On Teaching: The Art of the Fugue, part 7

Gavin Black, director of the Princeton Early Keyboard Center, Princeton, New Jersey, is preparing performances on Bach’s The Art of the Fugue on both harpsichord and organ for the next two concert seasons.

The Art of the Fugue, part 8

In the last few columns, I have started writing and thought of a suitable and effective name for each column somewhere along the way, even at the end of the process. However, today I was able to start with the title, because it is time to get back to writing about Johann Sebastian Bach’s The Art of the Fugue!

This is still an uncertain time, no less so than a month ago. And it is still true that there are things that are unknown as I write this that will be known when you read it: will there be a Major League Baseball season? Who will be the Democratic nominee for vice president? Will there be a post-Memorial Day spike in COVID-19 cases? Will Broadway theaters really reopen on September 6? And there are things that are unknown now that will probably remain unknown, at least with any certainty by then: will there be a second wave of the virus? What will Advent and Christmas be like—for church musicians, for retailers, for families? Will the practice of going to the movies survive?

When I wrote my first Art of the Fugue column a year ago, I could not have imagined that over the succeeding year I would be unable to practice or perform the work, so this column really was my only study of the piece. I also could not have predicted what the content of the columns would actually be. I thought that I could, but it turned out to be very different from what I initially planned. But that is all eerily appropriate. When Bach first set out to compose the work he certainly did not know that he would be forced to leave it incomplete or that he would not see it published. Uncertainty has to be an underlying theme of The Art of the Fugue.

I strive to organize some of my thoughts about how the uncertainties surrounding the order of the movements interact with my thinking about the work in general. In my column from May 2020 I wrote of “the basic definition of counterpoint, namely two or more things that are different from one another happening at the same time.” The second consistent characteristic of counterpoint as we usually know it—for me, just below the level of “definition”—is that things that are the same happen at different times. Paradoxically this is perhaps even more important in shaping our range of reactions to counterpoint: esthetic, emotional, intellectual, etc. It is the source of our need, when we analyze pieces of this sort, to know about and recognize themes, subjects, countersubjects, motifs: anything that happens more than once. And this phenomenon is entirely dependent on memory. We know that a theme has recurred because we remember it from before. This is true immediately when a fugue subject appears for the second time. That part is relatively easy. (And it is assisted by expectation, if we have been told that what we are hearing is a “fugue” and we have just heard a theme in one voice unaccompanied.)

When a theme comes from elsewhere, as in a chorale prelude, for example, the process of recognizing that theme takes on another layer or two. I find it interesting that chorale-based pieces have a kind of double life. They come across differently to listeners who already know the tune and those who do not. Presumably most chorale-based pieces have been written in the first instance for listeners who knew the tunes extremely well—almost instinctively. In a multi-movement chorale-based work like Bach’s Canonic Variations on Vom Himmel hoch da komm’ ich her the recognizability of the chorale creates unity that perhaps makes the exact order of the movements function differently than it might in another situation. That is not to say that it is not important: it is manifestly a somewhat different piece with the movements in one order from another. It is interesting that Bach did indeed present the work in two different orders: first, in a published version, with the most complex and imposing of the five variations as the closing movement; later, in an autograph manuscript, with that variation in the exact middle. (This could be a rationale for considering it possible that the triple/quadruple fugue from The Art of the Fugue could function not as a culmination but as an interior building block, way station, or destination.)

When a theme has been altered, perhaps through augmentation, diminution, inversion, or something else well defined, perhaps by just a small change or two, recognizing it becomes more abstract and mediated by the subconscious. For many contrapuntal works this creates a kind of layered structure in which as elements recur in different ways, they evoke different kinds of memory. The development of the sense of “this is answering that” or “this edifice has that kind of shape or structure” is a multi-faceted, interlocking, overlapping experience.

In my December 2019 column, I wrote that “the subject or fundamental building block of The Art of the Fugue is not ‘the AOTF fugue subject’ but is ‘the very concept of the melodic interval.’” That column develops some of the reasons I believe that this is so, which I will not repeat here. If this is true, it sets up a condition in which the layers and facets of what we recognize as we listen, what we rely on to create structure in our minds, is infinitely complex and varied. The more the status of recognized themes or ideas is different (some more obvious, some more subliminal), the more complex that structure will be. 

I believe this relates to the question of the order of the movements. Since these connections are so numerous—effectively infinitely numerous, since there is very nearly nothing within the universe of the piece that does not connect to other things—and since the nature of those connections is so varied and fluid, convincing, engaging patterns will form themselves in the listener’s mind regardless of the order in which the components are encountered. The structure is then not “x follows y, which follows z” but “a, b, and c are all connected.”

So the nature of the opening theme and the ways in which it is developed in the first contrapunctus set up this focus on any and all melodic intervals, which in turn creates an infinitely fluid set of ways of hearing connections and perceiving or synthesizing shape. This explains why the piece can be effective almost regardless of the order in which the movements are heard. And it is not just that it is effective: it takes on a convincing overall shape, a strong sense of arc, direction, and structure. This does not mean that Bach did not have an order in mind. It just explains perhaps why the piece works so well even though we do not and very likely cannot know what that intended order was.

I say “almost” regardless. I believe, based on this analysis and intuition, that it is important for Contrapunctus I to be first, since it sets the stage for all that follows. I also think that not placing the triple/quadruple fugue at the end makes the biggest difference among all possible ordering choices. I am reluctant to say that it is “wrong” or would not work, but I know that it would be a big statement to place it elsewhere.

Conventional ordering of musical content

With certain sorts of pieces convention gives us an expectation as to how the ordering of types of musical content will create shape. This is true of suites, sonatas, symphonies, and other similar types of works. These conventions are not ever absolutely fixed, and they vary with time and place. But there is no convention as to the ordering of twenty or so contrapuncti. The content must create the possibility of shape and arc if there is going to be such a possibility.

This thought leads to an idea that I have held for a long time, have never been able to implement, and will likely never be able to implement. It would be a logistic tour de force and extraordinarily expensive as well. The analysis above helps to explain why I think that it could in theory work. Consider a performance of The Art of the Fugue by twenty different performers or performing ensembles, each playing one contrapunctus in a different room. Each movement in its room would be played over and over, and the listeners would be free to wander from one room to another. Each audience member would create their own path and could come and go from rooms at any time. Timing considerations would make it close to impossible to hear all of the movements in any order without also hearing fragments, since each movement in each room would be a different length from the others. The experience for any listener would not be that of “hearing the piece” exactly, but of getting absorbed in it. This is not a type of deconstruction that I would suggest for, for example, the Goldberg Variations or most other multi-movement works. But I think that it would work beautifully here.

Some of the observations that I have been trying to pull together in the last several months’ columns have led this thought to occur to me. The Art of the Fugue is a fully, rigorously contrapuntal work, and that is a large part of the lens through which we think about it. There is a body of music out there that is clearly imbued with counterpoint, but that is not fully contrapuntal. For me one quintessence of this sort of music is the keyboard music of William Byrd. There are passages that are fugal, there are passages that are chordal, and there are fugal bits in which the number of voices seems to fluctuate, or in which it is not always clear which notes belong to which voice. There is music by Mozart, Beethoven, and many subsequent composers that fits this profile. I have always had a nagging tendency to be uncomfortable with this: is it counterpoint or not? Fundamentally there is no reason to consider this anomalous or problematic, though some of us do. Looking at counterpoint as an analogy to the structure of the world or of the universe in the manner that I described in my previous column, and noticing that under some circumstances entities other than complete, defined subjects can be fodder for contrapuntal development, both tend to mediate between and reconcile contrapuntal and non-contrapuntal textures and make sense of the sorts of pieces that flow from one to the other.

