Everyone knows what counterpoint is: two or more melodies sounding simultaneously. Everyone also knows, or at least has a sense, that counterpoint is a sophisticated business, abstruse, complex, and mathematical.
Yet it's also clear that you don't have to be a musical sophisticate to comprehend and even enjoy counterpoint; ask any child who's sung "Row, Row, Row Your Boat" or "Frere Jacques" as a round . The pleasure involved is actually rather primitive at base, a dizzy buzz in the face of simultaneity. It may be enhanced by the more meditated impression that, in following more than one melody at once, we're enjoying an expansion of our normal capacities: "I didn't know I could do that!" something in us exclaims. If the combination of melodies is an extraordinary oneat once very complex and very beautiful, saythe exclamation may be altered to "How did [insert genius here] contrive that?" On rare occasions, a combination will be so extraordinary as to suggest the miraculous ("How can that exist?"), a whiff of which, when it comes to the pleasures of counterpoint, is the most exalted of all.
If you ask the person in the street to free-associate with the name "Bach," "counterpoint" would probably be among the first responses. As usual, this oft-consulted sage is onto something; counterpoint is not only at the heart of Bach's music, but his mastery of it is unparalleled. (It wouldn't be going too far to say that no artist of any kind has ever been better at anything.)
Counterpoint is ubiquitous in Bach, so much so that on the rare occasions when he isn't writing multiple voices, he's implying them. The first prelude of The Well Tempered Clavier has just one voice in the sense of a literal line, its bass . This voice supports a superstructure of broken chords, or arpeggios (literally, "harplike" figures) , and yet these arpeggios imply no fewer than four voices of their own (each shown here in a different color) . A passage in the gigue from Bach's C-major suite for unaccompanied cello has a single voice, period, but Bach makes it sound like a playful combination of an upper and a lower one . 1
Bach's fundamental commitment to counterpoint gives rise to a wealth of particularized contrapuntal techniques. These can be grouped, biota-like, into two "kingdoms": the imitative and the nonimitative. The former gets more critical and analytical press, but the latter is just as worthy of attention. Its most obvious variety is one in which contrapuntal lines not only don't imitate one another, but don't resemble one another in the least. We've already considered Bach's best-known essay in such extreme disparity, the relationship (or lack of it) between chorale tune and original melody in his "Wachet Auf" chorale fantasy. As we saw earlier, these melodies aren't just very different in character, but seem (though only seem) to go about their business without the slightest regard for one another . In fugue 10 from the Art of the Fugue, a thrilling subject is joined along the way by the collection's preternaturally poised master subject . In Fugue 4/I of the WTC, Bach combines its portentous subject first with one secondary subject and then with another in the course of what develops into an enormous triple fugue. The secondary subjects don't just differ greatly from the main one but considerably from one another, a multi-heterogeneity that helps us follow all three subjects at once when, as the triple fugue concept dictates, Bach eventually presents them simultaneously .
As striking as such blatantly nonimitative counterpoint can be, it's a relatively special effect in Bach's work. More common is a kind of counterpoint in which the lines differ markedly, but not so greatly that difference is the point. It's subsumed instead within a single, composite conception. The opening of prelude 19/I in the WTC is a paradigm case of this practice. Each of its three lines has a distinct character. The topmost is tunelike , the lowest an exemplary bass . The middle line is sketchier than its companions, but no less coherent and rounded . Each of these lines knows its own mind, yet they collaborate in a gestalt that's otherand greaterthan the sum of their individual selves . In the fugal "Et in terra pax" chorus from the B Minor Mass, a lilting subject is joined by a running countersubject that, far from opposing or competing with the subject, wreathes it in jubilation. (The exuberant little yelp at the countersubject's conclusion is the sort of detail that elevates mere genius to sheer genius.) The subject of fugue 23/II in the WTC, its notes parcelled out and patterned like a constellation of stars , serves as the trellis for an extraordinary contrapuntal garland .
