Principles of Counterpoint
Fugue is usually considered the apotheosis of contrapuntal study.
In his article on fugue in The Forms of Music, the collection of his Encyclopedia Britannica articles, Donald Francis Tovey suggests that fugue is not so much a form as a set of textural procedures: The decision to write a fugue implies almost nothing about large scale form, compared to, say, a sonata. Even the seemingly obvious proposition that a fugue consists of an alternation of entries and episodes is contradicted by several fugues in the Well Tempered Keyboard, which have no episodes at all, e.g. the C major fugue in Vol. 1 and the D major fugue in Vol. 2. A sonata, on the other hand, despite enormous flexibility in the way the details can be realized, does dictate some major tonal (and, in certain periods, thematic) points of reference.
As in our other online books, we will not supply a substitute here for a full-fledged textbook on fugue (readers are referred to Gedalge’s excellent Traité de la Fugue). However, we will make some observations about the best approach to studying fugue, and also propose definitions of the components of a fugue which correspond to Bach’s actual practice, as opposed to scholastic models. (For a thorough musicological study of Bach’s fugal practice in the Well Tempered Keyboard, see Ludwig Czaczkes, Analyse des wohltemperierten Klaviers, unfortunately not available in English.) We will also make some comments about the artistic aspects of various fugal procedures. (Gedalge’s chapter, “The Musical Composition of the Fugue” is excellent in this regard, although it stays within the context of the school fugue.)
Before beginning, more about the school fugue (“la fugue d’école”):This practice form, particularly common in French musical pedagogy, is an artificial construction, corresponding to nothing in the standard repertoire. Its justification for existence is the fact that it gives the beginner a road map in planning his first fugues, and systematic practice in all the major fugal techniques. Pedagogically, however, this road map is overly standardized. It is best used only for the beginner’s first few fugues, and then gradually opened up to allow the student more individual choices. In the end, fugue should be approached as real composition. This implies that the formal design will be an outgrowth of the material, rather than a mold into which the material is poured.
Whatever the pedagogical system used, the study of fugue is best seen as an opportunity to explore the musical development of a given theme, and possibly a countersubject, in a concentrated way. It stimulates invention, due to its requirement that the composer constantly recombine a small bank of motives into convincing new melodies.
Normally, composing a fugue requires one to build a substantial musical structure without major contrasting ideas. Put differently, the success of a simple fugue depends entirely on the ability to build intensity, by imaginatively developing one main idea, in an imitative texture.
Let us examine the elements of a fugue, one by one. In all but the final section of this discussion we will refer to Bach’s practice as the norm.
The theme (subject)
A fugue should be a natural outgrowth of its theme(s). While it makes sense for a beginner to use subjects written by others, at some point it is important for the student to write his own themes. A good fugue theme has the following characteristics:
- It is concentrated, avoiding too many different motives; this helps it to have a strong and memorable character.
- It is melodically interesting enough to merit repeated, prominent presentation.
- it lends itself to fragmentation and to various sorts of imitation.
One of the goals in teaching fugue is to make the student (quickly) aware of the developmental potential of a given theme; this is another reason for requiring him to eventually compose his own.
Apart from the contrapuntal possibilities of a given fugue subject, its musical character will strongly influence the formal layout of the fugue. No fugal analysis is complete without considering the relationship between its theme and the way the composition is worked out. To take three striking examples:
- Bach, D major organ fugue, BWV 532: The virtuoso instrumental theme gives rise to a fugue whose primary characteristics are speed and élan. The highly repetitive subject is never presented in close imitation, and it is punctuated by a huge gap. The countersubject consists of the repetition of two simple motives. The interest of this fugue depends entirely on its modulations and on the excitement of the imitative “conversations”, combined with sheer speed.
- Bach, Well Tempered Keyboard, second book, fugue in E major: The subject is vocal in character, and derives its interest from the singing curve of each phrase, the close imitations, and the richness of harmony created by the combined lines. This fugue could be sung beautifully, as is, by a vocal ensemble.