Another less esoteric part of my recent Art of the Fugue experience has been that circumstances have led me to practice quite a bit on the piano. Using the piano in our home as a practice machine has been interesting. Vestigial memories of studying piano in my youth have come to the fore and have caused me to drift into doing things with volume that I am not very good at and that are not really relevant to this music.

But that raises a good question: does the “volume temptation” reveal things to me about the piece that are valid and that I can make use of, or is it just a distraction? Or is it actually misleading? These are not questions that I have never heard people ask. But they feel more vivid to me now as I have sat at the piano more in the last couple of months than I had in the previous many years. Many students do a fair amount of practicing on instruments other than the one on which a lesson or performance will take place: piano for harpsichord, piano for organ, electronic keyboard for either, or of course harpsichords and organs that are just very different from others. I have had too much tendency to see this as a necessary evil, to believe that ideally practicing on the exact right instrument is always better. I still believe that practicing on the performance instrument is better, more efficient, and that it should make up as significant a portion of practicing as possible. But since I am finding sitting at the piano to be enriching and interesting, I find myself rethinking all of this a bit. 

This is my twelfth consecutive column that is either about The Art of the Fugue or framed by my inability to write about The Art of the Fugue. It is my plan to put writing about the piece on the back burner, while getting back to actually working on the piece.

Just as I used the titling of this column as a boost of morale, I will reuse my mini-bio from 2019 below to express a bit of optimism about getting back to performance. It turned out not to be true then. Perhaps it will be now.

The Art of the Fugue, II

For discussion in this the next two columns, I offer the program notes I wrote for my first performance of The Art of the Fugue in May 1985. This performance, on the Fisk organ at Westminster Choir College, was one of my two graduate recitals. I prepared these notes over more or less an entire semester and had some input and help from my teacher Eugene Roan and from William Hays, who was the advisor for degree recital program notes. I have been pleased with this essay, and I have used it as partial program notes for subsequent performances. It has an integrity to its overall structure—thanks in significant part to Dr. Hays’s assistance—such that I have not changed it or excerpted it. Despite that, if I were to write these notes today, there are a number of things I would phrase differently.

It could be fruitful to use some of those theoretical revisions to frame future columns about the learning process, the evolution of my relationship with this work, and the relationship between my own work on this piece and teaching. Some of what I wrote about the order of the movements was too cut-and-dried, rather too simple, failing to reflect some of the complexities of what we do and do not know about the piece. In later columns, I will discuss that, including some new ideas.

History and form

J. S. Bach wrote The Art of the Fugue during the last years of his life, probably beginning work on what turned out to be his longest and most complex instrumental composition in 1743, leaving the opus incomplete at his death in July 1750. It was published in 1751 in Leipzig in a poorly engraved edition, the preparation of only part of which had been supervised by Bach himself. The publication was not a commercial success, and the project was soon abandoned by Bach’s heirs.

Copies of The Art of the Fugue circulated among musicians, however, from that time on. In 1799 a scholar referred in print to the work as “celebrated,” and both Mozart and Beethoven owned copies. The Art of the Fugue was studied extensively by musicians throughout the nineteenth century, and nearly twenty editions or arrangements were published during those years. The first known public performance of the whole work took place in 1927 in Leipzig under the direction of Karl Straube, one of Bach’s successors as Kantor of Saint Thomas School in that city.

The Art of the Fugue is a work of well over an hour in length, consisting of eighteen movements all based in one way or another on the same musical theme. This theme occurs in something like one hundred different forms throughout the piece. The first and simplest form of the theme is shown in Example 1.

The theme is closely based on the tonic triad of the key of D minor, or, looking at it another way, on the interval of a fifth, and on the idea of filling that interval in. The first gesture creates a perfect fifth; the next gesture fills in that fifth, in the simplest possible way. The rest of the theme provides the remaining notes needed to fill in the perfect fifth, D–A, by step, and outlines a diminished fifth, C-sharp–G. In the tonal world of Bach the perfect fifth is the source of security and repose, while the diminished fifth is a source of tension, unrest, and striving. The two are antithetical to one another. This antithesis, with the one side represented not only by the perfect fifth as such but also by all diatonicism, and the other side mainly represented by the chromaticism implicit in the diminished fifth, is a major source of direction, growth, and meaning throughout The Art of the Fugue.

The opening theme also contains, in significant contexts, all the intervals from the semitone to the perfect fifth. This is in spite of the brevity, compactness, and apparent simplicity of the theme. The use of such a theme creates a situation in which any interval, either open or filled in by step, can be used by the composer as a motive significantly related to the main theme of the work. This possibility for motivic interrelation is an important source of unity and coherence in The Art of the Fugue in spite of considerable variety and diversity.

Most of the movements of The Art of the Fugue are fugues or are largely constructed through fugal procedures. Four movements are strict two-voice canons. Bach did not designate any of the movements as fugues, but rather as contrapuncti. (He may well also not have been responsible for the title under which the work is known, since the title page was engraved after his death.) He seems to have been concerned in his use of nomenclature to suggest that the movements were not autonomous fugues such as the organ fugues or the fugues of the Well-Tempered Clavier (all of which are paired with non-fugal preludes), but rather stages in the working out of a musical idea, or a set of musical ideas, through a variety of contrapuntal techniques. Several of the movements, even apart from the canons, would probably not have satisfied Bach’s own definition of a fugue as such, because of serious irregularities in the construction of their opening sections. These irregularities, however, make perfect sense as stages in the contrapuntal development of the work as a whole. They serve invariably as responses to what has come before and as preparations for what will follow. These relationships are described in detail below in the comments on the individual contrapuncti.

The four two-voice canons (numbers 12–15) are lighter in texture and mood than any of the other movements and are simpler in construction. Coming after the most complex of all the contrapuncti, and before the movements in which contrapuntal ingenuity is carried to its farthest extremes, they provide for performer and listeners a moment of repose. This makes possible a renewal of energy and of momentum towards the climax of the final movement. Many individual Bach organ fugues contain within their structure a similar “relaxed” passage, which serves a similar function of providing a breathing space before the final climactic musical gesture. (Measures 121–139 of the Fugue in C minor, BWV 546ii, and measures 141–155 of the Fugue in E minor, BWV 548ii, are particularly good examples of this.) This suggests that The Art of the Fugue should be thought of not as a collection of fugues, but as one structure analogous to a single giant fugue. Further facts bear this analogy out (assuming it is not pressed into too detailed a form). The first movements of the work introduce the main musical ideas in a straightforward way, as does the exposition of a fugue.

