Analyze the role and application of counterpoint in a fugue by J.S. Bach, detailing the subject, answer, countersubject, and exposition, and explain the implications for overall structure.

The role of counterpoint in a fugue by J.S. Bach is central to its structure and expressive power. Counterpoint, the art of combining multiple independent melodic lines, is not just a decorative element but the very foundation upon which a fugue is built. In a Bach fugue, the contrapuntal relationships between voices create intricate textures and provide the means for the development of musical ideas.

The subject is the primary musical idea of the fugue, typically a short, memorable melody. It is the initial statement around which the entire composition is built. The subject usually has a distinct rhythmic and melodic profile. When the subject is restated in a different voice, it is termed the answer. The answer is usually transposed to the dominant key to maintain tonal variety. In a tonal answer, the intervals of the subject are adjusted to fit the new key, while in a real answer, the intervals of the subject are preserved exactly. The use of the tonal versus real answer influences the harmonic direction and the sense of tension and resolution within the exposition.

After the subject and answer have been introduced, the countersubject appears. This is a secondary melodic idea that is often introduced along with or immediately following the answer. The countersubject is designed to work contrapuntally against the subject and the answer. It has a contrasting melodic character to create a richer contrapuntal texture. The countersubject can be varied or altered throughout the fugue, but its presence reinforces the contrapuntal interplay and provides a recurring element that helps unify the work.

The exposition is the initial section of the fugue, where the subject, answer, and countersubject are first presented and established. The exposition follows a systematic pattern of entries: the subject is presented in one voice, the answer in another, the subject is presented in another, and finally the answer appears again. Each new voice entry builds on the preceding material, creating a fuller texture. The exposition can also include episodes – passages where the subject is not present in its entirety and is free to explore the harmonic and melodic potential developed from the main motives introduced earlier.

The implications for the overall structure of the fugue are significant. The exposition lays the groundwork for the entire piece, introducing the core musical materials and setting up the contrapuntal relationships between voices. The subsequent sections of the fugue are characterized by entries, where the subject (or a fragment thereof) appears in different voices and in different keys. These entries create a sense of continuity and development. Episodes between entries provide contrast and harmonic variety while exploring motifs derived from the subject. These episodes often create tension before the subject reappears. The interplay between the subject, the answer and the countersubject provides a structural framework. The subject's rhythmic and melodic profile are transformed through stretto (overlapping entries of the subject in different voices), inversion (reversing the direction of the melody), and augmentation (lengthening the note values), or diminution (shortening the note values) which are all different ways to develop the musical material. This continuous development of the subject and its related material creates a sense of organic growth that is characteristic of Bach's fugues. The fugue often ends with a final statement of the subject in the tonic key and often a codetta to confirm the tonal center.

For example, in Bach’s Fugue in C Minor (BWV 847) from The Well-Tempered Clavier, the subject is a descending motif of stepwise motion. The answer in the dominant is a tonal answer, with a slight adjustment of interval to fit the new key, The countersubject is a contrasting ascending line that works contrapuntally against the subject and answer. The exposition rigorously introduces the voices with each stating the subject/answer. The rest of the fugue involves several entries and episodes where the subject and related motifs are expanded in different keys. The fugue ends with a final statement of the subject in the tonic key and a clear cadence to confirm the tonal center. In a four-voice fugue the four voices enter in order during the exposition stating the subject/answer one by one, and this is crucial for building the texture and contrapuntal complexity. The use of counterpoint therefore, is not a mere embellishment but rather the core structural element that dictates the form and its unfolding, and the contrapuntal interplay between the subject, answer and countersubject defines a key principle of Bach's fugal compositions.