- Bach, Well Tempered Keyboard, second book, fugue in G major: The subject is in a lively, instrumental style, with a range of an octave and a half. Not surprisingly, this fugue is in 3 parts (the theme needs room to move freely without clutter), and culminates in a toccata-like climax (m. 62), which grows naturally from the light character of the theme.
Tonal answer exists for one reason only: to tonally unify the first group of entries of the subject. The desire for variety during repetition, as well as the ranges of the four basic human voices (high/low; female/male) explain why composers normally alternate tonic and dominant in the first entries of a fugue subject. Certain subjects, when transposed literally to the dominant, lend undue melodic prominence to other degrees (in particular, a prominent 5th degree at the beginning of the subject will emphasize the 2nd scale degree in the answer), or, in the case of a modulating subject, lead away from the tonic/dominant axis. Tonal answer signifies a subtle modification of the theme – which must not call attention to itself – permitting the opening group of entries, as a section, to emphasize only the tonic and dominant.
The important qualification “which should not call attention to itself” lies behind the rather abstruse maneuvers often suggested for finding a tonal answer: A compromise must be reached between the harmonic and melodic changes required, and maintenance of the subject’s clear identity. This is really just an elaboration of our notion, previously presented, of close and remote variants of motives: The composer searches for the place(s) where the change(s) required will be the least unsettling. In most cases, these places involve leaps and/or rhythmic stops. This technique is also relevant outside of fugue: Sensitivity to the degree to which motivic transformations call attention to themselves is important in building any form. The composer who misjudges where the listener’s attention is likely to go can never develop a subtle sense of formal balance.
The countersubject is a recurring counterpoint to the theme, often (but not always) adding its own motives. When present – it is not obligatory – it enhances and sharpens the profile of the theme through contrast, filling in rhythmic gaps, enriching the harmony through suspensions, etc. Normally it is in invertible counterpoint with the theme, allowing each to appear as a bass to the other. However there are examples in Bach of “pseudo-countersubjects”, which are not invertible but nonetheless recur: Bach simply avoids the problematic positions! Incidentally, Bach also sometimes uses recurring motives to accompany the theme without giving them the complete melodic contour of a genuine counter-theme. An example occurs in The G major fugue of the 1st book of the Well Tempered Keyboard, where what seems at first like a normal countersubject (m. 6 ff, top voice) in fact recurs only in part (.e.g. m. 12 ff, middle voice) in the following entry, and very sporadically thereafter.
In other words, Bach’s actual fugal practice is a form of free composition, where musical invention and momentum count more than academic rigor. However, it does not suffice to simply assert that “Bach was a genius”; rather, the student should aim to understand whyBach diverges from normal practice in such cases. Invariably, his solution is musically superior.
In the opening exposition of a fugue, normally each voice enters with the subject in turn, and then continues with the countersubject (if there is one) while the remaining voices come in. Once the countersubject is complete, the line continues melodically, normally without introducing significant new material. This plan creates a natural, textural crescendo. It is important that this cumulative effect be reinforced by the details linking the entries. In particular, fugal exposition is a good place to learn how to “keep the listener afloat” by avoiding dead spots which weaken the élan. The most common such error is to harmonize entries (after the first one, by definition not harmonized) with chords having no tonal momentum. The worst such chord is usually the tonic in root position. Bach’s practice is rather to create tension leading up to an entry, say with a rising line, and/or to have the new entry itself arrive at the same time as a suspension in another voice, thus making the new entry seem inevitable. It is rare for a voice to drop out for more than a beat or two during the exposition, since this would weaken the buildup of intensity. In particular, it is inadvisable to have a voice drop out at the moment of a new entry. This creates a hole in the texture and threatens to confuse the continuity of the individual parts.