The middle movements of The Art of the Fugue develop those musical ideas and others, with increasing complexity, contrapuntal and harmonic, and with increasing variety of texture. This is similar to the middle section (sometimes called “development”) of many fugues, especially, longer ones. The four canons fulfill the purpose described above. In the final three movements harmonic complexity is reduced, and anything even approaching the almost impenetrable density of Contrapunctus 11 is abandoned. In Contrapunctus 17, the original theme is reintroduced in a form closer to the opening of Contrapunctus 1 than anything that has been heard since Contrapunctus 4. This is analogous to the return of the initial subject that characterizes the final section of many fugues. The extraordinary contrapuntal ingenuity of Contrapuncti 16 and 17 (see below) is analogous to the increase in contrapuntal complexity that is found at the end of many Bach fugues, usually in the form of stretto.

Neither the first edition of The Art of the Fugue nor any of the eighteenth-century manuscript copies say on what instrument or instruments the work was meant to be performed. Over the years many different performing forces have been used, including piano, chamber ensembles of various composition, symphony orchestra, jazz combo, harpsichord, and organ. Many scholars believe that Bach actually meant the work for organ, some that he meant it for harpsichord, even though the posthumous title page says neither. The first edition was published in open score, that is, with a separate line for each voice. This was an old Italian and German way of presenting keyboard music used, for example, by Samuel Scheidt in his Tabulatura Nova (1624). It was certainly not the standard keyboard notation in 1750, but Bach had used it shortly before, in his Canonic Variations, BWV 769. The contrapuncti all fit very well under two hands and two feet, and with some difficulty under two hands alone. The pedal parts work as pedal parts: that is, they can be learned using the kinds of pedal technique known to Bach and his students, and when so learned they are comfortable (though occasionally challenging) to play. This would not be true of the bass lines of Bach chamber works or harpsichord works, by and large. The editors of the first edition chose to include a short additional piece by Bach, to compensate the purchaser for the incomplete state of the last movement. The piece they selected was an organ chorale, which they also presented in open score. It is thus likely that they assumed that the users of the work would be organists, even though they did not say so on the title page. It is also quite possible that Bach himself wanted musicians to use their own judgment as to how the piece can be realized in sound.

B-A-C-H

The third subject of the last movement of The Art of the Fugue is made up of notes that, in the standard German musical nomenclature, spell the name “Bach” (Example 2). In the German system, B-flat is called B, and B-natural is called H. Bach was aware throughout his life that the letters of his name made a plausible musical theme—it was certainly known to his musical ancestors as well—but he used it sparingly in his music. The only extensive use he made of it was in The Art of the Fugue. The final appearance of the B-A-C-H theme as the subject of a powerfully climactic fugue in Contrapunctus 18 is prepared by a chain of musical developments running through the whole work. This chain is best followed retrospectively. Before Contrapunctus 18, the B-A-C-H theme appears in Contrapunctus 11. Here, the four relevant notes form part of a lively and insistent eighth-note motive (Example 3). They do not stand on their own, but they are clearly present. This eighth-note motive, however, is an inversion of one of the main themes of Contrapunctus 8. That movement is thus revealed to have contained the B-A-C-H theme in a highly disguised form. The motive also occurs in once in Contrapunctus 8, casually, without repetition or development, in the bass voice at measure 143, transposed up a whole step. The first appearance of the B-A-C-H theme in the work occurs at the end of Contrapunctus 4, where the four notes form part of an otherwise meandering free chromatic countersubject to the main theme. This serves to underline the essential chromaticism of the B-A-C-H theme, and to tie that theme to the other chromaticism in The Art of the Fugue. The seeds of the chromaticism in the work, and thus the seeds of the B-A-C-H motive itself, are found, as explained above, in the initial statement of the main theme. The four contrapuncti in which the B-A-C-H theme is found (4, 8, 11, and 18) are by a considerable margin the four longest movements in the work, and each of the four is longer than the last.

To be continued.

Gavin Black, director of the Princeton Early Keyboard Center, Princeton, New Jersey, is preparing performances on Bach’s The Art of the Fugue on both harpsichord and organ for the next two concert seasons.

The Art of the Fugue, part III

This month’s installment picks up exactly where last month’s left off.

Notes on the Individual Movements

Contrapunctus 1 is a four-voice fugue on the original and simplest form of The Art of the Fugue theme (Example 1). There is no regular countersubject. The mood is forthright and powerful, but with underlying calm.

Contrapunctus 2 is a four-voice fugue on the same theme, except that the last notes are changed from eighth notes to dotted eighths and sixteenths (Example 2). This dotted rhythm pervades the piece. The sounds jump rather than flow from one beat to the next. The effect is, at least intermittently, unsettling. The dotted rhythms convey a suggestion of French musical style, though the piece is not fully worked out in that style.

Contrapunctus 3 is a four-voice fugue on the inversion of the original theme (Example 3). The piece moves slowly and gently, and the mood is quiet. The subject is accompanied (after the first entrance) by a chromatic countersubject loosely derived from the subject itself (Example 4). This countersubject also provides a short motive (Example 5) on which two episodes are based. This motive will return later in the work. In measure 23 the subject enters in a new form: with passing tones (Example 6). Although it is abandoned in the next movement this will become the most important form of the theme later on.

Contrapunctus 4 is also a four-voice fugue based on the inversion of the main theme (see Example 3). The theme is accompanied by a short chromatic motive, more explicitly dissonant than that of Contrapunctus 3 (Example 7). The movement is also characterized, however, by long episodes of almost exaggerated consonance. These are based mainly on a simple two-note descending third (which can be heard as a reference to the thirds in the original, non-inverted theme, although that theme is not explicitly present in this movement) and a four-note stepwise fourth, both ascending and descending, which is derived from the last four notes of the main theme, in both its inverted and non-inverted forms. There are several lushly beautiful internal cadences, almost hypnotic in their effect. The last measures conceal a single reference to the B-A-C-H motive.

Contrapunctus 5 opens with a statement of the inverted theme with passing tones, introduced in Contrapunctus 3. Before the end of the theme, this is joined by the non-inverted theme with passing tones in the bass (Example 8). This overlapping of themes, called “stretto” by theorists of fugue writing, is by no means characteristic of the opening of a fugue. Nor is it characteristic for the second subject entry to be an inversion rather than a transposition of the first. This is a gesture of union of the two essential forms of the main theme. They are the opposite of each other, but they can be joined together. In measure 53, an even closer union is suggested with a four-voice stretto at the interval of one quarter note. The inverted and non-inverted themes, however, are treated to this process separately (measured 53–57 and 65–69)! The mood of Contrapunctus 5 is alternately urgent and dreamy, or perhaps both at once. In the last measures of the movement the two versions of the theme are heard simultaneously, somewhat concealed by an unexpected six-voice texture.

Contrapunctus 6 is marked in Stile Francese. Thus it realizes the suggestion made by Contrapunctus 2. It is not easy to define what the “French style” is. For the purposes of this piece, however, it is enough to recognize that it involves the use of dotted rhythms, very fast short runs, and moderately fast longer runs. It also aims to create a feeling of power and grandeur in the context of rather static (non-flowing) rhythm. Native French pieces in this style would never have been fugues, so Bach was doing something new. This piece, like the last, opens with a stretto, at the interval of only one measure. The impatient quality of the second voice, created by its insistence on coming in early, is here intensified by the fact that is comes in “too fast,” that is, in diminution. The very rapid four-note runs, heard from measure 7 on, are a standard characteristic of the French style, but they can also be seen as a double diminution of the last four notes of the main theme.

