same as one another, what does that mean for a student or for how we teach? And if they are not, what does that mean? Is it possible that rhythmic notation means “these notes should be in any rhythmic relationship that would be expressed more closely this way than any other way?”

Another story from many years ago: I was coaching an amateur chamber music group—a violinist, a flautist, and a pianist who was exploring the harpsichord as part of this project. At that time, in my own work as a player I was exploring Sweelinck and beginning to discover some ideas about freedom of rhythm. At one point I played a segment of a Sweelinck toccata for the members of the group and drew their attention to a particular passage. This was one where I felt that the rhetorical force and expressiveness of the music could be enhanced by playing very freely. I had an approach to that passage that involved drawing out the development of harmonic tension and, when it was resolved, using timing to make it seem difficult to go on. That is a trite and inadequate description, but the point is that it called for freedom, and the musicians in the room found it very effective. I also played through the passage as written, and they found that almost embarrassingly boring and pointless, which was exactly what I intended to convey. So far, so good. But when we had finished this and were ready to get back to rehearsal, one of the musicians said, “Of course, you couldn’t do that with Bach.”

So, the question I had, and still have, is, “Why not?” Anyone may or may not appreciate the rhythmic choices performers exercise. In theory, there could be a plausible analysis of a Bach piece that concludes that free rhythm, or any particular approach to rhythm, might not be effective. And the question of what the composer himself would have thought always looms over our thinking about an issue like this. But none of that is what was meant by that remark that day. We discussed it, and the ensemble member specifically meant that since listeners’ expectations about Bach were pretty well formed already, unlike their expectations of Sweelinck, it would be imprudent to go too far in violating those expectations. To some extent, the way that music had been played and heard in then-recent decades had become part of the actual identity of that music. This seems to be another way in which something objective can gain a kind of privilege or priority that it might not have earned in any substantive way.

Another question that I will delve into more next month is, why rhythm? That is, what is the goal of having music organized into regular or somewhat regular micro units of time? Is it to create a sense of pulse or momentum? Is it something about comparability of experience throughout the duration of a piece or a movement? Is it very specifically about creating the palpable sense of a regular beat in the listener’s ear?

I mention it here to introduce an experiment I once conducted that was predicated on the observation that many people are of the opinion that Helmut Walcha’s recordings are rhythmically conservative. I obtained an interesting result.

I had always reacted to Walcha’s recording of Sweelinck’s Fantasia Chromatica as having both extreme steadiness and inexorable forward momentum. At the time that I did the experiment, I took it for granted that the way to achieve those qualities was to keep the beat very steady. But I must have suspected something that led me to investigate. I put on the LP and measured the beat at the opening. I then picked the needle up and dropped it a bit farther in. I measured again. I did it a third time, maybe a fourth. They were all quite different. This led to the intriguing notion that maybe inexorable forward momentum, and even the very sense of steadiness itself, might sometimes come from something other than regimented sameness of beat.

I have now repeated this exercise using more modern methods: the piece playing on my computer and an online tempo tool. The beat at the beginning hovered around 118, and later on it was over sustained passages as high as about 126 and as low as the mid 90s. That is very much like what I measured thirty years ago. I am intensely interested in the relationship between literal sameness of beat—or departures from that—and a subjective sense of steadiness, momentum, and pulse. Is it possible that sometimes a performance that features a doggedly steady beat comes across as uneven to listeners? If so, how can this be?

There are two games that I have played while riding in a car that both have to do with the use of time in music. First, if you are riding along a fairly busy two-way street or highway, pay attention to the sound of the cars passing on the opposite side. On a busy road cars will space themselves out almost regularly, since all else being equal, people pay at least a little bit of attention to following distance in front and behind. But there’s an emphasis on “almost.” The line of cars is never spaced out exactly evenly. So as the “whoosh” of each successive car goes by, see how far from even those sounds can be and if you can still accept them as conceptually even. How far apart can sounds 2 and 3 be—compared to the time between sounds 1 and 2—and can you still hear that timing as rubato or agogic accent rather than just discontinuity? For me there is a wider range of timings that I can assimilate to evenness than I would have expected.

If you can accept a stream of not quite even cars as conceptually even, is there anything interesting about the shape of the unevenness? Does it have any rhetoric to it, groupings or patterns of weak and strong beats, or impulses? One fascinating feature of this exercise is that each instance of it is ephemeral: you can hear it only once, never again the same way.

Another exercise I enjoy while driving is this: when going under an overpass, I try to experience the time in the shadow of that overpass as lasting forever. Since it will come to an end, usually in seconds or less than a second, I attempt to experience it as simultaneously brief and infinite. This feels even more intense if it is raining. It seems to me this has implications for rhythm in music. The ostensible rhythm, flow, motion of any increment of music is always about the next thing: when will the next beat come? But the state of being of each note, harmony, sonority, or beat, is also a thing that exists for as long as it exists, and that has identity and importance.

(It is probably best to do the above exercises as a passenger rather than as the driver, unless you are very sure that none of it will distract you from driving. I wonder how different any of this will seem if we have nothing but self-driving cars on the road!)

Are there other little slices of everyday life that might illuminate aspects of rhythm and the role of time in music? Rain is an interesting one. There are pieces of music that are inspired by or attempt to depict rain. What is the rhythm of rain like? It is more regular when it is fairly light. What about walking? How regular is that, and how does walking respond to outside conditions? Can one walk while hearing in one’s head a passage of music that moves at a pace different from the footsteps?

It occurred to me a few years ago that I often experience trills as containing some of that simultaneously brief and infinite quality that I get out of my second driving exercise. I sometimes suspect that that is what trills are essentially about, though that is speculative and unknowable. The fast and unmeasured quality of (usually) the middle portion of a trill seems to take it out of time, while, like all music, it is in the end bounded and defined by time.

I will pick this up again next month.

Further thoughts on
counterpoint II: a miscellany

I mentioned at the end of last month’s column that I have been rethinking some of my ideas and practices about teaching rhythm and counting. This is the result of recent experiences teaching students contrapuntal voices. I have noticed—more than I previously observed—that moving from an understanding of the rhythm of individual voices to an understanding of the way the rhythms of those voices fits together is not always easy or natural for everyone. It is helpful to address it directly, and I developed some new ideas about how to do so.

As outlined in my April 2012 column, my approach to teaching rhythm and counting involves ignoring time signatures, bar lines, and the “one-and-two-and” structure that we often employ. Rhythm is about ratios, and my experience has suggested that grappling with ratios directly—understanding that a dotted quarter note lasts as long as three eighth notes, that a half note lasts twice as long as a quarter note, and so on—is the best way for a student to become comfortable with rhythm. This is true for a beginner learning to understand rhythmic notation for the first time and also for a more experienced student who is having trouble with a particular rhythm or needs a refresher course. This has the advantage of being simple and completely accurate. I have seen much more trouble arise when students worry about whether a particular note is “on the ‘and’ of two” than I have seen clarity arising out of this way of describing rhythm.