The contrapuntal lines in these last few examples are at once collaborative and self-sufficient. Bach sometimes "descends" to a kind of counterpoint whose lines collaborate so closely that they relinquish self-sufficiency. A good example of such writing appears in fugue 13/I of the WTC. It takes its time doing so, though. The fugue's subject, an especially happy inspiration , is joined at first by a countersubject that's plenty self-sufficient, so much so that it performs a couple of pirouettes in its course . In the fugue's first episode, a new figure is introduced. (The inversion of this figure in the middle voice is a wrinkle Bach will presently make much of.) The subject now returns, joined by the pirouetting countersubject, but also, unexpectedly, by a rising line in the bass constructed from the inverted first episode figure . This line is too mechanical to stand aesthetically on its own; it's purpose-built, rather, to support the subject. (Which doesn't stop the ascent implied by its upper notes from making beautiful counterpoint with the subject .) Bach gives this rising line greater prominence when he restates it in the middle voice, again combined with the subject, a few measures later. The pirouetting countersubject quiets here into sustained notes that allow the rising line to be heard more clearly still as it knifes its way up through the heart of the texture . Like the rising line, these sustained notes yield character in the service of a larger, collaborative conception. 2
A pinnacle of collaborative counterpoint occurs in one of Bach's most familiar pieces, the Air from his D-major orchestral suite. So graceful is this piece's singing melody, or cantilena, that it's easy to overlook the support it occasionally needsand receivesfrom the lines that accompany it. This intermittent dependency is established at the melody's outset, where its opening note is held for an entire measure. Interest is supplied from below by the air's signature jumping bass and a change of harmony from I to VI . When the melody sustains another note a measure later, the second violin line, quiescent to this point, comes to momentary life beneath it. When the melody sustains yet a third note, the second violins quicken once again, a hint that they'll be playing a more active role going forward .
For the first six measures of the Air's second half, the melody needs no help whatsoever (which doesn't stop the second violins and the violas from making some expressive contrapuntal contributions) . Now come a couple of measures where the melody, though not requiring help, can use some nonetheless, given its relative inactivity . And help from the other voices is at hand. Contrapuntal relations don't get any more intimate than those here between the melody and the second violins, an interlinear pas de deux whose dissonances and resolutions suggest a series of embraces and releases . While the second violin line, like the melody, is almost self-sufficient, it's prevented from being fully so by its hesitations, little rhythmic blanks that the melody fills in . The viola line, for all the interest supplied by its successive syncopations, is also designed to contribute to the larger polyphonic conception . The bass is self-sufficient, but only in a steadygoing, basslike way . Bach's combining of these unelectrifying lines into so rich and passionate a wholethe Air's climactic passageis a piece of pure compositional wizardry .
Now for what must be viewed, fairly or not, as the main event in any consideration of Bach as a contrapuntist: a look at his achievement in imitative counterpoint. We're about to enter the realm of roundlike constructions, of melodies that echo from voice to voice.
When I say "melodies," I should add "in whole or in part." For it's often pieces of melodyfigures, motives, and the likethat Bach tosses among voices, rather than melodies in their entirety. In his C major two-part invention, Bach takes the first half of the piece's main motive and bats it back and forth between the upper and lower parts, cleverly alternating inverted and uninverted forms in the process . In the opening stretch of the G minor three-part invention , an arpeggiated motive bubbles up repeatedly in all three voices . In F minor three-part invention, Bach extracts a small motive from his subject and uses it to construct an entire, labyrinthine passage of three-voice imitation. In the first half of this passage, the motive is stated at its original speed in the lower two voices, but its statements in the topmost voice are slowed, or augmented. (They're also elided, the last note of one statement serving, with intriguing economy, as the first note of the next.) In the second half of the passage, Bach flexes his contrapuntal technique by swapping the augmented and original-speed statements in the topmost and middle voices . He even throws in an additional curve, as he transforms the motives in the bass from their original up-down shape to a down-down one (the first down being precipitously down) . As if all this imitative complexity isn't enough, the harmony it engenders, a succession of unstable chords, is intriguingly irresolute. 3 The result is a passage that stretches the musical mind.