Contrapunctus 7 is a fugue on the main theme with passing tones, sometimes inverted and sometimes in diminution. To this is added the theme in augmentation (Example 9) that is heard first in the bass and once in each voice. Almost every note in this contrapunctus is derived directly from the theme, by augmentation, diminution, or double diminution. The effect is one of extreme concentration and intensity, but also of rather jovial power and good humor.

Contrapunctus 8 opens with a short fugue exposition on a brand new theme, not derived in any way from what has come before (Example 10, but see under Contrapunctus 18, below). After several short episodes and a cadence in the tonic, a second exposition begins, based on the same theme and another brand new theme (Example 11), simultaneously. These two themes work themselves out with increasing intensity and exuberance, until they are joined in a somewhat hidden manner, that is, in an inner voice, by a new version of the main theme (Example 12). The intensity and exuberance, the latter quality created mainly by the almost exaggerated repetitions of the second theme (Example 11), continue to the end.

Contrapunctus 9 is a fugue on a new subject (Example 13) that is loosely derived from the main theme. Whereas Contrapunctus 8 opened with a descending whole-step (see Example 10), this movement opens with an upward leap of an octave. This is a contrast to Contrapuncti 1–7, all of which opened with the interval D–A or A–D. Contrapunctus 9 is a lively, flowing piece in which exuberance, though present, is restrained. The main theme enters in augmentation in the soprano voice in measure 35 and is heard several times thereafter.

Contrapunctus 10 opens with a fugue exposition on a subject (Example 14) whose rhythmic and melodic peculiarities create more of a sense of expectation than of fulfillment. The expectation is met with the entry of the main theme in the soprano voice in measure 23. What follows is a fugue on this subject, in which Example 14 serves as a countersubject. So many new motives are introduced, developed briefly, and then abandoned, that the movement becomes a celebration of fecundity (or perhaps of profligacy!). The countersubject (Example 15) that appears in measures 22–26, and does not appear again in this movement, will supply the subject of the second section of Contrapunctus 18 (see below).

Contrapunctus 11 is the centerpiece of The Art of the Fugue. It is probably the most complex contrapuntal creation of J. S. Bach, as well as one of the most deeply and powerfully expressive. It is a contrapuntal fantasy making significant use of at least ten different themes and including five fully worked out fugue expositions. It is related to what has come before and what will come after in more ways than it is possible to describe. All of the thematic material is taken from earlier contrapuncti, mostly from Contrapunctus 8, but also at least from 3 and 10. The richness of thematic material is reminiscent of 10, while the compactness with which it is used is reminiscent of 7. The sectional construction is reminiscent of 5 and 8. The compellingly powerful cadences suggest number 4. The most powerful of these cadences is followed by the introduction of the B-A-C-H motive, which looks forward to Contrapunctus 18 and also back to number 8 (see above under B-A-C-H). At measure 158, the main theme and its inversion are again, as in Contrapunctus 5, united in absolute simultaneity, this time without the concealment that marked this event in the earlier movement.

Contrapunctus 11 opens with a fugue exposition on the main theme, with passing tones and altered rhythmically as in Contrapunctus 8. The opening rhythm is the same as that of Contrapunctus 10. Following a cadence in measure 27 there is a loosely constructed fugal exposition on the two new themes: the inversion of the first theme of Contrapunctus 8 and a chromatic theme filling out the interval of a fifth. These themes are joined by a motive taken from Contrapunctus 3. This section ends with a cadence in measures 70–71. The next section is a short fugal exposition on the inversion of the opening theme, in which the motive borrowed from Contrapunctus 3 is also used. In measure 89 there begins a long, rather freely constructed fugue based primarily on the new eighth-note theme that incorporates the B-A-C-H motive. All of the thematic material that has been introduced so far in this movement participates in this section, which lasts until measure 158. At this point The Art of the Fugue theme and its inversion enter simultaneously in the soprano and alto, and then in the tenor and bass. The remaining thirty bars are concerned with a limitlessly joyous uniting of themes in all possible combinations. The main theme appears for the last time (in this movement) in the soprano in measure 180. The last note of this theme is also the last note of Contrapunctus 11 (see Example 16 a–j for all the thematic material of Contrapunctus 11).

At this point, power, grandeur, complexity, repetition, development, and intensity have gone as far as they can. The next four movements are relatively short, quiet, and simple, though not any less expressive. They are strict canons in two voices. Contrapunctus 12 is a canon at the octave based on an ornamented version of the main theme in triplet sixteenth notes (Example 17). It is a good-humored, even humorous, piece. Contrapunctus 13 is somewhat darker in mood. It is a canon at the twelfth, based on a close variant of the main theme (Example 18). Its rhythmic interest comes from an alternation between triplets and duple rhythms. Contrapunctus 14 is a canon at the tenth, based on a completely straightforward version of the inverted main theme. This is accompanied by a quiet, rather wistful triplet figure (Example 19). Contrapunctus 15 is a canon by augmentation in contrary motion. That is, the second voice enters with an inversion of the first voice, at half speed. The second voice falls progressively farther behind, and by the end it has presented less than half the musical material of the first voice. As if to be fair, Bach has arranged the work in two sections, with the upper and lower parts changing roles halfway. This canon is based on a chromatically altered version of the main theme (Example 20, p. 24). The effect is rather weird, both disquieting and humorous.

Contrapunctus 15 reintroduces a principle of inversion that was so important through number 11. Contrapuncti 16 and 17 carry this principle farther than before. Each is in two sections, and, in each case, the second section is a note-by-note inversion of the first. These are contrapuntal tours-de-force of the highest order, all the more so since in both cases the inversions sound just as natural, spontaneous, and free as the original forms. Contrapunctus 16 is in three voices, and based on a triplet version of the main theme, reminiscent in different ways both of Contrapunctus 9 and of Contrapunctus 12 (Example 21, p. 24). The style is that of a moderately lively trio sonata movement. Contrapunctus 17 is a four-voice fugue based on a version of the main theme that is identical to the original version (Example 1), except that it is in triple time (Example 22, p. 24). The texture is very thick. With many voice crossings, the contrapuntal nature of the work is somewhat obscured. The mood is dreamy and brooding. In measure 21 a new version of the subject enters (Example 23, p. 24), which is derived from a short countersubject figure (Example 24, p. 24), which was, in turn, derived from the last four notes of the subject itself. Each half of the work ends with a flourish in sixteenth notes.

Contrapunctus 18, the last movement of The Art of the Fugue, is a monumental work, surpassing even Contrapunctus 11 in scale, and equaling it in expressive power. It was left incomplete. Bach died either before he wrote the ending down or before he was even able to conceive of an appropriate ending to a work of this stature. In its present form, the movement has 239 measures. Any formally adequate continuation and ending would bring the length of the work to over four hundred measures, making it very likely the longest fugue from the Baroque era. This, however, was not to be.

