A Study in Fugues
A Study in Fugues
A Background; Fugues and Counterpoint
Canons and Counterpoint
A canon is a type of composition which features a melody that is imitated by one or more voices, usually at a set interval of time. The imitating voice(s) enter after the original melody has been stated, creating a layered effect as the parts build upon each other.
Canons can be found in various forms, including simple rounds (like “Row, Row, Row Your Boat” (Lyte 1852)), to complex contrapuntal works by composers such as Johann Pachelbel and Johann Sebastian Bach. In each case, the basic idea is the same: a single melody is imitated by multiple voices, creating a rich and harmonious texture.
Canons are often used as educational exercises, as they provide an excellent way for composers and musicians to study and understand musical structure, counterpoint, and melody. Additionally, they can also be used as musical puzzles or games, as listeners attempt to identify the underlying melody and figure out how it is being developed and transformed by the different voices.
Musical counterpoint is a style of arrangement in which two or more melodic lines are combined in a harmonically interdependent manner. The individual melodic lines, or voices, are distinct yet complementary, and the relationship between them creates a rich and complex musical and harmonic texture.
Counterpoint was developed during the Renaissance period and became a central aspect of Western classical music. The most famous practitioner of counterpoint was the composer Johann Sebastian Bach, who used the technique extensively in his works for keyboard, organ, choir, and orchestra.
In its simplest form, counterpoint involves combining two melodic lines in a way that they sound harmonically interesting when played together. This can involve using dissonance (clashes between notes) in a controlled manner to create tension and release, or it can involve the voices alternating and complementing each other in a more flowing way. Regardless of the specific approach used, the goal of counterpoint is to create a musical texture that is pleasing to the ear and that seamlessly blends the various melodic lines into a cohesive whole. When done well, pieces which utilise counterpoint are characterised by their rich musical texture and by the harmonic complexity that is produced (and often required) by this style. These works are amongst some of the most beautiful and intricate music ever composed.
In contrapuntal pieces, often no static (eg. ‘block’) chords are played except when emphasising particularly important musical and structural moments, ie. during cadences. The simplest examples of counterpart involve melodic leads alternating between an upper and lower voicing. Ie. between the alto and bass registers of a piano. However, even when the lead is passed off, the submissive voice is still playing a (less dominant / complex) melody - it is not harmonising to the lead voice (ie. playing chords).
A short-hand way to understand counterpoint is that: it is composed through the layering of constant countermelodies. Typically, a countermelody is used only to emphasise certain points. However, in counterpoint pieces, a countermelody (if you wish to label on voice as such) must persist throughout the whole piece, and must exist as a proper melody unto itself. Therefore, one disjunct way of composing (though not improvisation) points utilising counterpoint is to begin by composing a melody for one voice, and then to compose an independent though relatively harmonically similar / euphonic (if desired) melody in a second (etc) voice. Although, at no stage should the additional voices be seen as an 'accompaniment' instead of a proper melodic voice so-called. This is what the traditional harmonic 'rules' of voice-leading attempt to avoid. Ie. This is why it is traditionally discouraged to write melodies which involve parallel motion by fifths or by octaves. Because, these intervals are incredibly strong, and they make it much harder to maintain the independence of the voices. Ie. You don't want independent voices constantly moving by the same interval, in the same direction, at the same rate, or you'll tend to perceive them as one (harmonic) voice (in the same sense that you perceive a chord as one voice even though it has 3+ notes).
The Fugue
The greatest and most complex form of contrapuntal arrangement is the fugue.
A fugue is a musical piece characterised by the delayed restatement of a musical subject of motif in additional voices to produce a counterpointal texture. The subject is introduced by one voice, and then imitated in one or more voices, often at a different pitch level. For example, Bach's “Little” Fugue in G Minor (BWV 578, c. 1703) begins with a single (tenor) voice. For the first ~8 bars, this voice performs the fugue's initial motif.
The voices then continue to enter in a systematic manner, each presenting the subject and developing it in their own way. Ie. Once the subject has been completed by the first voice, it will then continue into melodic development of the motif. As the first voice leaves into development, the second voice (in the example, this is the piano's soprano register) enters and restates (ie. exactly repeats) the motif (albeit transposed, nearly always down a fourth to the dominant) whilst the first voice continues to play independently.
Once the second voice finishes the restatement, it too enters a phase of (independent) melodic development. Once this development is considered sufficient, the first voice will state another motif (all whilst the second voice is still playing independently). Once the first voice finishes the second motif, the second voice will restate it whilst the first voice engages in melodic development and the process repeats. Often a third (or more, often 4 voices in Bach) voices are introduced, producing an incredibly rich musical texture and an incredibly complex arrangement. Often when new voices are added, they begin with the first motif (of that movement) regardless of which motif was last stated by the first voice (as is the case when Bach adds a fourth voice via organ pedals, he does so by beginning it with restatement of the first motif).
