The orderly, sequential entry of the subject in each and every voice of a fugue.
All fugues begin with an exposition and the term most formally applies to this first exposure of the subject heard in each voice, one after the other. When such an orderly and complete statement of the subject in all voices occurs elsewhere in the fugue, it is called a counter-exposition, that is, a new exposition which is different than and thus "counter to" the initial exposition.
Here is the complete exposition of a four-voiced fugue, from Bach's Well-Tempered Clavier, Book 1, No. 1 in C major:
Despite the requirement of having each and every voice sing the subject, an exposition has a wide range of possibility and variability as evidenced by Bach's 48 fugues in the Well-Tempered Clavier. Though the number of entries must be at least as many as voices in the fugue, typically, there are more. (clarify) The order of the voices can vary as can the distance between each entry. Some fugues even have the entries overlap, creating stretto as early as the exposition. Often, multiple subject entries fall into pairs owing to the melodic and harmonic qualities of "call" and "response" or, subject and answer.
The exposition includes more than just the subject however. Once the first voice has completed the subject, it must continue with new material that is not the subject but new, independent material that plays well along with the subject in another voice. This is the essential requirement of counterpoint. When this material has a strong identity and recurs throughout the fugue, it is called a countersubject. Countersubjects frequently occur as early as the exposition, often in the first voice against the subject in the second voice. In this way, the exposition often reveals all of the essential material that informs the remainder of the fugue.
Credit goes to Siglind Bruhn for her excellent diagrams.
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