The movement opens with a fugue in four voices on an apparently new theme (Example 25). The first four notes of this theme, however, are identical to the first four notes of the main theme (with passing tones). After the fourth note the theme doubles back on itself, in an allusion to the mirroring process whereby the previous two movements were created. Thus this theme is a version of the first half of The Art of the Fugue theme. This new theme can also be derived directly from the second half of the first subject of Contrapunctus 8 (see Example 10). Furthermore, the very first two notes of Contrapunctus 18 are an exact echo of the last two notes of Contrapunctus 17; and the whole of Example 25 is essentially the same as the flourish that ended the first half of that movement. The fugue built on this subject—one of extraordinary grandeur—lasts 115 measures. It overlaps with a new fugue on a new subject (Example 26). This subject is derived closely from a theme that was introduced briefly in Contrapunctus 10 (see above and Example 15). It also includes, on strong beats, all of the notes of the main theme that are missing from the first theme of Contrapunctus 18: it completes that theme. This fugue is livelier in character than the first fugue. The first theme does enter, however, five times as a kind of cantus firmus. The last two such entries are in stretto with each other. This fugue ends in measure 193. A third fugue now begins, on the B-A-C-H theme in the simplest, most straightforward possible form (Example 27). This section brings a return to the majestic quality of the first section, but it is much shorter. In measure 233 it gives way to a return of the second subject in (partial) stretto with itself. This is joined one measure later by the first theme in the bass, and, one measure later still, by the B-A-C-H theme in the tenor. The three themes are united; the main Art of the Fugue theme is present only by implication or by transfiguration. At this moment the work ends. There is no cadence, no resting place. It ends as abruptly as could be. We are challenged to accept that the meaning, quality, and completeness of a work of art, as of a human life, do not come from the timing or nature of its end, but from what it is and what it does.

The Art of the Fugue, Part 1

Before delving into the principal topic of this column, I must first briefly revisit the subject of the last two columns, which dealt with aspects of the practice of listening to music. Shortly after I finished the May column, I was in New York City for the day, and I happened to notice, walking along one of the avenues, some bins of used LPs outside an antique store. I had a few minutes to spare, so I started leafing through the boxes. Midway through I saw a record of Brahms’s First Symphony. This is a favorite piece of mine, and part of my program for that day was to hear a concert performance of it at Lincoln Center. I pulled it out to take a look, as I wanted to know who recorded it. But there was nothing: no orchestra name, no conductor, no date, no recording venue, no clues.

I had just written of my experience noticing that students and other listeners have a habit of seeking out recordings online and listening to them without noticing anything about who the performers are. I presented this as being a characteristic of the structure of modern listening technology and a strong and well-accepted modern ethos. But it is interesting to be reminded that it also is not a new concept. This Brahms LP, monaural as far as I could tell, is an artifact reflecting the view that it is perfectly acceptable to listen to a performance not only without noticing who is playing, but also without having any way to find out.

Some of the implications of this would be fascinating to explore at greater length, and I will write more about it at some point. When we listen to a performance, especially when we listen to the same one repeatedly, what do we feel about letting that particular way of performing the piece shape our way of defining it? Do we think about this consciously and give the performance permission consciously and deliberately to affect us in certain ways? The history of this has been more complicated than I was remembering when I touched on it before. That in turn ties in with questions of authenticity, which we tend to think of as being about composers, and authority, which can come from any number of places—writings, performances, teachers—and which can influence us with or without our being aware of it.

Questions concerning Bach’s The Art of the Fugue

For this month’s discussion, I turn to Johann Sebastian Bach’s The Art of the Fugue, BWV 1080. There are a myriad of issues surrounding this monumental opus that open windows into our thinking about authenticity and authority in particular, as well as many different aspects and dimensions of what we do as performers, listeners, students, and teachers. With its length, complexity, and importance in the arc of the work and career of Bach, crucial questions about the work are unanswered and perhaps unanswerable.

So, what is The Art of the Fugue?

It is a work written by Bach over the last decade or so of his life, consisting of many movements—about twenty, but that is one of the areas that can be looked at a number of different ways—each constructed contrapuntally, some as fugues and some as canons. The movements are all based to some extent and in some way on a particular theme. The piece was published shortly after Bach’s death in an engraved edition, and while Bach certainly composed the bulk of the music, the work was completed by others. There are also surviving earlier manuscripts of some but not all of the work.

The theme mentioned above is found in Example 1. The theme in this form opens the first movement, which is a four-voice fugue on this subject. The first movement is the only one to open with a simple statement of the theme in exactly this form and the only one that is based primarily on this form of the theme. The variants of the theme that form the basis of the other movements include inversions, diminutions and augmentations, rhythmic variations, and versions with added passing tones.

One question that intrigues me, and that I will broach here and come back to in the course of these articles about The Art of the Fugue, is, why this theme? One answer could be, why not? After all, Bach wrote fugues on a large number of different subjects and must have improvised fugues on many, many more. However, I think that it is worth interrogating the ways in which this theme in particular might have lent itself to the extended and varied treatment that constitutes this long work. The Art of the Fugue theme was not, as far as we know, or as far as I have ever heard, taken from somewhere else. (As, for example, the theme of A Musical Offering was, or as the themes of all chorale-based pieces are.) Bach wrote a number of other fugues on themes that are largely based on a minor triad, like this one. That is true of the subject of the fugue from the Fantasia and Fugue in G Minor, BWV 542, and, in an even more thoroughgoing way, of the stand-alone Fugue in G Minor, BWV 578. The Canzona in D Minor, BWV 588, is based on a subject that could in fact qualify as a variant of The Art of the Fugue theme (Example 2).

(If that piece were dropped into the middle of The Art of the Fugue it would be quite possible to justify it, at least as far as themes and motifs are concerned, as part of the work. It would seem an interesting variant that the semitone by which the subject departs from the confines of the perfect fifth is the one going up, whereas in the original subject it is the one going down, and that the two of those outline the notes that give the minor mode its harmonic flexibility or instability.)

I have heard or read suggestions that this theme or subject is so simple, basic, plain, that it is astonishing that Bach could construct a massive edifice upon it: that his ability to do so is a particular proof of the power of his genius. I do not disagree with that conclusion, in that it took a genius to create this work. However, I am not inclined to agree with the premise. It seems to me that constructing this theme was indeed part of the genius: that it is specifically and purposely designed to carry the weight of all that was developed out of it, and potentially more. I will come back to this later on.

That brings us to one of the most famous and important things about The Art of the Fugue: that it is incomplete. The movement we regard as the final one, while already the longest in the piece, breaks off in the middle of a measure. It is not a neat ending; it is not the end of a section—just an abrupt crashing from music into silence.

The reason that fugue is incomplete is that Bach died before he could finish it. Perhaps, he had it composed in his head. It seems likely that Bach or any composer would have had to have a fairly strong idea as to where a big contrapuntal structure such as this movement was going before venturing on starting it. It is a complicated fugue with multiple themes. But that does not mean that he had worked out the ending in detail.

In any case, we do not have the last measures of this movement, and therefore we do not have all of The Art of the Fugue. This creates a set of dilemmas for performers. Should one simply break off, playing all and only those notes that we have, allowing the “ending” to be jarring? Or should the performer or performers play one of the many endings that composers, scholars, and performers have composed over the last hundred years or so? Or should one look for a nice closing cadence as close as possible to where the piece currently ends and stop there? The fundamental fact is that none of these portrays Bach’s true intentions.

I have always favored the practice of ending abruptly. This preserves a certain “purity” of playing only Bach. It also forces us to confront in the most direct way the fact that things do not always go the way we want. That breaking off is beyond jarring: it can be deeply distressing and filled with anguish. It is an ending determined, as endings often are, not by any person but by death itself.