This is an over-simplified summary of an approach that differs from other approaches in emphasis, since the learning of note ratios is always fundamental. This is not the place for a complete restatement of what I wrote six years ago, or to explain why I believe that it is a good approach. You can find it in the above-mentioned issue of The Diapason, which I have posted at http://gavinblack-baroque.com/Diapason column April 2012.pdf.

When a student needs to work out the rhythm of a passage, I suggest counting individual notes in relation to a suitable short note value. For example, if that short unit is the eighth note, then each eighth note is counted as “one,” each quarter note is counted “one-two,” each half note “one-two-three-four,” and so on. This does not involve denying that those beats also relate to positions in a measure, but it just deemphasizes that at first. In learning the rhythm of a single line or passage, the texture of which involves all of the notes, voices, chords, etc., being in the same rhythm as one another (say in the manner of many hymns), this approach is very successful.

Therefore, it is also successful when learning individual voices of what will be a contrapuntal texture, regardless of its eventual level of complexity. So the following two “voices” (the pitches are not important) could each be counted as indicated in Examples 1 and 2. As with any rhythmic counting, the numbers have to come at an even pace.

Each of these examples contains nineteen eighth notes, and the two of them fit together. But the sort of counting that I have suggested would be extraordinarily hard, if not impossible, to follow in Example 3.

The first-order solution is that the rhythm of each voice should be internalized before the voices are put together. This is a goal worth striving for in the practicing process. However it seems sensible to readmit traditional measure-based counting at this point.

I have purposely created an example that is not divided into measures. But it could be, like Example 4, which then allows for the regular (“one-and-two-and”) approach to counting.

I believe that it is a good idea for this sort of counting to be secondary. To the extent that working out the rhythm of each voice separately is something that must be approached consciously and deliberately, I think that my per-note way of counting is useful. In addition to being sensible and effective in itself, it can promote an awareness of the importance of the middle and ends of notes, whereas “one-and-two-and” counting, especially with counterpoint, can focus attention disproportionately on the beginnings of notes. As soon as the measure-based number for a note has been spoken or thought, the next step is to see what notes in other voices correspond to the next counting numbers. This can shift a player’s overall focus toward a more homophonic feeling.

Thinking about applying the normal measure-based templates to counting multi-voiced pieces has made me more open to other ways of keeping track of the rhythm of pieces that are not derived from a strict awareness of the voice structure. For example, there is net or overall rhythm. Sometimes, while the rhythm of each voice in a contrapuntal passage is different from all the others, the surface rhythm is quite simple. That rhythm might be thought of as a new note every eighth note.

Here are a few examples of writing in which this concept might make things easier. Example 5 is from Bach, Art of the Fugue, Contrapunctus 8. Example 6 is from Mendelssohn, Prelude and Fugue, op. 37, no. 1 (fugue). Example 7 is taken from Bach, Well-Tempered Clavier, Book II, Fugue #2.

In the last of these, the net rhythm is more complicated, but still simpler than each individual rhythm. It is interesting that the first three measures of each piece have the same overall surface rhythm, which in turn is the rhythm of the fugue subject. In the opening measure, the fugue subject is alone, in the second it is combined with a countersubject that does not change the surface rhythm, and in the third measure only the surface rhythm alludes to or quotes that subject.

If the printing is neatly aligned, it is possible to derive all of the rhythm from the surface rhythm by how things line up. Sometimes it is useful to picture a cursor line going steadily from left to right and to identify whatever it touches as what is to be played at that moment. But this should all be secondary, understood to be forms of assistance, not a primary way of understanding rhythm.

Some observations

Here are a few more observations about counterpoint, some of which may be a bit abstract:

It is natural to think of counterpoint as being something that is easily differentiable from harmony. After all, there is counterpoint that exists outside the norms and expectations of tonality or functional harmony. But for most of the music that we play on the organ and other keyboard instruments, pieces that are rigorously contrapuntal are also completely embedded in harmony. This is something essential about the meaning of the counterpoint itself: that is, the way that the flow of contrapuntal voices/melodies comes across to us, and therefore the way that the directionality of the different voices interact with one another function, is determined crucially by where the voices are moving harmonically.

For this discussion, I have invented an exercise. Try playing a piece of contrapuntal music with all but one of the voices transposed, and not each transposed by the same amount. The simplest case would be something like Example 8, from the opening of the first Bach Two-Part Invention.

This is going to sound discordant, to put it mildly. But that reaction or judgment is not necessary. The point is that it sounds radically different from the “real” version, although, analytically, a good deal of the counterpoint itself remains unchanged. A little bit of this exercise, possibly even just the amount that I have written out here can be eye-opening for a student. (It is not bad as an odd sort of sight-reading exercise either, and it is much harder with more than two voices.)

A bit of exposure to this oddity can also make the ears more sensitive to the ebb and flow of harmony-based motion or pressure in contrapuntal works. Even though the interaction of voices is always shaped in some way by the separate voices’ harmonic direction, the evident force of that phenomenon is stronger in some places, yet weaker in others. It is also more divergent in some places and more in sync in others. The places where the force of the harmonic direction of different voices is most in sync are cadences and moments that feel cadential.

Many years ago, probably when I was still in high school, an older friend of mine recounted something that a friend of hers had said. This friend once removed was an extremely erudite and thoughtful music listener, thinker about music, and keyboard player. The comment that was relayed to me was: “Helmut Walcha’s problem is that he is too focused on the counterpoint.” I immediately felt that I was learning something—not so much about music or playing or performance, but about attitudes toward musical work. The first thought that occurred to me was that Helmut Walcha was not the one with the “problem!” The phrase “Helmut Walcha’s problem is . . .” really meant, “The reason that I do not like Walcha’s playing is . . . .” That comment and my reaction to it at the time taught me that many people take their own reactions for objective truth, and that it made me uncomfortable. That in turn has shaped a great deal of my own approach to my development as a player and teacher.

What does that comment tell us about counterpoint? I have never thought that it was an apt description of anything I heard in Walcha’s playing. That may be in part because I grew up listening to him, and, at that time, his approach was my own default. But it has made me sensitive to the question of what it would mean to pay too much attention to counterpoint and too little to other things.

My guess is that this commentator was probably referring to Walcha’s relatively steady approach to rhythm, and also, just as importantly, to his focus on consistently phrasing subjects. (For what it is worth, he was also reacting to hearing recordings, not to any conversations with Walcha or to any direct knowledge of the roots of his approach.)

This leads to a host of questions:

Is it correct that relatively steady metronomic playing is associated with bringing out counterpoint clearly, whereas playing that is rhythmically free is associated with harmonically derived music, or with an approach to playing music that is more focused on harmony?

What about the business of “subject phrasing?” If a phrase, theme, or motif happens more than once in a piece, is it important to articulate and phrase it consistently? Does doing that enhance the extent to which the counterpoint seems like counterpoint?