It is worth noting here that whilst the second voice almost always features an exact (albeit transposed) restatement of the musical subject, this principle is less essential for later voices. In this context, I should explain the difference between a real answer (or restatement) and a tonal answer. In the context of a fugue, a “real answer” and a “tonal answer” refer to two different ways in which the subject can be imitated in another voice.
A “real answer” is a direct transposition of the subject to a different pitch level. In other words, the subject is played at the same rhythm and intervals, but starting at a different pitch. For example, if the subject starts on a C, a real answer might start on a G (the dominant).
A “tonal answer” is similar to a real answer, but with one important difference: the intervals between the notes of the subject are altered to create a more harmonically rich texture. The goal of a tonal answer is to create a smooth and seamless transition between the subject and the answer, so that the listener is unaware of the change of key (as is typical between the entry of each new voice). Knowing when to use real and when to use tonal answers is a key aspect of fugal composition and, when understood well, helps to create a rich and complex musical texture without over replicating or ‘saturating’ the subject.
As can be inferred from what has been said so far, the fugue was developed during the Baroque period and became a staple of the keyboard repertoire. Some of the most famous composers of fugues include Johann Sebastian Bach, who wrote many of the greatest examples of the form, and George Frideric Handel, who used fugues extensively in his keyboard and choral works.
Fugues are often praised for their intellectual and technical demands, as they require a deep understanding of musical structure and harmony. They are also admired for their musical beauty, as the interplay of the voices creates a rich and complex musical texture. Whether you are a composer, performer, or listener, studying and understanding fugues is considered an essential part of a complete musical education.
Fugal Essentials
At a base level, a fugue has three, and only three, necessary components:
One (or a number of) Subject(s)
A Response or Restatement of the Subject
A Period of Independent Melodic Development
Musical Subjects
In music, a ‘subject’ is a relatively short, dense musical idea. Usually occupying no more than 8 bars and always in one key (for later melodic and harmonic cohesion). An ideal subject is (a) memorable and (b) melodically dominant. A subject is memorable if it can act as an overall announcement and summarisation of the melodic development and harmonic synthesis of the whole piece. The subject is melodically dominant if it can be individually perceived in a dense polyphonic context.
The subject can be found in many different types of music, from classical to popular, and it can take many different forms. In some cases, the subject may be a short, simple melody that is repeated throughout a composition, as in the case of a folk song or a pop hit. In other cases, the subject may be more complex and developed, as in a classical symphony or a jazz improvisation. In all cases, the subject is an essential element of musical composition, providing a central point of focus and a basis for development and variation. It is often what gives a piece of music its memorable and recognizable character.
Musical Responses
A musical ‘response’ can be either an exact or approximate restatement of the subject, in a closely related key, which follows the subject in a separate voice after it has been wholly or partially stated. This delayed re-statement of a subject in separated voices, one after the other, is the essence of all fugues. Indeed, much of the difficulty of Fugues comes not only from the typical polyphonic complexity of the baroque period but also the challenge of harmonising (near-)identical subjects in successive voices in a melodically pleasing and interesting manner
Melodic Development
Melodic development is a section in many musical forms, notably sonatas. It plays much the same role in fugues with one notable difference. Once the first voice finishes its subject and enters into melodic development, it must wrestle and attempt to harmonise with the next voice which has immediately begun its restatement of the subject. Thus, in fugues, the initial outset of melodic development is particularly hard to compose until all voices have finished their restatement.
Once restatement has been conducted by every voice, the composer may choose to move to a new subject, either isolating the first voice or continuing melodic development in the others whilst this occurs, or they may choose to continue melodic development on this subject as long as they desire.
Transposed Subjects; Similarity in the Circle of Fifths
The exceptional texture and particular difficulties of fugues result from attempts to synthesise voices, particular during the subject-response portions of a fugue. This is the section where any voice is currently announcing the subject or its restatement and will continue until all voices have restated the subject. The reason for the particular difficulty in this section is because fugues, typically, transpose the subject between each restatement. For example, if the subject is initially stated in F Major, it is quite rare for the restatement to be in the same key. Rather, restatements traditionally would occur first in the dominant of the subject (ie. C Major) and then (where there is more than one voice) in the secondary dominant (G Major) or back in the initial key (F Major).