There is no way to maintain that this troubling breaking off is what Bach intended. I have had colleagues point out to me that by playing only and all of what we have on the page, we guarantee that we are doing something that Bach could not possibly have wanted. And every completion that has been attempted has been predicated on some analysis of what Bach might have been planning. Therefore any one of them has a chance at least of being similar in concept to what Bach would have envisioned. If nothing else, the length of the piece gets closer with each added measure to wherever it would have ended up if Bach had been able to finish it. And the abrupt breaking off is replaced by a normal ending. In between, the further working-out of the counterpoint might well be something like what Bach would have done. That is presumably the goal for those who have written such continuations, and each person has brought knowledge, care, and analysis to that project. But it is not Bach’s ending, and the piece is no longer just a Bach piece.

The first published edition, supervised by some of Bach’s surviving family and colleagues, chose a version of the third plan. The printed edition ends with the last solid chord, so to speak, before the spot where the manuscript source breaks off. This is an A-major chord in a piece in D-minor and indeed sounds like a dominant. It is a chord, and the rhythmic structure of what has preceded it gives it some solidity. But it does not sound stable, which raises an interesting question about authority. This is the approach apparently sanctioned by those closest to Bach. What authority do we give to that? What do we know or believe about how likely it was that they got that idea from J. S. Bach himself?

We tend to believe that this movement, an ostensible triple fugue that was very likely intended to end up as a quadruple fugue, was clearly meant to be the last section of the overall work. It certainly looks the part. However, we do not know for certain that if he had had several more years, Bach would not have added much more. Perhaps this triple (quadruple?) fugue would have ended up as a centerpiece rather than a culmination. Or perhaps it was really intended to be a centerpiece even without more movements. We do not have absolute certainly about the intended order of the movements, only very well-informed guesses.

Speaking of performance: we also do not know for certain what Bach’s intentions were for the performing forces that are brought to bear on this work. The surviving manuscript sources and the first edition say nothing about what the music is “for.” It is all in open score—four staves for a four-voice piece, three for a three-voice piece, and so on. There are no instrument names or any words on the pages of music talking about instrumentation or performance. There is a significant amount of evidence that this was probably intended to be a work for keyboard instruments, though that is not absolutely certain. And accepting that, it is less clear whether it was for harpsichord or for organ. There is a tantalizingly similar amount of suggestive evidence for each. Another real possibility is that the piece was intended equally for each of those two instruments. There was a long tradition of writing music that fits that profile, mostly from the generations before J. S. Bach. And there is a great deal about this piece that suggests that the composer wanted it to resonate in part as a throwback to those older generations.

The question of what instrument or instruments to use in playing a piece is crucially relevant to performance, to put it mildly. And since this series of columns is really about my own efforts to grapple with The Art of the Fugue as a performer, I will return to this in considerable detail.

The Art of the Fugue, part 6

I recently had a concert performance anxiety dream. This one was specifically about The Art of the Fugue, and it followed a common pattern: I was aware I was to perform the piece, and I sat down at the organ, but when I looked at the score, I realized that I had never seen the music before. I started to panic, but also to try to scheme: could I get away with sight-reading this? How slow could I play it without giving away what was going on? Could I leave out the hardest bits? I remember looking around hoping to see something happening that would force the concert to be abruptly canceled. The music on the page was not actually The Art of the Fugue. It was dense, chordal, and chromatic. The typeface looked like something French from the late nineteenth century. But within the dream I accepted that this was indeed the work that I was supposed to be playing. I had just somehow neglected to learn it.

I have this kind of dream once or twice a year, but I am pretty sure that these dreams usually involve playing at someone else’s behest, in chamber music or in a church service, not solo performance. My Art of the Fugue project is something that is being carried out only at my own behest. My decision to do it is a choice to focus on something that is specifically important to me and that I am utterly convinced I can only do well if I am very committed to doing it the way I want it to be done.

So why the dream? My only thought so far is that I am trying to remind myself that I should really learn the thing! There are always traps in any project, and maybe the trap with this one is that my sense of “ownership” over this work, my awareness that I have been studying it and interacting with it for nearly fifty years, my experience of being comfortable with it at a conceptual level will disguise how hard it is to play and lead me into complacency about the business of learning the notes. Maybe that is what the dream was about, or maybe not. Yet, any interpretation of a dream that reminds me to work as hard as I can to become as certain of the notes as possible is worth paying attention to!

Connections

The connection between the sense of understanding, inner commitment, ownership, and the nitty-gritty practical learning of a piece is a complicated one. At one level, there is the fact that, for almost everyone, focus and concentration are easier and more successful working on something we love than on something that we do not care about, or at least not very much. This is why I want students who endeavor to become more accomplished players to work on music they care about. The translation of time and effort into growth as a player is most efficient this way. Students do not know what music is out there as well as I do or another teacher does. The process of helping them to explore things that they do not know interacts in complicated ways with the goal of working on music they love.

Yet, I am wondering about something deeper. If one feels a deep bond with the music one is working on, what dangers does that invite? I believe there is a risk of assuming that the bond with a piece or a composer or a repertoire is in itself enough to make something come across to listeners. If I feel something while I am playing and listening, am I interpreting the music a certain way because of my love of the music? Is this hiding the necessity of doing something interpretively from me?

Overall structure of The Art of the Fugue

I now turn to some thoughts about the overall structure of The Art of the Fugue. There are two salient facts I would like to discuss. The first is that one of the movements is incomplete—the one that most of us take to be the final movement. Second, that there is some uncertainty as to the order in which Bach intended the movements to be played. There has been a lot of serious, thorough, and fascinating scholarship about both of these matters, each of which has a different set of interactions with performance. Regarding the final movement, there is the question of whether to play it as it has come down to us, breaking off abruptly, or to alter it in some way, either by completing it or by ending it at an earlier point than the break-off. As a set of choices this is probably more about philosophy than about scholarship or research. Scholarship, research, and analysis can, for those who wish to perform a completed version of the piece, elucidate different possibilities for that completion. There is also, however, the question of whether to play it at all. This is a matter that has to be addressed by a combination of scholarship and philosophy. It used to be fairly common for performers or scholars to entertain the possibility that this movement was actually never meant to be part of this work. I believe this is a rarely held view nowadays, and indeed the musical connections between this movement and the rest of the work are convincingly strong. To be fair, though, there are one or two quite recent recordings of the work that omit the long, unfinished fugue.

Anyone who sets out to perform The Art of the Fugue has perforce to end up making a decision about an order in which to play the pieces. And, speaking of recording, I have just completed a quick and random survey of a dozen or so CDs of the work, and no two recordings present the movements in the same order! This includes performances on organ, harpsichord, piano, and by chamber ensembles, such as saxophone ensembles, consorts of viols, and so on. These are mostly recent recordings, subsequent to the burgeoning of Art of the Fugue scholarship that we have seen over the last few decades.

(Further regarding recording: nowadays it is easy to set up a recording of the piece with each movement in a separate file, and with playing order to be determined by the listener. This could be as a set order or as a random “shuffle.” So I should say that any performance of The Art of the Fugue presupposes a choice by the performer about movement order. Only a live performance does so of necessity.)

There are two underlying causes of uncertainty about movement order. The first is there are two primary sources for the piece: a manuscript of most of the movements in Bach’s own hand from the mid-1740s, and the edition published in 1751 by Bach’s heirs. These sources have different numbers of movements and are presented in different order. The second source of uncertainty is we do not know how much input J. S. Bach himself had in the preparation of the printed edition. Therefore we do not know that he had signed off on the order represented there.