We've been looking at motivic imitation in Bach's inventions, but the practice pervades his work. If the example we've just considered is a knotty one, the Trio Sonata in E-flat major for organ has a specimen, sounding for all the world like a hurdy-gurdy, that presents no difficulties whatsoever . In the chapter on melody we encountered two examples of motivic imitation from the Well Tempered Clavier: a memorable one from fugue 2/I , and a magical one from fugue 7/I . A thirdthe most remarkable in the entire WTC, if not in all of Bachappears in fugue 5/II.
Or rather, is fugue 5/II. This fugue's brief but lovely subject is worth pondering in its entirety note its semi-symmetry (two motives matching in rhythm but not in shape, separated by a fulcrum of a suspended note) and the fact that it begins, unusually, on the subdominant (IV) but what I want to focus on is the motive that concludes it . Bach insinuates this motive into the voices that accompany the subject's first few statements . On the basis of this much, you'd suspect this motive will play an important role going forward, but you'd have to be a serious seer to guess just how important this role turns out to be. Here's the entire fugue; to help you hear the motive's many occurrences, I've transcribed the piece in "open score"i.e., placed each fugal voice on its own staff, where it's easier to followand surrounded each occurence of the motive with a red box . As you can see, the piece is riddled with the motive. Several passages are in fact built, like a construction of lego blocks, out of little but the motive . You might expect a piece making such extensive use of a single component to sound manufactured. This fugue is striking, on the contrary, for its sustained suavity. A look at the first lego-like passage helps explain this seeming contradiction. In most of the motive's occurrences here, its passing note makes a seventh with a note in another voice . Each of these sevenths is part of a seventh chord. Seventh chords are harmonically plush, and their prevalence in this passage softens the edges of Bach's joistwork. As with this passage, so with the other two lego-like ones (as well as, to a lesser but still considerable degree, with the fugue as a whole). 4
To turn from Bach's imitation of motives to his imitation of full-fledged melodies is to turn, first of all, to his fugues. Every fugue begins with an exposition, a section in which a subject enters successively in multiple voices. (The word fugue comes from the Latin fugere, "to flee," a reference to the subject's "flight" from voice to voice.) This process is particularly easy to follow in the exposition of fugue 10 from The Art of the Fugue. We've already encountered this fugue's subject; the octave leap that launches it makes its entrances impossible to miss . In the first chapter we touched on the subject of the fugal "Kyrie" chorus that opens the B Minor Mass. This beautiful subject engenders the most magnificent of expositions; one listens with mounting awe as the music builds, voice by voice, in richness and momentum .
Beyond their presence in fugues per se, fugal expositions pop up in a number of other genres (including some seemingly unlikely ones) in Bach's work: in some gigues, for instance, like this exhilarating one from the fifth partita for clavier . They're also found in some of his concerti grossi (concertos for multiple solo instruments). In the exposition of the fourth Brandenburg's fugal finale, Bach treats us to one of his most winning surprises, an "extra" subject entry in the solo flutes, whose beguiling timbre, which he's saved for this moment, soars above the churning orchestral strings like sunglow gilding the top of a cloud . Bach begins some concerto grosso slow movements with what might be called a semiexposition. The subject in these movements enters only in the solo instruments, the orchestral ones being content to sit out the imitative action. The slow movement of the "Double Concerto" for two violins opens with an exquisite example of this procedure .
For all its importance in Bach, fugue isn't the last word in his imitation of entire melodies. This distinction belongs to a kind of imitative piece called a canon (as in "law"; an apt connotation, as we'll see). In a canon, one statement of a melody, termed the dux (Latin for "leader"), is imitated by another, the comes (Latin for "follower"). Canonic imitation calls to mind a subject's flight from voice to voice in a fugal exposition. In an exposition, however, the subject statements occur in succession, whereas in a canon the comes overlaps with the dux, a more difficult arrangement to contrive.