The reason why this transposition makes fugues difficult is because of the differences in the key signatures themselves. For example; the key C Major and its Dominant G Major are separated by one tone; the tritone of C Major (F#). As such, if I transpose a melody from C major to G Major and restate it, if there are any B’s in that melody (which would become F# following transposition) the piece will sound harmonically discordant. The same is true working backwards, if I want to compose a melody in G major which will sound okay alongside a melody in C Major, I want to avoid all F#’s (the 7th degree of G). This limitation is necessary to preserve the rich polyphonic texture and restatement style of fugues.
Below is an image of the circle of fifths. When composing subjects that will be transposed by fifths (ie. from the tonic to relevant dominant), it is important to understand how accidentals are added between keys. Moving clockwise around the circle (in major keys), it is always the tritone of the initial key that becomes the accidental for the next key. Furthermore, this tritone actually represents the seventh in the new key. Normally this would be rather trivial in close key changes however it is essential for fugues. This is because, when two key signatures related by a fifth are played sequentially, the major seventh degree of the latter clockwise key can also be perceived as the tritone of the initial key. Ie. if you wish to avoid tritones between the keys, you must avoid the seventh degree, at least in the transitory cadence(s). Note; moving counterclockwise, you must avoid the minor second, but that interval is less common, and thus less problematic to begin with. For example, when transitioning a melody between C and G, you must avoid F# and C# in the bridging cadences.
Below I have outlined two keys, a tonic and a dominant side by side two and bolded the notes in each key which are not present in the other (and thus to be avoided in cadential phrases between the keys). In other words, degree 4 is to be avoided in the initial key as it is only presented in it’s augmented form (the tritone) in the subsequent dominant key.
Thus degree 7 is to be avoided in the initial key when using 5th-wise major melodies.
Note: In Minor Keys; it is the Major 6th (augmented 6th) (rather than the tritone) that becomes the new accidental fifth-wise. Further, in inversion to the major keys, in minor keys it is the second degree (the diminished degree) that appears as the new note in the dominant key.
Thus Degree 6 is to be avoided in the initial key when using 5th-wise polytonal minor melodies.
Rules For Counterpoint: Gradus Ad Parnassum
Gradus ad Parnassum (Steps to Parnassus [Mount of the Greek Muses]) (Fux 1725) is a music theory treatise on counterpoint written by Johann Joseph Fux in the early 18th century. Even today it remains perhaps the most influential work on music theory and education written in the western classical tradition, and perhaps in any tradition. In particular, the text's considerations of counterpoint were essential for the development of the common practice period classical musical mannerisms. In this way, its stylistic influences continue right into the modern day.
The treatise itself is divided into three parts, distinguished by the aspect of musical consideration they discuss:
The first section of the text introduces quite basic notions in music theory such as musical notation, scales, temperament, intervals, and chords. Although trivial compared to the later sections, the particular conceptions of music theory taught in this part are essential for comprehension of the concepts discussed in the subsequent section.
The second part of the work focuses on counterpoint in itself, a critical aspect of Western classical music composition. Fux presents a system for the categorisation of counterpoint into different ‘species’ or kinds, separated by the intervallic types and relative motions involved between the musical lines. These types are fivefold: Note against note; Two notes against one (in the bass); Four notes against one (in the base); Notes offset against each other (as suspensions, ie. syncopations); florid counterpoint. Construction in each species of counterpoint involves melodies which adhere to strict rules about intervallic and rhythmic relationships between the voices. This system requires that composers follow specific rules about the progression of harmonies and the spacing of voices in order to achieve a harmonically pleasing result.
The final section of Gradus ad Parnassum (Ibid.) is dedicated to actual composition and arrangement, and takes great pains to show how the principles of counterpoint can be applied to create complex musical works. Here, Fux discusses how to incorporate multiple melodic lines and how to manage the various elements of a composition such as harmony, rhythm, and structure. By applying the principles of counterpoint, composers can achieve a harmonically rich and dynamic sound that is characteristic of much Western classical music.
Overall, the work was widely influential in the development of Western classical music, particularly in the Baroque and Classical periods. Many of the greatest composers of the time, including Johann Sebastian Bach, Wolfgang Amadeus Mozart, Joseph Haydn, and Ludwig van Beethoven, studied and widely used the techniques presented in the treatise. The strict rules of counterpoint and composition presented in Gradus ad Parnassum (Fux 1725) served as a foundation for much of the music of the time, and continue to be studied and admired by composers and music theorists today. With this background, let us now consider the rules themselves, at the end of which I will analyse their application to Bach’s Fugue in Gm ‘Little’ (Bach, c. 1703)
For Melody:
The melodic rules to be adhered to in writing contrapuntal melodies are as such:
The final note of the Cantus Firmus (‘fixed subject’) must be approached by stepwise motion.
