Polyphonic procedure in which a motive (subject) is exposed in an initial tonic/dominant relationship, then developed by contrapuntal means.
A fugue generally consists of a series of expositions and developments with no fixed number of either. At its simplest, a fugue might consist of one exposition followed by optional development. A more complex fugue might follow the exposition with a series of developments, or another exposition followed by one or more developments. Fugues that are tonally centered will expose the subject without venturing out of an initial tonic/dominant constellation.
A. Main Idea of the Fugue and How It Is Stated
- Subject: Melody that comprises the primary melodic/rhythmic material of the fugue. Subjects typically have two parts: the "head" is calculated to attract attention either by unusual rhythmic or intervallic emphasis, while the "tail" is typically more conjunct, rhythmically uniform, and sometimes modulatory. The head and/or tail itself may employ variation of one or two smaller motives or figures...each comprised of a characteristic rhythm and/or interval.
- Answer: Subject imitation which immediately follows the first statement of the subject: in a different voice and usually fifth higher. Answers are a subclass of subjects which bear certain interval characteristics in relationship to the subject as it was originally stated.
- Tonal Answer: An answer that typically (though not always) stays in the same key as the subject. To do this it is necessary for the intervals of the subject to change somewhat. In a tonal answer "do" and "sol" switch places: The position occupied by "do," in the subject, becomes "sol" in the answer and vice versa. Analytical technique: Subjects having many skips (disjunct) that focus upon the tonic and dominant scale degrees lend themselves to a tonal answer.
- Real Answer: An answer that is a transposition of the subject to another key, usually the dominant. Analytical technique: Subjects having mostly steps (conjunct) that don't focus upon "do" and "sol" lend themselves to a real answer.
Countersubject: Substantive figure that sometimes recurs immediately following the subject or answer (in the same voice). Countersubjects serve as counterpoint to subjects (or answers) sounding simultaneously in a different voice. Not every fugue will have a countersubject. Some fugues may have more than one countersubject.
False Subject: Some people use the term "false subject" to describe an entry of the subject (or answer) that begins but never finishes. This term should be reserved for instances where the subject appears to enter, breaks off, then follows immediately with a complete statement. Most other instances of incomplete subjects are developmental and should be termed "imitation."B. Main Sections of the Fugue
- Exposition: Portion(s) of the fugue consisting of subject(s) with at least one answer, and possibly countersubject(s). To qualify as an exposition, the subject (or answer) must appear in all voices and answers must be in the proper relationship (tonal or real) to subjects. The exposition normally concludes immediately after the subject (or answer) appears in the last voice. Expositions may defer the cadence until after a codetta. Differentiation between exposition subtypes is based upon the order in which voices enter (as compared to the first exposition) and whether or not the subject has changed.
- Re-Exposition: An exposition, following the initial exposition, in which the voices enter in the same order as the first exposition.
- Counter Exposition: An exposition following the initial exposition in which the voices enter in a different order than they did in the first exposition, or the subject of the new exposition is a contrapuntal variation of the original.
- Double Exposition: Exposition utilizing a brand new subject (i.e. not contrapuntally derived from the first). If the new subject is unique, then the fugue is a double fugue (or, in the case of three subjects, triple fugue).
- Developmental Episode: Section in which motives from the exposition are treated in sequence, modulation, contrary motion, double counterpoint, stretto, augmentation/diminution, pedal, etc. Episodes are generally terminated by a cadence and may follow one after the other. Developmental episodes characteristically begin by departing from the subject, to fragment or vary it in some way, but gradually building up to a restatement of the subject in at least one voice. These statements of the subject are typically not in the tonic/dominant relationship of the exposition and are called "middle entries" (or in German Durchführung). Episodes typically do not enunciate the subject in all voices.
- Coda or Codetta: Concluding segment of a section (codetta) or of the entire fugue (coda). Codas and codettas often sound as if they are something added after the structural end of the section or work. The function of codettas is often modulatory (to return the tonality to the key of the subject after an answer at the dominant). Not all fugues have these.
A. Tonal Variation
B. Contrapuntal Variation
- Modulation: Repetition of a motive in another key. Bach typically arranges his fugues around closely related keys (major and minor keys immediately adjacent to each other on the circle of fifths).
- Mutation (also called "change of mode"): Statement of the subject or answer (or any other primary material) in the opposing mode. A subject first stated in minor and later stated in major is said to have "mutated."
- Stretto: Entry of a motive in a second voice before the first voice has finished its statement. Motive can mean subject, answer, countersubject, or any other substantive melodic/rhythmic entity in imitation.
- Augmentation/Diminution: Statement of a motive in rhythmic durations that are proportionately doubled or halved.
- Pedal Point: Suspension of one pitch, often the bass, in such a manner that it is alternately consonant then dissonant with the chord progression. Fugues often conclude with episodes of pedal point.
- Retrograde: (rare) Statement of the motive's pitches in reverse order.
- Melodic Inversion: (Contrary Motion) Statement of a motive where interval directions have been made to move in the opposite direction of the original motive. If the quality of the intervals is preserved the motion is said to be the "mirror inversion."
- Sequence: Repetition of a motive at another pitch level, usually up or down a step. Each repetition is called a "leg." Sequences in which each leg itself contains a sequential pattern are said to be nested. Bach's sequences tend to be of this latter variety, with the overall sequence comprised of two or three legs, each leg comprised of two subsidiary units. For example: study the sequences in the mirror fugues of Art of Fugue. Sequential episodes seldom appear in fugal expositions but are frequent accouterments to developments.
- Contrapuntal Inversion: (Double/Triple Counterpoint) Reappearance of a pair of voices (double ctpt.) or trio of voices (triple ctpt.) in which registers have been reassigned in such a way that the voices have crossed and the interval relationship between voices is fundamentally altered.a. Types of Contrapuntal Inversions:
b. How to Calculate Type of Contrapuntal Inversion:
- At the Octave: Fourths become fifths, unisons become octaves, etc. While parallel 4ths sound fine, they do not invert contrapuntally, and double ctpt. at the octave avoids them. See the Canon per Augmentationem in contrario Motu from the Art of Fugue for an example of double counterpoint at the octave.
- At the Tenth (8va+3rd): Parallel motion tends to be avoided altogether. This is because intervals that parallel acceptably in one texture (e.g. 3rds & 6ths) become unacceptable when inverted (8vas & 5ths). Study the Canon alla Decima of the Art of Fugue.
- At the Twelfth (8va+5th): With the exception of 3rds (which remain 3rds), acceptable parallels become unacceptable when inverted at the 12th. Thus, in the Canon alla Duodecima of the Art of Fugue (which features this type of double ctpt.) the composer uses many parallel thirds.
- Determine interval that the lower voice has been moved UP
- Determine interval that the higher voice has been moved DOWN. Note: if the voices have not exchanged registers, the higher voice becoming the lower and vice versa, then contrapuntal inversion has not occurred.
- If steps 1 and 2 are each octaves, then the double counterpoint is at the octave. Otherwise, add the results of steps 1 and 2, then subtract 1.
c. How to Calculate What Intervals Become After Inversion:
- Double counterpoint @8va: Subtract the interval (before inversion) from 9 to get the interval after inversion. For example: a 4th before inversion will become a 5th after inversion.
- Double counterpoint @10th: Subtract the interval (before inversion) from 11 to get the interval after inversion. For example: a 4th before inversion will become a 7th after inversion.
Double counterpoint @12th: Subtract the interval (before inversion) from 13 to get the interval after inversion. For example: a 4th before inversion will become a 9th after inversion.