Is it important, or is it good or bad, for the notes of different voices that are written ostensibly at the same time to happen at exactly the same time? Could there be something good or necessary about notes being somewhat staggered at times? The reason that I associate this question with the Walcha-related anecdote is that in his playing, notes do line up rhythmically to a great extent. Furthermore, the above-mentioned commenter believed that Walcha was too focused on counterpoint. But there are many players and theorists who believe that counterpoint is enhanced by staggering: it is difficult for the ears to follow separate lines if the notes of those lines coincide with one another exactly. Which of these is true? Does it depend in part on habits of listening? Is it really about clarity of counterpoint, or is it about some other aspect of style?

How does this relate to thoughts about basic counting in counterpoint expressed above? I suppose that I feel most interested in the notion that any melody including a contrapuntal voice has some logic by which it might be stretched out or otherwise inflected rhythmically. Also, two voices that occur at the same time might have different logic. They might not line up all the time, but not because the phenomenon of their not lining up should be sought out, rather, because the internal logic of each has them manifesting slightly different timing. This may be why I am more interested in counting individual voices that are somewhat removed from each other and from the template of measures. But then all voices have to be together enough in the end so that the piece is coherent.

I present all of this as questions or ideas that are interesting to think about or, more to the point, to engage with in conjunction with students. None of it is hard and fast; none of it represents answers or conclusions.

One last topic: notes that we play have overtones. On the organ some of those are the inherent overtones of the pipes, while some are the higher stops added for a particular purpose. Especially in the former category, there are often overtones that, while they blend in enough to make the sound coherent, can also be heard as faint separate notes. And the rather cool thing is this: different overtones are relatively audible on different notes. So in playing a theme you might hear the twelfth over the first note, the fifteenth over the second note, the twelfth again, then the seventeenth, and so on. In Example 9, in this melody, you might happen to hear the higher (gray) notes as very faint, audible accompaniment to the “official” theme.

This is a kind of shadow overtone counterpoint that is always present, at a greater or lesser degree of audibility, and is present separately in multiple contrapuntal voices when they combine.

Gavin Black is director of the Princeton Early Keyboard Center, Princeton, New Jersey. He can be reached by email at [email protected].

Gavin Black is director of the Princeton Early Keyboard Center, Princeton, New Jersey.

Further thoughts about rhythm, part 4

After writing my last column and in the course of searching for further ideas about rhythm, I came across this quote, which was new to me and I quite like. It is from a review of Rhythm and Tempo by Curt Sachs published in 1953 in Journal of Research in Music Education, written by Theodore F. Newmann:

The essence of all musical expression is that of creating a feeling of organic movement. And that factor, above all else, which contributes to a work’s organic unity, its total feeling, is rhythm. Rhythm is not a matter of divisions of time and accent but rather a relation of tensions—the preparation of new expectations by the resolution of former ones. It is rooted deeply in the framework of all living organisms. It permeates the very body of music—its tempo, melody, harmony, and form. Rhythm is, in fact, the most vital element in music.

I might quibble about the absolute terms in which some of this is expressed. Rather than “the essence” I might say “an essence” or “one essential part.” Despite my focus on rhythm right now, I would not say that it is “the most vital” element in music—certainly one of the most vital, and one that, defined broadly enough, must be present.

What I really like is the part about expectations. This is a powerful way of framing the theoretical background for my second mode (as per last month) of determining rhythm in music. The ebb and flow of tension, the creation of expectations—and then the fulfilling or the subverting of those expectations—is the source of the rhythm of my second sort. And this operates at every level, from a whole piece to the motion from one short note to the next. Newmann’s quote is a beautiful expression of the notion that rhythm can be derived from and described as something other than counting beats. For me, this description also establishes that the expectation-based sense of unfolding rhythm is primary. It is not just a system of departing from the rhythm created by the beat structure. If anything it is the deeper, more essential source of rhythm.

How is it possible to reconcile the simultaneous existence of these two radically different ways of defining, projecting, and perceiving rhythm in one piece of music? I do not know if I have a thorough or systematic answer to that. Both forces being discussed here are real, and therefore they do coexist pretty much regardless of whether we know why. Not having an answer to how or why that works may reflect something about how flexible it is in actual practice. It may also relate to the phenomenon of people responding differently as listeners to the choices made by performers. After all, expectation is a listener’s experience. And if a piece or passage of music creates and then fulfills or subverts expectations, then the actual flow of that experience is very likely determined in large part by the listener’s experience, training, philosophical stance, temperament, and so on. Different listeners will hear what is being performed in a way that is literally different.

If a piece of music is ostensibly measured and structured in a regular meter, then any overt conflict between the two approaches to rhythm would take this form. The intuitive, listening-based approach might well lead to something other than playing the rhythm as written, and playing the rhythm as written might lead to ignoring the ebb and flow of tension and expectation. One axis along which a solution to this conflict might exist is that the sense of pulse is remarkably strong and resilient.

My experiment with Helmut Walcha’s recording of Sweelinck, as described in my December 2021 column, page 11, is a manifestation of this resilience. As I wrote then, my reason for testing out the literal metronome steadiness of the recording in the first place was that the performance came across to me as remarkably and inexorably steady. So it was interesting and telling that the metronome reading varied so much across the piece. I have conducted similar tests on other recorded performances. For example, about eighteen different recordings of the beginning of the first movement of Charles-Marie Widor’s Sixth Symphony yielded similar results. Some of my odd driving-based experiments about time have this same import.

A Chopin anecdote

There is an anecdote about Chopin that I have come across in writing twice over the years, though I cannot remember where exactly. One version framed the story around his mazurkas, the other around his waltzes. It works equally well for either, as they are in ¾ time. The point is that a friend of Chopin’s said to the composer that when he played one of his own waltzes there were four quarter notes in each measure, not the notated three. Chopin denied this. The friend made him sit down and play and kept strict track of the timing, and Chopin was indeed taking four quarter notes worth of time between bar lines. This story implies three things: that Chopin was playing very freely, holding some supposed quarter notes for as much as twice as long as they “should be” held; that there was nonetheless something that could be identified as “the quarter note;” and that Chopin himself intuitively accepted what he was doing as being a manifestation of three quarter notes per measure.

It seems also possible that if the intuitive, expectation-based approach to rhythm is applied thoughtfully in a way that is really derived from careful, involved listening, then it might enhance the sense of regular pulse while being less regular than a purely beat-based rhythm would be. If the sense of momentum that is associated with the concept of regular accent-based meter is as robust and as difficult to destroy as I suspect, and if an expectation-based shaping of rhythm can make the ebb and flow of intensity the most compelling that it can be, then it could make sense that the latter approach would make the accentuation patterns that define regular meter more rather than less convincing.

The actual results of the two approaches do not always have to be very different from one another. For some performers and listeners experiencing repertoire, the listening-based intuitive approach might well yield results that are quite regular in their actual rhythms.