The harmonic plan of the exposition is simple: In a tonal fugue, the first two entries are always in the tonic and the dominant, respectively. However, Bach does not always limit himself to simply continuing the alternation: At times the third entry will be in the dominant, and the fourth entry will return to the tonic. Again, his overall plan is governed by artistic decisions and not by convention.
Finally, a word about interpolations between entries within the exposition: Such mini-episodes, sometimes called codettas, serve several goals:
- They break up the squareness of equidistant successive entries.
- They sometimes allow for smoother modulation (to the dominant or the tonic, as the case may be).
- They can be used to create more momentum for the coming entry.
An episode is a portion of the fugue where the subject does not appear as a melodic whole. Most episodes are constructed as harmonic sequences, and draw their motives from the theme and countersubject. Due to the predictable nature of any sequence, episodes frequently “lower the temperature” of the fugue, allowing the form to relax, to breathe. Often the next voice to present the complete subject is absent during the episode, making its return with the entry a special event. This type of episode is therefore also thinner in texture. As mentioned above, occasionally Bach writes fugues with no episodes whatsoever.
Fugue originated in the old vocal motet, where each line of text engenders a separate point of imitation. Most fugues reflect these origins, in an alteration between entries of the subject, possibly in groups of 2 or 3 voices, and episodes, where the subject is not present as a whole. While the first exposition in the fugue normally brings in all the voices one after the other, these “internal expositions” are much less predictable, both as to number of entries (sometimes only one), and as to tonal organization. It would be formally monotonous to restart each inner exposition with only one voice, so the voices which do not have the subject usually continue in free imitation, instead of stopping. The countersubject, if there is one, may or may not be present.
Finally, inner entries normally explore tonal areas away from the tonic, often reached through sequential modulation in the preceding episode. In this sense, the episodes and the inner entries form a sort of development section, although not in the sense of sonata form; in the latter, the pace and range of modulation would normally increase; this is not normally the case in fugue.
As for stretto, there are two points to be made.
First, while the convention of a series of increasingly close stretti in the school fugue can indeed create suspense, such schemes are not at all the norm in Bach’s fugues. There are many Bach fugues which use stretto here and there, in no particular order of closeness of imitation (e.g. the fugue in Eb minor from the 1st book of the Well Tempered Keyboard.) As well, Bach seems to have conceived of a special type of fugue, consisting entirely of stretto imitation. As examples, see the fugues mentioned above as having no episodes: the C major fugue in Vol. 1 and the D major fugue in Vol. 2 of the Well Tempered Keyboard.
Second, a tip: Part of the preparation for writing a fugue involves studying its subject for its motives and their potential for development, as well as looking for possible canons. In looking for canons, a useful starting point is the search for sequence within the subject: A subject which opens with a sequence automatically allows a few canons where the entries of the following part simply double the sequence unit at the third or sixth. Since the most audible part of any canonic imitation occurs at the beginning of the imitation, even if the canon breaks down after the opening, the effect can still be successfully. This rule applies even if the sequence is camouflaged,
The second motive (a2) of the theme here is simply an ornamentation of the first (a1). The underlying sequence is clear.
The pedal point, an obligatory part of the school fugue, is, once again, not a norm in Bach. Tonic pedals do often end Bach fugues, simply because they are a good way to stabilize the fairly even harmonic and modulatory rhythm typical of all fugues.
Ending a fugue
The difficulty of ending a fugue lies in stopping its momentum. Here are several common ways to accomplish this (they may also be combined):
- Harmonic stasis: arrival on a dominant and/or a tonic pedal, as a means of braking the forward motion.
- Dissolving the imititative texture: The forward movement engendered by the continuous counterpoint gives way to a simpler texture, easier to bring to a stop.
- Climax and resolution: By definition, a climax is a high point, followed by a resolution. Once the fugue has reached its culmination, the cadence provides the final resolution.