For example, it is possible for a performer to assume that, since the people who finished the preparation of the 1751 edition were close to J. S. Bach and worked with him directly on at least much of the project, the order that they created is probably the correct order. But this would be at best just an assumption, and there is a significant amount of scholarship to call it into question. A further complication is that the published edition contains elements that almost everyone agrees are not really part of The Art of the Fugue at all, at least not part of the integral structure of the work as a whole. For instance, there are alternate versions of two of the movements and a chorale prelude, the latter manifestly added to the end of the publication as a kind of compensation for its incompleteness and a memorial to the composer. I have never encountered or even heard rumored any performance of literally every note in that original edition in the order in which those notes are found there. So what were these early editors up to, and what did or didn’t they understand about the composer’s intentions?

You will probably not be surprised to read that we cannot answer that question. As I said, there is a lot of good scholarship on the matter, but that scholarship is often in disagreement. Every argument rests on assumptions, and while many of those assumptions seem sound, none of them seem rock-solid certain.

As a performer I must come up with an order. I notice, reading my 1985 program notes, that I then completely accepted the order that I had grown up with. This is the order that Helmut Walcha used in his recording, and I used the same order. This is almost the same as the order used by the early twentieth-century scholar Wolfgang Graeser, who is credited with rediscovering The Art of the Fugue, and whose edition was used as the basis for the first known public performance of the work, which took place in 1927. As best I remember, I did not particularly know at the time there were issues about movement order, since much of the scholarship on the matter has been carried out since then.

I found considerable structural logic in the shape of the piece with this order of movements. For example, I considered the placement of the group of four canons, with their two-voice and therefore relatively light texture right after the dense and complex Contrapunctus XI, to be aesthetically and structurally significant. However, the canons as a group are not placed after Contrapunctus XI in either original source, and they are differently placed in the two sources. Other performers find logic in spreading them out around the other mostly four-voice fugal contrapuncti. This uses the lighter texture to define groupings in the successive fugal pieces. Some performers place them after the long unfinished fugue, displacing it as the ending, and instead ending the work using the lighter textures to create a relaxed sort of coda.

I also found logic in the gradual introduction of the B-A-C-H motif, successively more open and clear in Contrapuncti IV, VIII, XI, and XVIII. Likewise, I find logic in the immediacy of the motion from the flourish ending Contrapunctus XVI—the four-voice mirror fugue—to the opening subject of Contrapunctus XVIII of which it is a very close pre-echo. This happens only if the appropriate section of Contrapunctus XVI is the last item before Contrapunctus XVIII. I also found the juxtaposition of Contrapunctus VIII—the first one whose subject opens with an interval smaller than a fourth—and Contrapunctus IX—the first one whose subject opens with an interval greater than a fifth—to be meaningful. This is only a juxtaposition if those two movements are indeed next to each other.

There are a number of questions that arise out of this complicated picture. If I see a convincing overall structure in this piece with a particular movement order, does that mean that this is likely to be the “correct” order, the order that the composer had in mind? (I will answer this one: no.) Or am I simply imposing that sense of structure because I am accustomed to that order and fondly want to believe the piece has a meaningful overarching shape? (Maybe.) Is it possible that if we really knew for sure what order Bach had in mind we would see that order gave to the work an even more compelling overall shape than we experience now?

Musical entities that are not just clearly one unit have a variety of shapes. To stick with Bach for the moment, The Well-Tempered Clavier is one thing (a collection of completely separate pieces, but ones constructed along similar lines to one another); a suite or partita is something else (a piece clearly intended to have an overall shape, but made up of movements that are in different forms from one another, each of which could stand as a piece in itself); The Goldberg Variations is something completely different yet a piece made up of clearly separate movements, but also with a clear unifying principal among the movements. Das Orgelbüchlein falls into the same category as the WTC. How about Clavierübung III? Unlike all of the above, it gets part of its unified or structured feeling externally, through the associations that we have with the hymns on which the pieces are based. It is also given a circular structure by the prelude at the beginning and the fugue at the end.

Which of all of these would seem very different if the order of the component parts were changed? It would probably depend on the specifics of the changes. For example, if in The Goldberg Variations, the return to the opening “Aria” were taken away from the end and placed somewhere in the middle, that would feel like a drastic change, probably a change for the worse. If the imposing “Variation 16” were moved away from the position that it occupies in the middle of the work that would probably seem odd and weaken the overall shape of the piece. Moving other movements around might make considerably less difference. In Clavierübung III, if the closing fugue were moved into second position right after the prelude, leaving the work to end with the four duets, the structure of the whole would seem radically different. If the four duets were distributed throughout the piece rather than placed in a group at the end that would seem like a real change as well.

In all of these cases we have reason to believe that we know what order Bach intended. With The Art of the Fugue we approach the same sorts of issues from a less anchored place.

I will continue this discussion next month, attempting to tie it in as closely as possible to some of the specific compositional details of The Art of the Fugue and, in turn, to performance as such.

To be continued.

Further thoughts about rhythm

In the very late 1960s and early 1970s, when I was first interested in the organ and listening to a lot of organ music, I had an LP recording of late-nineteenth-century French organ selections, pieces and excerpts of pieces by several composers, played by several different organists. It was a miscellany, a real sampler. All of the pieces were new to me then, as were the organs and the performers. The piece that impressed me most was the first movement of Charles-Marie Widor’s Sixth Symphony, played by Xavier Darasse. I should probably say that the passage that impressed me was the opening of that movement (Example 1). I remember quite clearly, fifty years later, that I listened to the opening of that track over and over again. I probably destroyed that part of the LP, but it helped to solidify my love for the organ; thus it was worth it!

Several weeks ago, as I decided to write a column or two focusing on various aspects of rhythm, that passage started going through my head again. It has scarcely been out of my head since, except when I have been listening to something else. Every passage of music has some relationship to the concept of rhythm. In spite of my early love for it, the first movement of that Widor symphony is not one that I have performed or analyzed. But thinking about it now and finally analyzing it a little bit, I think that there are all sorts of interesting things occurring with the rhythm of those first few measures. In particular, there are fascinating relationships between the rhetorical and theoretical aspects of rhythm, and this is part of what I address in this column.

The rhythm of the striking opening chords of the movement is demonstrated in Example 2. The rhythm of those powerful chords, and of the melody that is their treble line, is treated as a motive throughout the movement. But the surface rhythm, the rhythm of new notes, whatever voice or part of the texture they come from, is demonstrated in Example 3. That extra quarter note is a passing note in the pedal part, the bass line.

This is all straightforward, just a description of what is in the score. But it is fascinating to me that I hear two things going on at once, two different ways to describe the rhythm of this passage, both valid and meaningful. This seems to be a wrinkle in the relationship between counterpoint and rhythm. Lines of counterpoint, in the way that we usually conceive of them, are almost always characterized in part by having at least some rhythmic difference one to another. But here we have a passage that is seemingly homophonic. But the fact that the rhythm of that last quarter note of the second measure comes across as being on a lower level of importance rhythmically injects an element of counterpoint. This is subjective, but it is an interesting confirmation that the bass/pedal line is used over the next few measures to open the passage into more and more palpable counterpoint, as shown in Example 4.