In musing a bit about how to introduce this idea to students, I start by quoting something that I wrote in my column from April 2015, page 17:

I often suggest to students the following practice tool. Once they have identified a spot where they want to make a rhythmic gesture (usually of the sort that might be described as “rubato”) they practice that gesture, in the privacy of the studio, in as exaggerated a manner as possible: take the risk of executing a gesture that is utterly tasteless, mannered, “schmaltzy.” This is to counter the fact that we usually only visit the gestures that we think we want to make “from below” (so to speak), that is, only compared to and judged in comparison to not making such a gesture, or to a modest version of the gesture. This stems from and then reinforces a philosophy that teaches a kind of reluctance about such gestures. If you hear a rhythmic inflection from both sides, you get a different sense of exactly how it might be effective.

When I wrote the above, I invoked the word “rubato.” Now I would be inclined to avoid that, since it is a term that relates the flow of rhythm to the underlying regular meter. The relationship is real, but I would want to encourage the student to move away from considering that relationship to be primary or of overriding importance. In applying this practice suggestion with many students over many years I have observed that almost everyone finds it easy to identify spots where they intuitively want to create rhythmic gestures that are not identical to what a strict reading of the metric notation would suggest. While these gestures fluctuate in number from piece to piece, it is more common for the student to believe that their interpretation of some of them are wrong.

Experimenting with notable works

As an experiment, try giving a rhythmic inflection different degrees of magnitude—including the highly exaggerated, but not limited to that—and try to experience and analyze what is happening with each different shape. For example, consider the very beginning of J. S. Bach’s Toccata in D Minor, BWV 565. Play the opening mordent and then listen for when you think that the following scale passage “wants” to come in. Try taking longer next time. What has changed? Have you lost the moment when the impact would have been greatest, or does the added suspense increase the impact? Does it sound like you caught a sort of wave of expectation when you initiated the scale passage, or did you miss it? If you missed it, could there be another one coming up, different yet also fulfilling?

Questions like these are easy to highlight with a moment in an intrinsically free passage like this, but they are potentially present all the time. Returning to the pieces that I invoked at the beginning of this series of columns, I would say that the first barline in the opening of the first movement of Widor’s Sixth Symphony is also a place where the dynamic of this sort of rhythm is clear. A student can play it metronomically and then try various timings, listening as closely and open-mindedly as possible. What is gained or lost by various different timing possibilities? What does the timing at that barline imply or suggest or even necessitate about playing the second measure itself? Would the answers be different if this were not the opening of the piece? Is the answer to that question different if we stipulate either that it was a recapitulation or, on the other hand, that it was a new idea introduced in mid-piece? Does the use of this motif later in the movement as written affect choices here at the beginning?

It is interesting to listen for intuition-based timing of quick notes within a texture. In Example 1, in this Mozart Piano Sonata in A Major, K. 331, how can we determine exactly where to play the sixteenth notes?

My own approach to figuring this out would be to leave those notes out at first, making the rhythm of the first half of each of the first two measures the same as the rhythm of the second half. I would play this a few times, just like that, as if it were the piece. Then I would play it a few times hearing the sixteenth notes in my head, but not playing them and then add those notes back. I often do not know and cannot seem to determine as a matter of counting and calculation where I want to place short notes after dotted notes. To use a piece I have played a lot, I have no idea exactly what ratio I give to the pervasive dotted rhythms in Bach’s Prelude in E-flat Major, BWV 552i. I can only hear whether they do or do not seem to do the right sort of job in creating momentum from one long note to the next.

When one asks a student to listen carefully to every moment in a passage and to discern as much as they can about rhythm through that listening, interesting things happen. Some of these are advantages while others are pitfalls. One great advantage is that this tends to keep the student from rushing. Not everyone has a tendency to rush, but most of us do at least some of the time. One reason for rushing is that we pay the most attention to the things that we must do physically—and for organ and harpsichord that means initiating the notes and releasing them. As soon as one note is safely played, there is some pull toward either the moment of release or the moment of initiating the next note. The sort of listening that I am describing here focuses the attention as much on the middle of a note as on the beginning and end.

Listening this way tends to lead to slower playing. That has been true for me, to a fault some of the time, though it is a perfectly natural outcome of this approach. The way to counter this tendency is through awareness and honesty. This listening approach is both a practice technique and something to bear in mind while actually performing. As with any other sort of practice, it is important and good to utilize it at a slow tempo, but as with any slow practice, it should be ramped up to the desired tempo in a systematic way.

It is natural to prepare for the moment when you believe the next note should occur and then play that note. This makes that moment in the music late by definition and leads to slow playing. It is probably acceptable for this to happen early in practicing a passage, as long as you are aware of it and work on evening it out as you get to know the music better. I have also been guilty of this in performance, not recognizing it clearly enough in practicing, and therefore not working to smooth out the process.

This is analogous to something that can happen when using a metronome, when one listens for the metronome beat and then plays the note. Something like this can also happen with releases. If you hear the instant at which you want to release a note and then release it, the release is late. A release that is executed by pushing off—down, then up—will also be late if the downstroke is timed to the desired moment of release. There is a sort of leap of faith when executing something, whether the timing of the beginning of a note, as we are discussing here, or of the end, without waiting to confirm that the looked-for time has arrived.

A few miscellaneous points

What is the meaning of any given note value? We know the answer: a quarter note is equal to two eighth notes or to half a half note; a whole note is four quarter notes, etc. But here’s another definition: a quarter note (or whatever note value) is whatever a listener will hear or accept as a quarter note. This relates directly to the Chopin story above.

Watching a TV game show recently, I heard the host say to a contestant, “Time is ticking away!” Why? Why is time defined as made up of discrete entities, ticks of a clock? Is it just because old clock technologies produced audible ticking? Or is there really something more fundamental about time that requires us to think that way? This vignette from a fraught moment in a game reminds us that thinking of time as ticking tends to create a sense of impatience or even panic.

The beating of the heart is sometimes evoked as a possible source for the notion of beats in music. This makes sense because the heart does beat fairly regularly in the absence of a medical issue. That beating is audible to the one whose heart it is, though not usually in a way that rises to the level of conscious awareness. If you quiet the outside world, you will easily hear your heart beating. But no heart beats as absolutely steadily as the beat we hear from a metronome. If the heart is a model for musical beats, maybe it should confer some flexibility onto those beats. If the heart is the model for a beat-derived sense of musical rhythm, breathing should be the model for a sense of rhythm based on expectation and fulfillment.

What is rhythm? That question has recurred to me as I have thought about and written about rhythm over the last few months. This is a prime example of putting the cart before the horse. After all, how have I been thinking about rhythm without first sorting out what it is? But there is no one answer to this very fundamental question, and we all work with rhythm without having established a clear definition.