There are two types of multiple fugue:
- A full fugue is presented on one subject; its final cadence is linked to another fully worked out fugue on a new subject, and so on, up to three, or occasionally four subjects. Once the separate subjects have been developed individually in their respective fugues, the last fugue ends with one or more presentations of the combined subjects, which of course have been composed from the start in invertible counterpoint. This final, culminating combination may be set off by cadential punctuation, always a special event in the normally continuous fugal texture. If not, its arrival will be prepared by a buildup of intensity in register, texture, harmony, etc. This final synthesis neatly brings the normal, accumulative nature of the fugal form to a peak, and at the same time adds a powerful psychological boost, stimulating the listener’s memory with the return of thematic material from a previous section. This is by far the most common kind of multiple fugue.
- All the subjects are presented simultaneously, from the first entry on. The effect is only slightly more dense than a fugue with a strong countersubject, since the only really noticeable difference occurs in the very first entry.
The most important issues: flow and momentum
As we have repeatedly remarked above, Bach is no slave to formulas. When he deviates from more schematic solutions, it is always interesting to consider why. The answer is usually to be found in the way the changes either make the line more interesting, enrich the harmony, or somehow increase the élan of the piece as a whole. Given that fugue functions mainly by accumulation, rather than through strong contrasts, in each Bach fugue it is fascinating to trace the waves of growing intensity, and then to remark how details of construction are refined to reinforce punctuation, to vary the melodic line, to drive more powerfully into an entry or a climax, etc. Real mastery of fugue shows itself, finally, in mastery of musical movement. Each Bach fugue has its own character, growing out of its material, and is a unique composition with its own form.
Fugue today: new possibilities
One might wonder whether anything can still be done with this venerable form. In the twentieth century, both Hindemith and Shostakovitch composed imposing collections of fugues, for piano. As a personal answer to this question, I composed twelve preludes and fugues for piano, in 2008. Each fugue contains some technical novelty. Examples include, among other possibilities:
- a slow, serious fugue, ending with an exposition in reverse.
- a stretto fugue, in which each stretto increases the size of the leaps in the subject.
- an energetic fugue with a very wide ranging subject: It is sometimes shared between the two outer voices.
- a fugue with a non-monophonic subject.
- a fugue on a subject in an odd, symmetrical mode, leading to an unconventional answer. It ends by fragmenting into silences.
- a large imposing fugue, with several entirely homophonic episodes.
(Links are to performances of these preludes and fugues. Links to the entire set can be found here.)
Dramatic Fugue: Beethoven, Mahler, etc.
Fugue is not inherently a dramatic form, built around sudden, major contrasts. Rather, its intensity results from accumulated rhythmic, harmonic and contrapuntal momentum. Since sonata forms develop through contrasts, and fugal forms develop by more or less linear accumulation of intensity, the combination of the two can engender intriguing large structures, with varying degrees of élan.
Composers have, in fact, experimented using fugue in hybrid forms. For example, the fugue in Bach’s giant E minor Prelude and Fugue for organ (BWV 548) combines fugue and concerto principles. Following the powerful final cadence of the first fugal section, the monumental middle section alternates fugal passages and more brilliant, toccata-like writing. The work is rounded off by the literal return of the first section. Note the difference between this kind of movement, framed by one recurring fugue, and limited to one subject, and the typical toccata style in Buxtehude and other predecessors of Bach: The latter virtually never bring back fugal material previously heard: Their works therefore typically have a more improvisatory quality.
Beethoven introduces fugues into sonata style in many of his late works, exploring the possibilities of dramatic interruption. In these pieces, a climax in the fugue leads to an inconclusive stop, followed by a return or an elaboration of some previously presented (non-fugal) material, often from another movement. Eventually the fugue returns and leads to a more powerful culmination.
These possibilities are carried farther by Mahler, e.g. in the final fugal movement of his 5th Symphony. Beethoven’s dramatic use of interruptions and contrasts within the fugue is here expanded in time scale, and with the full orchestra. The principle is the same, but the emotional range is wider.