If we do not know where notes are coming from rhetorically, then our sense of what the rhythm of a passage is can only be the surface rhythm. Turning that the other way around, if we notice hierarchies of rhythmic importance in different notes within a texture, that may be a clue as to some of what is going on rhetorically and contrapuntally in that passage.

For me, this is a new and slightly different way of looking at the relationship between rhythm and counterpoint. This means that I have not yet worked out how to help students apply it to pieces they are working on. There is a lot of music—from Byrd fantasias through Beatles songs, and including a lot of organ repertoire from all time and places—that is clearly not fully, rigorously contrapuntal, but in which counterpoint keeps breaking through. It is probably true that the vast majority of tonal music falls into this category. But nonetheless I have always had a problem feeling comfortable with it conceptually. Is it counterpoint or is it not? I understand that this is just the imposition of a rigid framework. But still, the concept that I am sketching out here seems to be able to help me get more comfortable with counterpoint flowing in and out of a piece or a passage.

My second answer to why this passage started going through my head has to do with the relationship between rhythm and rhythm as rhetoric. The rhythm of the notes at the beginning of this piece is well-defined and clear. But what is that rhythm doing? In a lot of circumstances, the rhythm of the first measure—just two half notes—would suggest a downbeat and an upbeat. But the way that I hear this first measure is something different and harder to describe. I hear each of those chords as a kind of world in itself, neither coming from anywhere rhythmically nor leading anywhere, but just rather insistently being. It is as if the second chord has so much gravity and weight that it refuses to be in a hierarchical relationship to the first chord or to the downbeat of the next measure. This is subjective, my way of hearing it. Assuming for the moment that this is correct or at least meaningful, is it about rhythm? A plausible and normal answer to that would be no. The rhythm is what it is, and everything else is a different aspect: affect, aesthetic, sonority, the push and pull of the harmony, interpretive choices, and so on. But it strikes me that it might be more interesting to expand the concept of rhythm to include more about what the rhythmic pulses and impulses are doing.

I believe that this concept or image could be interesting and helpful to students. It might provide a way of broadening the comfort zone of some students who are making choices about how to precisely execute rhythms on the page. If so, that would probably be through allowing choices about freedom of rhythm, bending and stretching the notated rhythm to feel more like an essential part of the rhythm itself. It might also provide me or any teacher with a way of helping the student to think about interpretive rhythmic choices without simply suggesting details of those choices to the student. I am now eager to work with a student on this piece!

In a way, I have put the cart before the horse, describing some ideas that occurred to me once I decided to write a column about a certain subject. As a result of one conversation with colleagues and a few interactions with students, I revisited a few of my ideas about rhythm, the relationship between notated rhythm and rhythm in practice, and certain practical matters about teaching rhythm. The latter include metronome use, counting, how to approach counting during slow practice, and a few other matters. The core of what I plan to write about is a concept that intrigues me: the possibility of deriving rhythm fundamentally, though only in part, from something other than the notation. I will talk about this at length in my next column.

I finish this column with a few more circumscribed yet fascinating points. The first is an anecdote from well over thirty years ago that has stayed with me all this time. I was then beginning to look for ways to participate in chamber music, and I connected with various colleagues as best I could. There was one player with whom I had a session or two of running through pieces and with whom I started to talk about giving concerts. As I came out of one such session, another colleague caught sight of me and said concerning my rehearsal partner, “You know, she really can’t count.” I did not know what to say, and I ended up with something like “Umm . . . ok,” and I did not let that comment affect my decision to go ahead and work with this fine and interesting musician. That comment was false in experience. This player had no more tendency to misread a rhythm or waver in rhythm or tempo than anyone else. But she was someone who often played purposely and quite freely. I learned a lot from her in this respect. I was left wondering what the real source of the uncalled-for carping criticism was. Did that other colleague have a bad experience with the person with whom I was working? Or misremembered or mistaken her for someone else? Or had it been a case of mistaking intentional, interpretive rhythmic freedom for an inability to count?

This latter idea always intrigues me and can be confounding. How do we know whether something that we hear (as to rhythm, for the purposes of this discussion) that departs from the most literally accurate is a mistake or a purposeful gesture? What different attitudes do we bring to such an event if it is one or the other? Is there a gray area in between? The attitude that we bring to mistakes that our students make is pretty clear—it is part of our job to point them out and help the student to understand what the problem is, how to correct it or to avoid similar things, and so on.

But what if the student says, “No, I meant to do that?” There is a strong pull to ask why and to accept that any deviation from what seems to be on the page is all right if there is a good reason. The discussions that arise from grappling with situations like this can be very fruitful indeed, but I have always thought that it is too restrictive. I hope that the concepts I will discuss next time can be used to help students understand what is going on rhythmically when they feel the pull to do something other than what seems to be the literal meaning of the notation but cannot express why. Furthermore, I hope that this can also help teachers address this situation with students without simply dictating outcomes.

I am reminded of a review that I read once of Joseph Szigeti’s recording of the Bach sonatas and partitas for violin. I apologize, as I have been unable to find this review, an unusual issue for the internet era. I am certain that it was written by B. H. Haggin. As best I remember, Haggin liked the recording and got a lot out of the playing and the pieces. But he also felt that the pieces themselves were not very interesting, that Szigeti, with his rhythmically free and idiosyncratic approach to Bach interpretation, had made great music out of pieces that were intrinsically dull exercises. I believe that this assessment was not as unexpected at that point in history as it would be now, though I do not agree with it. It seems to cast an interesting light on rhythm in performance. The reviewer’s perspective was that the player’s striking rhythmic choices were what we might call “wrong” in the sense that they were not really based on anything intrinsic to the music. They were imposed upon the music and thereby made the music great when it really was not. Is this a good situation or a bad one? How would we react if we thought we heard this happening with our students?

The last item that I mention here is an observation that I made while driving recently. I rolled down the window and shortly thereafter heard the sound of two cars passing me in quick succession going the other way. There were two whooshing sounds probably about 0.7 seconds apart. There had not been any cars ahead of them for a while, nor were there any following behind. So the two sounds were isolated. Although translated into the terms of musical rhythm this was just two notes out of any context, and it immediately evoked for me a very specific moment, namely the opening of Beethoven’s Sonata in E Minor for piano, opus 90 (Example 5).

The two car sounds seemed to deliver the rhythm of this opening gesture. Why and how? Two notes in a row is so commonplace in music that it is almost silly to evoke a specific instance of it as a thing in itself at all! Even if the two cars sounded clearly like an upbeat and a downbeat, that hardly narrows things down at all. And it cannot be harmony or sonority—what I heard had no pitch or harmony, and the sonority was that of a couple of cars. So what made that passage come into my head? I do not have an answer, but it adds to my sense that there is something more to rhythm than what we see notated on the page or can describe in words.

A special note: I will be playing selections from J. S. Bach’s The Art of the Fugue on harpsichord as part of the New York City concert series Midtown Concerts on Thursday, October 28, at 1:15 p.m. at the Church of the Transfiguration, 1 East 29th Street, New York, New York. I take the liberty of mentioning this since I have written extensively in these pages about my Art of the Fugue project. This will be the first public manifestation of that project and my first public concert in two-and-a-half years. If any readers can make it, I would of course be overjoyed to see you there