I searched the internet with phrases such as “What is rhythm?” and “rhythm in music.” I was not looking for any answers as such, but to get an idea of some of the “headline” ways in which any sort of definition of rhythm is encapsulated. The results were very interesting. There were two basic kinds of results: 1) simple descriptions of some of the mechanics of the way that rhythm is depicted in some sorts of music, such as “Rhythm in music is the regular motion of half-notes, quarter-notes, etc.” or “Music has a regular beat sometimes indicated by a metronome marking;” and 2) complex but certainly intriguing philosophical discussions of concepts of rhythm.

One set of answers to the question of rhythm is contained in the fact that we work with rhythm when we make music. Music moves through time. Rhythm is predicated on the phenomenon of time passing as we listen to music or create it.

Music moves through time in a way essential to its nature. The same can be said for dance, though I have less experience talking to people who have had a deep involvement with it. Drama—live theater, television, movies—also moves through time, but in a way that seems meaningfully different as much of what is going on is dialogue. The semantic content influences the way one experiences the flow. (This is in part also true of vocal music.) Painting, drawing, sculpture, and other visual and graphic arts do not move through time the same way. Time passes as one experiences that sort of artwork, but visual focus is up to the viewer, as is shifting of that focus as a viewer encounters the work of art. There is no set time that the rhythm of viewing a painting, for example, will occupy overall, and no set ratio between times allotted to different parts of the whole. There is also no set order in which those parts will make themselves felt to the viewer, including any aspects of repetition.

I have always thought it fascinating that if you hit the pause button while a CD or any music file is playing, there is silence, whereas if you do the same on a video recording, a still image is displayed. Music has no existence without the passage of time, without motion and change.

Would a piece of music that existed in time but never changed seem to have rhythm in any sense we would recognize? This is an abstraction, since we do not encounter music that never changes. Maybe the closest we could come would be to play a note on an organ and hold it forever. But even that would change. There is always a miscellaneous fluctuation in the sound or in the way that the sound reaches the ears. Maybe a computer-based instrument could create a sound that really would never change even at the level of what the most sensitive instruments could measure or any ears could hear. If someone were to listen to (part of) such a piece would there be anything that they would experience as rhythm? Maybe not, but what occurs to me is that they would experience the inner rhythm of their own shifting reaction to the unchanging sound.

I see from framing the last thought the way I did that I want something to happen in order to accept that the quality of “rhythm” is present. So possibly rhythm is not just music moving through time, but things happening—things that we can hear—as music moves through time. I understand that this is sort of obvious and phrased in a deliberately simple or even naïve way. Maybe it is even really a tautology: if we are listening and something happens, we notice it. Since it happens across time, there is rhythm to it, if we define rhythm broadly or just decide to apply that word to that phenomenon.

But as the first set of answers that I found in my internet search reminds us, there is something specific about how we use rhythm, ​​not necessarily what rhythm is in most of the music that organists, pianists, harpsichordists, or other classical musicians grapple with. That has to do with regular beat, which presumably means that the “something happening” happens at regular time intervals, and some sort of way of grouping that regularity. This is expressed with a naïve pretense that we do not know it perfectly well already. It points to a vast set of questions as to why this should be standard. How did it evolve? What does it do that is different from what we might be able to do with musical sounds not organized that way? Are there necessary relationships between this regularity and other aspects of music in this range of styles: melody, harmony, texture, and so on? To answer these questions is not the point here, since the truth is that no one knows, although there has been a lot of interesting research and speculation.

So where does any of this leave us as to the teaching or grappling with rhythm in our own playing? Grappling with rhythm means deciding when to play notes. The thought that has formed in my mind over the last few years is that there are two fundamental ways to do this, and they are opposite. They are both important, but very different in how straightforward they are to describe.

The first of these is the normal one for most music that we play. It is the one that we engage with all the time, the regular beat and meter phenomenon. I like to describe it this way: there is a regular beat that exists outside of the piece and before the beginning of the piece; the notes of the piece will fit in with that beat once the piece begins. That beat defines one note value, and the other note values are all clearly defined in relation to that one. There could be many other ways of describing this same thing. And to avoid its being a caricature, we should add that although the regular beat exists before the piece begins, it is in a sense a separate entity from the piece itself, and to a large extent governs the motion of the piece.

This is consistent with this concept that the actual notes sometimes deviate from the place where that regular beat says they should have been. That then becomes a matter of taste, of judgment, of awareness of a composer’s intentions and so on. Although this deviation is normal and frequent, it is defined as an aberration, and therefore it is often felt to require specific justification.

The second, opposite pole is a lot harder to describe. At any instant in a piece of music, there is some prevailing sound—a note, more than one note, the dying away of released notes, the ambient room sound, any combination of these things and others—and something about that sound will suggest when it feels right for the next sound to happen. That suggested moment is the appropriate time to play or sing the next note or notes. This perceived sense may sometimes be caused by the phenomenon of notes having come at a certain regular pace up until that moment. But it may also be caused by other factors that have nothing to do with regular pulse. These could include something about the inner behavior of the sound in the time since anything was last expressly played, shaped by the nature of the performing medium or by the room acoustics; anything about harmonic development and the ebb and flow of harmonic tension; something arising out of the desire for a certain kind of forward momentum; a need either to sustain a mood or to change the mood; and so on.

Each of these two poles can be found to a greater or lesser extent in any piece or passage that we play, and they interact in an infinite number of ways. The second idea is often most obviously at work in recitative. This only starts with “official” sung recitative. It also includes instrumental passages that imitate recitative and are marked as such—for example, the section in the first movement of Widor’s Sixth Symphony that immediately follows the opening about which I wrote in my September 2021 column (pages 10–11), or sections of Bach’s Chromatic Fantasy and Fugue, BWV 903, or Toccata in D Minor, BWV 565.

More interestingly, perhaps, it includes passages that strike us as being in recitative style without being marked as such. This applies to many other sections of those same Bach pieces, for example. The opening of BWV 565 is full of fermatas, unmeasured and written-out arpeggios, and tempo changes, all of which add up to making it an absolute necessity to find some source other than a grid of beats and accents to figure out when each successive note is best played.

The opening of Bach’s Fantasia in G Minor, BWV 542, is not aesthetically like the opening of BWV 565. It has no fermatas, tempo changes, or other direct suggestions that the rhythm suggested by the mapping of the notes onto the regular meter is not perfectly viable. When I played this piece decades ago, that was how I derived my sense of when the notes came. I recall being very focused on counting it correctly. Now, when I sit and read through it, I am more focused on listening to each sonority and trying to feel or intuit when what comes next should occur. This is not in the absence of an awareness of how the different note values stack up against one another or where the strong beats are. In this case it supplements that. The feeling of pressure or momentum to move to the next note or cluster of notes comes in part from an awareness of the imperatives of the beat structure. For example, everything about the opening chord and the act of moving away from the opening chord comes from sonority, ideally including whatever I can hear of room acoustics.