The difference between such hybrid forms and, say, the insertion of a fugato section into a sonata movement, has to do with overall proportions (the fugue is not just a small episode in a non-fugal whole), the material used (the non-fugal sections may use contrasting material, whereas in a fugal episode in a sonata, usually the fugato theme is derived from the movement’s main material), and the fact that the fugue returns as a major formal turning point.
Another critical aspect of such dramatic hybrid forms is the way the fugal sections are knitted into the whole: The fugal and non-fugal sections are connected with propulsive transitions, rather than squarely marked off with solid cadences.
In this example, from the my 5th symphony, a vigorous fugue arrives at a climax (the cymbal crash in m. 88), which serves as a turning point into a contrasting section, in a simpler, stratified texture. This section is not based in any obvious way on the fugal material; its goal is rather to increase the music’s emotional range.
Finally, let us mention an intriguing possibility: using fugal textures in the orchestra simultaneously with non-fugal continuity, in two different planes. This area remains largely unexplored.
Canon is a venerable form, with roots in folk music, children’s rounds, and art music going back many centuries.
Most textbooks on counterpoint enumerate the various sorts of canon. For each type of imitation there corresponds a type of canon; it is not necessary to repeat the list here. However not all types of canon are equally musically interesting or useful. Some are so abstruse as to be nothing but musical puzzles, of mainly recreational interest. The less audible the imitation within a canon becomes, the less likely it is to find application outside of such musical games.
The most common sort of canon, by far, is that which is usually presented as the simplest: the two part canon at the unison or the octave. However its simplicity is deceptive. It is easy to see and to hear, but it poses a serious problem of harmonic monotony. The reason is obvious: the following voice is always repeating the same pitches as the leader, which, in turn, suggests the same harmonies yet again . If this harmonic stasis is not overcome, the canon becomes an endless and aimless harmonic circle. There are three common ways around this problem:
- Using third related harmonies to avoid repeated chords.
Notice how the arrival on B in measure 3 of the leading part, implies an E minor chord, instead of another C major chord.
- Reinterpreting passing notes as chord tones and vice versa.
Notice how the A – an accented neighbor note – in m. 3, becomes a chord tone in measure 4.
- Adding a free part, most often in the bass. In effect, this is a way of making the first two solutions more easily audible.
Other canons, found with some frequency, include two part canons and canons by inversion, at various intervals, often with added basses.
An unusual form of canon, which seems to have been invented by Brahms, may be called the “variation canon”: here the following part is an ornamented version of the leading part. A beautiful example can be seen in the Brahms-Paganini Variations for Piano, Book 1, Variation 12.
PASSACAGLIA AND CHACONNE
The passacaglia and the chaconne are continuous variation forms. The variations tend to be largely contrapuntal; each variation develops its own motive(s) in imitative or stratified texture while repeating the basic melody (passacaglia) or harmonic progression (chaconne).
As in any set of variations, the difficulties with the overall form are caused by the potential monotony of multiple adjacent sections, always of the same length and in the same tonality. The best solution to this problem is to create irregular groups of variations, through similar motives, textures, progressions of note values, etc. Such grouping allows the creation of higher, asymmetrical formal units, mitigating the obvious periodicity of the form.
In this example, from my 3rd Symphony, the passacaglia theme (here in the bass) is presented, first in a fragmentary way (in the introduction), under a slow oboe melody. In Var. 1 a more active flute melody appears above the theme. Var. 2 follows, with a more energetic string texture. Thus, the three presentations create a progression in musical momentum.
After a series of grouped variations, a major contrast of some sort usually follows.
- The pedagogy of counterpoint
- Stylistic assumptions
- Voice leading
- Compound line
- Melodic structure and ornamentation
- Motives and coherence
- Neutral lines
- Harmonic definition
- Classifications of contrapuntal texture
- Invertible counterpoint: a special case
- Counterpoint and orchestration
- Specific instrumental idioms and motives
- Passacaglia and chaconne
- Counterpoint in non-polyphonic forms