If the final note is approached from below, then the leading tone must be raised in a minor key (elsewise the final motion will be full-stepwise (because of the minor 7th in a minor key) and not half-step wise, producing a less resolute ending in the upper voice and diminishing the classical voice-leading of the chord)
Melodies may move (up or down) by a third, fourth, fifth, sixth, or octave
Melodies must not move (up or down) by intervals greater than an octave (ie. a 9th), or by a seventh (particularly a major seventh), or by a dissonant (ie. minor second) or aug/dim interval (ie. tritone)
Movements of a fourth or larger should be followed by a change in direction, ideally by a step
Movements of a fifth or larger must be followed by a change in direction, ideally by a step
Each Cantus Firmus should not contain more than two leaps larger than a fourth
Avoid two leaps in the same direction
Avoid more than two consecutive leaps, even when opposed
Avoid a large leap in the same direction as a preceding step
For Harmony:
With regard to harmonic construction, that is, the construction of the off-voice whilst the Cantus Firmus is being enabled, the rules are as such:
The counterpoint must begin and end on a perfect consonance.
Contrary motion (ie. motion in the opposite direction to the melody at that time) should dominate.
Perfect consonances between the voices (ie. Perfect Fifths and Fourths) must be approached by oblique or contrary motion (so as to not unify the lines and thus diminish their melodic independence)
Imperfect consonances may be approached by any type of motion.
The interval of a tenth should not be exceeded between two adjacent parts unless by necessity (ie. to avoid perfect consonance)
Build from the bass, upward.
For Key Structure:
Separate from the restrictions on key and harmony imposed by classical rules, and perhaps a little outside the scope of Gradus Ad Parnassum (Fux 1725), are the classical stipulations regarding the keys used and the transition between the keys in fugal pieces.
In Western music (with the potential exception of fusion and jazz-derived traditions), a key refers to a system of organising and understanding tonality, or the perceived sense of hierarchy and tension created by the relationships between musical pitches. With the exception of modes, which may or may not be viewed as keys, a key is typically defined by its tonic, which is the central pitch or note around which the other pitches in the key revolve. This central note provides a sense of stability and resolution, and serves as a point of reference for the other pitches in the key. For example, in the key of C major, the tonic note is C and the tonic chord C Major. All other pitches in that key are characterised by their intervallic quality (ie. major third, perfect fifth, major seventh), which is to say by there relationship to the tonic C.
In addition to the tonic, a key is composed by a particular set of pitches, all of which are arranged according to the diatonic scale associated with the key. This scale includes seven pitches, each of which is a specific interval distance from the tonic. The order of these pitches is determined by the specific mode of the scale, such as the major (Ionian) or minor (Aeolian) mode.
The hierarchy of pitches is often used to create musical phrases and harmonic progressions that provide a sense of movement and direction. This concept of harmonic progression applies not only to motion in one key (ie. a chord progression) but movement into another key (a transposition or modulation). In this regard, as with notes in a key, not all keys are equally related, or not all keys are equivalently euphonic to a specified reference ‘tonic key’. Generally, without the use of some quite exotic cadences found primarily in Jazz, keys are most harmonious with their scalar neighbours. By this I mean that, with regard to the circle of fifths, keys situated next to each other (ie. a tonic (eg. Em) and it’s clockwise dominant key (ie. Bm) or its anti-clockwise subdominant (ie. Am)) are easiest to progress through as they share all but one note (as previously discussed in the section on transposed subjects). In light of this, the motion from one key to another has a particular emotional effect depending on the relationship between the keys. Because of this classical composers, particularly those composing in the forms of Sonata and Fugue, used an almost completely standardised progression of keys throughout the piece.
The Fugal Progression
In a fugue, the typical progression of keys follows a well-defined pattern, which is known as the ‘fugal exposition’. The fugal exposition establishes the main musical subject of the fugue and sets the stage for the subsequent development of the music.
The fugal exposition typically begins with a statement of the main subject in the tonic key, which establishes the primary tonality of the piece. This is followed by a statement of the subject in the dominant key (that key which is a perfect fifth above the tonic). This statement is known as the answer, and it usually follows the subject closely, with the same intervals and rhythm, but starting on the dominant pitch.