Bach wrote nowhere near as many canons as fugues, but when he did write a canon it was often extraordinary. Some good cases in point are found among the canons in the Goldberg variations. These thirty-two pieces are all built over the same master bass line. To show how this method of construction works, here's the beginning of the Goldberg bass , here's the beginning of the (noncanonic) Variation 1 , and here's how the latter fits over the implied former . Bach must have been feeling his oats even more strongly than usual when he decided not only to cast nine of the variations as canons, but to have the dux and the comes in them begin at successively greater intervals from one another: unison, second, third, and so on, all the way up to the ninth.
The easiest Goldberg canon to follow is the one "at the sixth," i.e., where the comes begins a sixth above the dux . Canonic writing doesn't get much clearer than this. Nor much more appealing (though its appeal is greatly enhanced by the sprightly bass Bach adds beneath the canonic voices). 5 Not that this sunny piece is shadowless; I'm thinking of a brief stretch in its second half, where the template of the master bass directs the tonality from G major to E minor. Until now the piece has been marching blithely along in a succession of foursquare phrases . As the harmony darkens, the phrase structure responds by becoming less settled; these are the only measures in the piece where the dux, the governing voice from a rhythmic standpoint, syncopates across the bar line, twice. The resulting unrest is heightened by a little influx of urgency as a quarter-note ascent in the canonic lines quickens into an eighth-note one . These rhythmic rufflings notwithstanding, the piece flows so gracefully that it's easy to forget it satisfies not only canonic requirements, but the requirement of fitting over the implied master bass . Nowhere in Bach do comparable exertions yield a more effortless-seeming result.
The Goldberg canon at the seventh, on the other hand, seems anything but effortless: its sinuous intricacy all but flaunts the contrapuntal labor of its construction. But it was just Bach's luck (though luck grows, as Ben Franklin said, from the soil of preparation) to hit in the course of this labor on an extraordinary harmonic possibility. Here's the canon's first half; among the chords in this stretch are some strange, almost queasy-making ones . You may recognize them from our primer on harmony as augmented triads. As if Bach doesn't already have enough on his plate in this piece, he manages to work repeated appearances of this weirdest of chords into its canonic stringencies.
Maybe the most remarkable passage in all of the Goldberg canons occurs in the one at the fifth. This canon is in "contrary motion," i.e., the comes turns the dux upside down
. The passage in question is its concluding one . The dux begins it with an arresting upward leap, followed by a scalar sweep back down. The comes answers, as the principle of contrary motion dictates, with a plunge followed by an upward sweep. In the piece's next-to-last measure the dux makes a second scalar descent. The comes must again rise in responsebut will it? A corresponding ascent would lift the comes not to the stable tonic, where the dux has settled, but to the suspensive fifth scale-step. To which it nonetheless climbs, not caviling to overshoot the last beat of the piece in the process. Is this move an emblem of fateful necessity? Cold logic? Godlike unconcern? Or is it not an emblem of anythingjust a contrapuntal process running its course? The comes's last note, hanging high in the sonic heavens, is no more forthcoming on the matter than the solitary star it calls to mind.
Some time in his mid to late twenties (no more precise dating has proved possible), Bach began working on one of his characteristically ambitious projects: a compendium of 182 chorale settings for organ. For unknown reasons, he completed only forty five of them, but these have come down to us in a collection known as the "Orgelbuchlein," or "Little Organ Book." Among these settings is one of the chorale "Liebster Jesu, wir sind hier" ("Dear Jesus, We Are Here"). Here's the chorale, first, in one of Bach's four-part harmonizations . One imagines Bach tamping some especially choice tobacco into his pipe as he prepared to create an organ setting of this beautiful tuneand suddenly noticing that it almost works in canon with itself at the fifth: almost, because the canonic lines clash in spots . A practical idealist when he had to be, Bach solves the problem by altering the lines: where dissonance was, euphony is . If the piece nonetheless suggests something of a struggle within canonic constraints, the suggestion lends drama and even poignance (like Houdini's struggles in his chains) to a remarkable feat .