So how does this concept differ from a simple acknowledgement that it is possible to play ordinary measured music either rather strictly or more freely? This is a common though not uncontroversial subject to debate, be it in a friendly or heated way. One point I like about this approach is that it seems to be true. That is, whether it is something a composer intends or a performer tries to bring out, it is palpably the case that each moment in a piece of music has some sort of inherent momentum. It can vary in strength and be perceived differently by one person and another. After all, this momentum is something that arises in part from what a given performer or listener wants to do with a piece, their prior aesthetic predilections and tastes, and a host of other factors. This is never something that can be defined by one person for another: it must be heard and felt.

One way to demonstrate that this intrinsic momentum is real is to stop a piece at a random place. A jarring quality that experience creates comes from unfulfilled expectation or broken momentum. That momentum cannot be just that of a regular beat pattern—they stop all the time, and it does not bother us.

Another matter that I like about this concept is that it ties in with the notion of playing a piece as if you were improvising it. Even if your sense of the directionality and momentum of the sonorities does not lead you to do anything very different from a fairly strict rhythm, as soon as you start listening for that momentum you are behaving like someone who is creating the piece.

Also, just as a matter of my own discipline, trying to listen like this prods me not to let my attention and hearing faculties wander too much, to pay attention and to care about each note. This is one of the key points in working with students around this idea.

Next time I will write in some detail about that: how to introduce this idea to students, what some if its advantages are, and what some of its pitfalls are, how to help students reconcile this approach with whatever amount of respect to the notated rhythm is due, if that ever rises to the level of conflict. I will end this series about rhythm with notes on a few stray ideas and observations about rhythm that are either germane to the above ideas or interesting on their own account.

Repetition I

Recently I have been thinking about situations in which something in a piece of music happens more than once. This encompasses out-and-out repetition—which may be written out or may be indicated by repeat signs—true da capo, the structural element known in some circumstances as “recapitulation,” recurring sections in rondo forms, and any recurrence of a passage identically—even when it is not da capo. This also includes the ubiquitous practice of using recurring motifs—fugue subjects, other motifs treated contrapuntally, any sort of leitmotiv, the repeating bass line of a chaconne or passacaglia, and so on. I want to share some of these thoughts in this column and the next.

These columns will be filled with questions and speculation about a number of different aspects of repetition and how we come to expect it. I will also offer a few practical thoughts about what the concept of repetition and recurrence means for teaching, learning music, and performance. The value of this speculation for teaching lies mostly in the possibility that students may find it interesting, and perhaps it will lead them to further exploration. I also have thoughts about ways in which focusing on repetition and recurrence can help with the practicality of the learning process.

How many times shall we repeat?

Let’s begin with a curious example of repetition. Namely, why did Bach write the first Prelude of the Well-Tempered Clavier like the passage in Example 1 rather than like in Example 2 or Example 3?

In Bach’s composition, the first half of each measure is repeated exactly to form the second half of that measure. (It is so fully exact that he could have used repeat signs.) This pattern persists until the ending, where it is altered to lead to a satisfying cadence. We are deeply accustomed to the piece that is created by this patterning: it is one of the most familiar in the repertoire. But a piece on either of the other two models would have been a perfectly valid musical entity as well. If he had written it one of those ways, we would be deeply used to that.

Or would we? Maybe the “real” version is somehow actually better. I have played through the entire piece with both of these patterns, and it is my feeling that they work. Each one suggests something different about performance. The most striking of these differences is that the three-times-per-measure version feels to me like it should go a lot faster than the other two. It ends up reminding me of this passage from the longer Praeludium in E Minor of Bruhns, at measure 95 (see Example 4). If Bach had written version 3, we might speculate that it was influenced by this passage, which he may have known.

I tend to play the once-per-measure version slower and with more rhythmic freedom than either of the others. It is interesting that a mainstream analysis of the piece, which would be an analysis of harmony, since it is such a pure chord-based piece, would be essentially identical for each of these three very different pieces. I doubt that we can answer the question of why Bach wrote it one way rather than another, or indeed whether one way is better. It seems pretty clear that one repetition is very different both from no repetition and from two or more repetitions. Would writing the chord pattern four or five times in a row be as different from three as three is from two? At what point would it become ridiculous? If each chord pattern were repeated sixteen times, it would be a particular kind of extreme statement: outside the realm of Bachian music or most of what we ever encounter, but valid nonetheless.

So what about repeats as such? We accept it as normal that in many pieces of music, something—a definable passage that has a pretty clear beginning and an end, and not typically just a half-measure—will happen twice in a row. This is a defining trait of late Renaissance and Baroque dance forms: which usually consist of two sections, sometimes more, each repeated, like A-A-B-B. We accept that as routine, but, as with the Bach prelude, we would never expect each such section to be repeated more than once: A-A-A-A-etc.-B-B-B-B-etc. Same question, here applied to bigger increments of music: why not? I mean, why not in both directions? Why is twice in a row effective, and why would more than that not be? The same question applies to other places in common musical structures where repeats are routine, such as the first section (and sometimes other sections) of works in sonata-allegro form.

Tension and release

The most interesting effect and purpose of a repeat does not arise from or during the repeat itself, but rather at the moment when the repeat does not happen that second time. That is, the repeat sets up a tension (about whether it is indeed going to happen yet again, and whether it will end), and the motion to the next section relieves that tension. This tension is a fiction that we allow ourselves to experience, or that the music allows us to experience. After all, we know that the repeated material will not go on forever. We know this for certain if we know the piece, and we know it essentially for certain even if we do not. But there is something in listening to music that allows our reaction—our appreciation—to feel surprise even when listening to something that we know intimately.

I strongly suspect that this dynamic is one of the explanations for the persistent tendency for all of us to feel that “taking the repeats” is a different matter for the first section of a piece in binary form than for the second section. When we come to the end of the second section, first time, the suspense-and-relief scenario has just happened once, and our appetite for it is perhaps satisfied. Also, we know that when the second section gives way, whether after a repeat or not, it will give way to the silence following a cadence, and then perhaps another movement or another piece. That is a less compelling change of direction. This may also tend to explain the almost universal practice of not taking repeats in da capos of the sort represented by the return of the Aria at the end of the Goldberg Variations of Bach.

This may also tend to explain why repeats of sections happen only once (AABB). If part of the point is to set up the moment at which the repeat gives way to something new, it is important for the repeat not to wear out the listener’s interest. Once makes the point, more than once risks irrelevance.

I once heard an experienced concert pianist, playing from memory, repeat the opening section of the first movement of Beethoven’s Waldstein Sonata twice (that is, play the section three times). My impression was that he accidentally started the first ending after the second time through, and that led inevitably to replaying the whole section. I remember having the following reactions: first of all, that it was nice to hear the passage an extra time, just because it is wonderful music; but second to feel a little bit of panic that it would never end!