After the initial statement of the subject and the answer, the fugue proceeds through a series of modulations, which involve shifting from one key to another. The modulations in a fugue typically move through a sequence of related keys, with each new key providing a sense of contrast and variation to the musical material. The modulations in a fugue are typically structured in a specific pattern (tabulated below). The final section of a fugue typically involves a return to the original key, with a restatement of the subject and a recapitulation of the musical material from the exposition. This provides a sense of closure and resolution to the piece, bringing it to a satisfying conclusion.
Little Fugue in Gm
To finish this writing, I will briefly apply the rules for counterpoint above to Bach’s famous Fugue in Gm (‘Little’) (I cannot urge you enough to listen to it for yourself) (Bach, c. 1703). Below is the first thirteen bars of the piece, in its original arrangement for Organ (with three voices, including pedals):
To begin with, it is worth noting the timing of the entrance of the voices, which is quite unusual insofar as fugues go. The first voice enters on the first beat of the first bar, this is quite standard. The second voice enters on the first beat of the 6th bar, something quite unusual given the typical 4 bar phrase structure which dominates music. The third voice enters on the third beat of the 12 bar, something stranger still (given that the entrance of the voices is thus irregular; ie. the second voice took 6 bars to enter, and the third took 7 ½ from the second). Here it is important to note that, unlike many fugues, the timing of the subject and the timing of the voices are not the same. The subject takes 13 bars to complete (as shown by the minim in the first voice) whereas the voices enter every 6-8 bars, leading to the subject being stated in multiple voices at once.
Even so, we can see the characteristic standardisation of the subject between the voices. This is easily demonstrated by comparison of the first three entrances (third voice has been standardised / brought up to first beat for ease of comparison) (up to the first five bars of each voice):
The First Voice:
The Second Voice
The Third Voice
Thus it is clear that the fugue utilises an irregular albeit properly repeated subject.
Similarly, if we examine the cantus firmus itself, analysing the first voice for simplicity (and inferring the subject to end at bar 13), we see the rules of fugal melody reflected. Let us consider them one at a time:
The final note of the Cantus Firmus (‘fixed subject’) must be approached by stepwise motion. Further, if the final note is approached from below [it is], then the leading tone must be raised in a minor key.
This is perfectly reflected by the final step from F# to G (final beat of bar 12 to first beat of bar 13) (F# being the major seventh / raised leading tone in the key of Gm), a semitonal movement.
Melodies may move (up or down) by a third, fourth, fifth, sixth, or octave
Not broken in the subject
Melodies must not move (up or down) by intervals greater than an octave (ie. a 9th), or by a seventh (particularly a major seventh), or by a dissonant or aug/dim interval (ie. tritone)
Never done
Movements of a fourth or larger should be followed by a change in direction, ideally by a step
Seen many times, ie. in bar 3 beat 3 and in bar 4 beat 4 (second semi-quaver)
Occasional broken, ie. in bar 4 beats 1 and 2 (D-G-A and D-A-B)
Movements of a fifth or larger must be followed by a change in direction, ideally by a step
Broken in bar 4 beat 2 but otherwise followed
Each Cantus Firmus should not contain more than two leaps larger than a fourth
Broken
Avoid two leaps in the same direction
Followed
Avoid more than two consecutive leaps, even when opposed
Followed except at bar 3 beat 4 → bar 4 beat 2
Avoid a large leap in the same direction as a preceding step
Followed
As for the harmonic and key structure guides: All the harmonic rules are followed, the key structure is different. The piece never technically leaves Gm and, although there is continuous and heavy use of accidentals throughout, can be analysed as not containing any relevant transposition. This is something which is not necessarily a hinderance for such short a work (insofar as Fugues go), usually taking only ~4 mins to perform (in comparison to works like his other Fugue in Gm (BWV 542 “Great”, 1720) (which he improvised) at ~11 mins, his lesser known Prelude and Fugue in Eb (BWV 552 “St Anne”, 1739) at ~15 mins, or his two Toccata and Fugues in Dm (BWV 538 “Dorian”, c. 1708) and (BWV 565, c. 1704) at ~12 mins and ~9 mins respectively). Thus concludes our discussion of fugues.
Reference List
Lyte, EO 1852, Row, Row, Row Your Boat
Bach, JS c. 1703, Fugue in G minor, BWV 578
Bach, JS 1720, Great Fantasia and Fugue in G minor, BWV 542
Bach, JS 1739, Prelude and Fugue in E-flat major “St. Anne”, BWV 552
Bach, JS c. 1708, Toccata and Fugue in D minor, BWV 538
Bach, JS c. 1704, Toccata and Fugue in D minor, BWV 565
Fux, JJ 1725, Gradus Ad Parnassum, viewed 28 February 2023, <http://www.opus28.co.uk/Fux_Gradus.pdf>
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