A piece by Bach doesn't have to be a canon to contain canonic writing. To take a very familiar instance, the two-part invention in F major opens with a stretch, heard chiming from cell phones of late, in which a dux and comes combine to produce a delightful lattice of sound . (Bach "cheats" in canonic terms in measure 8, where the comes departs ever so slightly from the dux: a liberty that, as tiny as it is, he would never permit himself in a canon proper.) In the two-part invention in B-flat major, Bach takes the head portion of its opening motive and spins a modest length of canon that, if quieter than the F-major invention's, gives comparable pleasure .
While patches of canon can be found throughout Bach's work, most of them occur in a province we've already visited, that of his fugues. They arise there from a technique called stretto, an imitative layering of subject statements. In fugue 7/II from the WTC, for example, Bach combines a subject entry with a stretto entry beneath . Stretto plays a prominent role in a fugue we've already looked at, WTC 5/II. It figures here as a kind of subplot, a story of ever-increasing contrapuntal intensity. Stretto enters the fugue unobtrusively in measures 5 7, where a dux statement of the subject nearly concludes before a comes statement answers; not much layering here 6 . Layering is beautifully present, however, in the piece's second stretto (in measures 14 16), where the comes enters before the dux is even halfway complete . Bach ups the imitative ante in measures 21 24, where he unfurls a three-entry stretto . 7
In measures 27 29, Bach changes the imitative game altogether by springing a two-entry stretto whose comes enters on top of a dux that's barely begun. Such a "close spacing" of stretto entries is a surefire (and time-honored) technique of contrapuntal intensification . You might think you're hearing three entries here, but what sounds like an entry in the alto is actually an incomplete, or "false" one (a common bit of imitative chicanery) . Having teased us with the illusion of a closely-spaced three-entry stretto, Bach delivers the actual, amazing goods in measures 33 35: all three subject statements here are complete . This being a four-voice fugue, a sadist might ask Bach to cap its contrapuntal wonders with a closely-spaced four-entry strettoa challenge he in fact steps up to, near the fugue's conclusion, in a climactic cascade of subject-statements .
In a few of Bach's fugues, stretto isn't just a subplot but the main storyline. An example of such a "stretto fugue" is WTC 1/I, seventeen of whose twenty two (!) subject statements are involved in stretti. The piece sounds even more stretto-happy than it is, as Bach adds five false stretto entries to the mix. Here's this remarkable fugue in its entirety, with the subject statements shown in color (including the false entries, which I've enclosed in brackets) .
A stretto fugue as remarkable, in a different way, as WTC 1/I is WTC 8/I. Not that the straightforward exposition of its austerly beautiful subject hints at anything unusual to come . In measures 19 22, Bach slips a two-entry stretto into the proceedings , a neat enough trick, but nothing extraordinary (yet). In measures 24 26 Bach gives us a three-entry strettoor seems to. Only the topmost voice actually states the subject in its entirety; the apparent entries in the bass and the middle voice both prove false. The former trails off partway through into a running line; the latter, a "free" augmentation of the subject (one in which only some of the subject's notes are lengthened), comes closer to completion, but still falls a couple of notes short . Complete subject entries return in measures 27 30 with the fugue's second two-entry stretto. Whereas the comes in this stretto's predecessor occurred above the dux , it here occurs below it , a second stretto relationship in the piece. (One such relationship in a fugue would be enough for many composers . Two, as we'll see, won't begin to satisfy Bach in this one.)
Measures 30 32 introduce a new wrinkle, as the subject appears in inversion in the topmost voice . In measures 36 38 an inverted entry occurs in the middle voice, where it's harder to hear but, like those medieval frescoes painted into recesses where only God can see them, nonetheless there . Bach is nothing if not systematic, so it's no surprise when he presents yet a third inverted entry in the bass in measures 45 48, thus completing a downward cycle of such entries through the fugue's three voices. What is surprising is his addition here of an inverted stretto entry in the topmost voice (the third stretto relationship in the piece) .