This experience makes me ponder, how does our enjoyment of a particular passage play a role in interpreting or perceiving repeats? Does this apply in particular to pieces that are short enough that repeats will not try anyone’s patience? In the Bach variation set on the chorale O Gott du frommer Gott the composer follows the structure of the chorale melody as far as repeats are concerned, repeating the first half of each variation, but not the second, at least in most variations. In Variation V and Variation VII he does indeed repeat the second half as well, violating the relationship between the chorale melody and the shape of the variation. Why? Well, I noticed practicing the piece that the opening measures of each of those second halves was especially charismatic, unusual in texture, and harmonically rich. These are moments that I would find myself practicing over and over again just because they are so cool to play and hear. I had to remind myself to practice other bits that needed practicing more. I noticed this before I stopped to think about those repeats. But I wonder whether Bach put in those extra, musicologically “wrong,” repeats just because he liked that bit of music! This is a kind of non-rigorous, taste-driven choice making that we do not associate with JSB. But perhaps we are wrong not to do so.

Do repeats generally have to do with symmetry or logic? I think that we often assume that they do or that they should. If the allemande of a given suite is A-A-B-B, then presumably the courante and the other movements will also be. Otherwise there would be an imbalance. When playing the above-mentioned Bach variations, I have been aware of a pull either to omit the extra repeats in the two variations that have them or to add repeats to all of the “B” sections. And this would be in apparent direct contravention of the wishes of the most august of composers. I once had a student who was working on the Goldberg Variations, and who proposed to omit almost all of the repeats, but to take one pair of repeats, namely in Variation VII. He had an interesting registration idea for that movement, which required four rather than two sections. I have to admit that this bothered me: no logic or symmetry, just going with an aesthetic choice. It is just an interesting light on what I (we?) sometimes want out of repeats.

To play the repeat or not, that is the question.

This brings us to something pretty concrete and specific: “should I take the repeats?” Essentially we never find ourselves wanting to omit any part of a piece that is not a repeat. I have never had a student ask me about a through-composed piece, “should I or should I not play measures 9–16?” or anything like that. Even though we accept repeats, we clearly do not quite accept them. That is, we do not accord the notes indicated by repeat signs absolutely all of the status that we give to other notes. We give this higher status to passages that are a lot like, or exactly like, earlier passages, including actual da capos. Only when a few measures of material are repeated right away, and with such exactness that it can be notated by a repeat sign, do we consider it an issue whether to play those notes or not. In fact, we likely think that way only if the repeat is actually notated by a repeat sign. Once in a while a repeated passage, a literal, full repeat, is written out. In such a case I do not often hear a student, or anyone, suggesting cutting out the similar bits.

For an interesting side note, in his notes for his 1950s recording of Scarlatti sonatas for Columbia, Ralph Kirkpatrick said of his not taking the repeats in the pieces, almost all of them in binary form with repeats indicated, that if listeners wanted to hear them again, they could play them again on the record! This probably indicates as much about his skeptical attitude towards recording as it does anything about his approach to repeats.

As for myself, I recognize a strong tendency to want to take all repeats, in whatever way they are indicated by the composer. I think that this has to do in part with my having spent my life absorbing the ethos of “the composer is always right.” That ethos has grown pretty steadily over the last couple of centuries and has found one expression in certain aspects of the early music movement. And I believe there is a certain logic to that. If a composer put in repeats, why shouldn’t it just be routine to play them? I have this stubborn feeling that there really is not any reason to single those notes out for omission. I also recognize that this could be a different story for pieces that were written in a style that made repeats a matter of routine—early dance movements, say—than for pieces written with no assumptions about repeats—nineteenth-century sonatas, for example—as to which composers made choices about repeats on a custom basis.

I also notice that, of course, repeats make a piece longer. That may seem trivially obvious, but the magnitude of a piece in time is a valid part of its aesthetic. The difference in length between a Haydn or Mozart symphony and a Mahler or Bruckner symphony is of course not even close to the whole difference. But it is not meaningless or insignificant either. If you take no repeats in one of the Bach French suites, it becomes a short piece. There is at least one recording of all six of those pieces, by Thurston Dart from 1961, that managed to fit the whole collection onto one LP. Independent of anything about the playing or the instrument (clavichord) or the recorded sound, that presentation of the whole collection as being that short feels different from a recording twice as long that would have had to occupy at least two LPs. (I say at least because it was a rather long LP.)

Making a piece long by repeating each of its parts is not the same thing as making a piece long by composing a longer amount of new material. And one thread over the long arc of western classical music has been to look for ways to create bigger structures without literal repetition. But in making a choice about whether to take repeats or not, the effect on overall length is part of the equation.

In keeping with some of my thoughts above, I do suspect that when I hear a performance of a piece that features repeats as part of the structure, but are ignored, I usually feel disappointed by the time the second section appears. If the suspense created by the repetition has not been allowed to build, then the resolution of that suspense through moving on to something new cannot have the power that it was meant to have.

To be continued.

The Art of the Fugue, part 7

I begin this column with an account of something that happened in a recent lesson, something surprisingly germane, by complete coincidence, to what I had already been planning to write about this month. I notice recently that between when I finish one column and when I am due to start writing the next, something often crosses my path, completely by chance, that is relevant to what is coming next in the column. Sometimes it provides an example, other times an interesting sidelight. Often it essentially ratifies and strengthens what I had been planning to write; yet sometimes it suggests a bit of a change.

Before I get to my story, it occurs to me that the fact that I encounter these coincidences as often as I do is itself a commentary on the philosophy and purpose of the column. My hypothesis is that everything is relevant to performing music and to teaching others about music and performance. I believe everything we encounter in life might be potentially fruitful for our music and our teaching.

This series of columns on The Art of the Fugue is meant to be an account of my working process. Part of that working process is to be aware of anything that might lead to an interesting thought about the piece or the performance. If we notice things and assume that they will be interesting and relevant, we cannot lose. If they are relevant, the gain from having noticed them is clear and describable; if they are not, the thought process sparked by noticing them is still rewarding.

The teacher-student experience

A week or so ago a harpsichord student of mine brought up The Goldberg Variations, partly as a step in thinking about whether to work on the piece, partly to ask a few questions. What is the overall structure of the piece? Did the composer write the movements in the order in which we see them? Does the order matter in performance? If so, why? Is it about individual transitions or something else? We talked about all of this a bit, mentioning “official” sources of structure and continuity. These start with the basic fact that each variation shares some version of the same underlying harmonic pattern and includes the phenomenon of every third variation’s being a canon. We talked about the formal layout of the canons, in which each one is a canon at a one-greater interval—Variation 3 a canon at the unison, Variation 6 at the second, Variation 9 at the third, and so on, up to Variation 27, which is a canon at the interval of a ninth. We discussed, but did not resolve the question of whether this formal layout is palpably meaningful to a listener, or is just meaningful as a formality.