Having broached the musical topics of false entry and inversion, Bach now combines them. In measures 47 50, an inverted entry is joined by an inversion of the freely augmented false entry heard earlier . Bach presents still more false entriesa fusillade of themin measure 52, followed by a counterfusillade in which false entries are inverted . As if to remind us that entries in this fugue needn't always be false or inverted, in measures 57 60 Bach finally grants us, high in the topmost voice where it can't be missed, a complete, uninverted statement of the subject . This statement, presented in the subject's original (and the fugue's overall) key of D-sharp minor, serves as a kind of way station, a signal event that slides, via a preparation like a deep breath or a pitcher's windup, into the fugue's most remarkable passage yet . A lot going on here: a subject entry in the middle voice, to begin with, joined by an entry in the bass in which every note is exactly twice as long as in the original subject: an entry, that is to say, in strict augmentation (rather than the free augmentation heard earlier). The original-speed entry concludes while the slower, strictly augmented one still has its latter half to go. Above this latter half, the subject appears at original speed yet a second time (in inversion no less). Against odds a supercomputer would break down in sobs at having to calculate, a beautiful subject has joined in perfect counterpoint with both the first and second halves of its strictly augmented self.
Not content to have contrived this mind-boggling passage (which contributes a couple more stretto relationships to the fugue), Bach proceeds to surpass it . Ranking great moments in Bach is like ranking miracles in order of impossibility, but this passage seems to me among his greatest. It's constructed much like the preceding one: a strictly augmented subject entry (this time in the middle voice), its first half fitting with one original-speed entry (this time in the bass), its second half fitting with another (this time in the topmost voice). There are other structural differences between this passage and the preceding one (contributing yet more stretto relationships to the piece), but the essential difference between them is that this one is surpassingly beautiful. Its beauty has a number of sources: a shift in key from minor to sun-emergent major; the vaulting leap with which the strictly augmented entry begins (more effective here in the middle voice than its predecessor was in the bass, more as though sounded on a trumpet, if a subtle, silver one); the dancing passage in the topmost voice (especially its ornamented apex); the eloquent underpinning in the bass (especially the motif sequencehugely expressive in its mix of upward yearnings and great, register-delving plummetsthat concludes it) . . . . In its fusion of complexity and beauty, this passage is, for me, the climax of the piece.
It isn't the official one, though. This occurs just before the end of the fugue, in a third passage like the two just discussed . A strictly augmented entry here appears in the topmost voice, thus completing an upward cycle of such entries through the fugue's voices (a cycle that mirrors the downward voice-cycle of inverted entries heard earlier). As in the two preceding passages, each half of the strictly augmented entry is joined by an original-speed one, the first in the bass and the second in the middle voice (contributingall together nowyet a couple more stretto relationships to the piece). The main difference between this passage and the previous twoa difference that makes this passage the technical, if not the aesthetic, capper of the setis the surprise reappearance, in the middle voice, of the freely augmented subject. In the first three measures of this passage, Bach actually presents the subject in three different speeds at once: original-speed, freely augmented, and strictly augmented. Note that the freely augmented entry isn't false, as it had been in its earlier appearances (its concluding D-sharp is supplied by the bass). Only now, under the severest of contrapuntal constraints, does this version of the subject attain completion, something it's hard to imagine any composer but Bach asking of it under the circumstances. 8 What would be too much for even Bach to have asked is that measures meeting such enormous technical demands should also be beautiful. Nor would I claim that these measures are. Like a pyramid of tightrope walkers, Bach's construction of simultaneous, differently paced subject entries is all careful stepping, grim faced business in performing its astounding feat. 9
Bach's son Carl Phillip Emanuel said his father could instantly see all of a fugue subject's contrapuntal possibilities. I don't know about instantly, but Bach certainly saw his share of such possibilities in the subject of the fugue we've been considering. 10 In presenting so many of them, did he stress intellect at the expense of emotion? Perhaps, though the display of intellect here induces a wonder that's an emotion of its own.