Then we tried a live-action version of something similar to an experiment I described in a recent column, about electronic listening. I played a movement, or the last few lines of one, and my student turned some pages at random. I then started the movement on the new page, randomly chosen, after the one that I had just completed. We did not go through the whole piece in this fashion—not enough time, and other things to do. But we got a good sample of what it felt like to move from one variation to another when that transition is neither what the composer had in mind nor what we are used to.

The result was that each transition sounded fine (“worked”, nothing jarring) but also sounded odd. The oddness came from the confounding of fixed expectations, fixed by years (or for me, decades) of listening. It is hard or impossible to sort out what each of those transitions would have sounded like to someone who had never heard the piece before. I strongly suspect that they would each have been just fine, but I cannot be sure of that. This was a reminder that structure comes in part from expectation. The kind of expectation that comes simply from having heard the work before can be very powerful. But I assume (or perhaps I hope) that expectations created intrinsically by the music are even more powerful.

Sources of overall structure in The Art of the Fugue

It is my working hypothesis that in The Art of the Fugue there are two main sources of overall structure. The first is the nature of the theme itself and the way that it sets up other themes and musical gestures to be meaningful. The second is the recurrence of specific themes. I mentioned both of these in my 1985 program notes that were reprinted here in two recent columns. But I want to delve into both of them in more detail in this column and in next month’s. Of these two ideas, the recurrence of specific themes is, perhaps, the more clear-cut. It is not remotely unusual to include as a source of continuity and of structure. This includes all of the uses of the main theme and its variants: the top level of what we get by contrapuntal analysis. But there is also more below the surface.

My thoughts about the nature of The Art of the Fugue theme are perhaps more speculative. That theme starts on a note, goes up a perfect fifth, goes down a major third, then down a minor third, then down a semitone. After that it goes up by that same semitone, and up and down by step until it ends. Every musical theme, of course, has some pattern of intervals that I or anyone could describe in words. What is striking to me about this theme is how comprehensively, systematically, almost encyclopedically, it lays out all of the intervals that define tonal music. They occur in what could plausibly be considered their order of importance, there is essentially no redundancy: almost no interval is repeated until each interval has been heard (the exception is the semitone), and the inversions of intervals are assumed rather than stated.

My own experience as a listener has been that throughout the piece I hear any interval as a reference to this theme. This starts right away, as soon as there is anything to hear other then the theme itself. The scale notes that make up the counterpoint to the second entry of the theme at measure 5 fill in the opening interval of the theme itself (Example 1).

Does a listener hear it this way? Does a listener spontaneously think, “that is that same theme” or “that is a reference to that same theme?” I believe I do, and that I began to do so after repeated listening years ago, before I thought that I might have any idea why.

The chromatic countersubject in Contrapunctus III seems to arise out of that original semitone that is the interval from the fourth note to the fifth note of the opening theme. The leaping thirds that are one of the characteristic gestures of Contrapunctus IV seem to answer the thirds that make up measure 2 of the theme. Contrapunctus V plays around further with the idea of the third. First of all, when the theme comes in the second time, it is a third away from the note that accompanies it. (In Contrapuncti I–IV, this has always been the interval of a fifth.) Second, this movement contains a pervasive gesture that is the interval of a third filled in by step, and there are also a lot of parallel thirds!

The thirty-second-note flourish that pervades Contrapunctus VI, and that is a principal justification of the heading of this movement as being “in French Style,” sounds like a reference to the last four notes of the opening theme. The mordent-like figure that enters Contrapunctus VII in measure 3 is related to the semitone-based gesture in the opening theme going (in Contrapunctus I) from the end of measure 2 into measure 3. For me, the nature of the opening theme itself turns the whole piece into a tapestry of familiar, known, referential themes and gestures.

A complement to this is the gradual introduction of the octave into the picture. The opening subject, while displaying all of the discrete intervals of its tonal language, almost pointedly fails to encompass an octave. However, the first thing that happens after the filled-in reference to the opening interval that I pointed out in Example 1 is a drop of an octave: the first octave in the piece. When the second voice to enter (the soprano voice) finishes its statement of the fugue subject, it immediately makes an octave leap: one not in any way required by the counterpoint or harmony (Example 2).

Then, later on, whereas each of the first seven contrapuncti begins with the interval of either a fifth or a fourth, Contrapunctus VIII beings with a step, a very striking change. While the compass of the theme of each of the first seven movements has been never more than a sixth, the compass of the opening subject of Contrapunctus VII is a tenth, but one that could also be seen as outlining an octave. (The first and last notes of the subject are an octave apart, and the note that creates the tenth is off the beat and somewhat ornamental.) Then Contrapunctus IX begins with the brand-new gesture of a leap of an octave. The compass of this subject is a ninth, and that of the next movement a tenth, so that we are in a region of expanded compass of themes.

All of this is enough to make me feel that it makes sense to say that the subject or fundamental building block of The Art of the Fugue is not “the AOTF fugue subject” but the very concept of the melodic interval. This in itself does not create structure in the sense of linear shape. But it establishes the conditions for the creation of that structure.

The repetition, recurrence, and referencing of identifiable individual themes is the foundation of counterpoint, and probably the major defined source of contrapuntal structure. At the level of “this theme is the inversion of that theme” or “this theme is the same as that theme, but with altered rhythm,” this is fundamental and definitional. And there is an abundance of that sort of correspondence in The Art of the Fugue. What interests me beyond that is the more fleeting or hidden thematic connections. There are quite a few of these in the piece. The phenomenon that I have tried to describe above (the tendency of this piece to permit any interval, even in an isolated occurrence, to seem meaningfully thematic) is a background against which it becomes clear, I think, that small, individual thematic connections are meaningful to a listener and almost certainly intentional on the part of the composer. Some of the ones that stand out to me are as follows:

This seems to be an isolated event in the bass line in measure 35 of Contrapunctus IV. The notes seem like a filler in a kind of quasi-cadence (Example 3).

All of the eighth-note motion in the movement prior to this has been by step, as is almost all of it after. This is then picked up in Contrapunctus X (Example 4). The figure that enters in the tenor voice at the beginning of this example is passed back and forth among all of the voices and then culminates in an exact statement (at a different pitch level) of the theme from Contrapunctus IV cited above.

The beginning of the main theme of Contrapunctus X (Example 5) is referenced in passing in the bass voice near the end of Contrapunctus XI (Example 6) and again in the middle of the final movement (Example 7).

That latter one is an “answer” rather than a literal quoting of the motif. Are these parallels valid? It could certainly be argued that the fragments of themes that I am pointing to here are just routine cadential figures or other tropes that are too non-specific to be meaningful. I do not hear them that way. Again, I think that the structure of The Art of the Fugue subject itself predisposes any motivic entity to be significant, and the ways in which some of these fleeting motifs are deployed seems too systematic to be non-intentional.

As I said above, I think that there is more to be gained by assuming that correspondences are real than by resisting hearing them that way. Next month I will bring forward several more examples—perhaps even more important to the actual overall structure of the work as a whole—and talk more expressly about that structure.

To be continued.