How did Johann Joseph Fux organize counterpoint instruction in Gradus ad Parnassum?

In his influential 1725 treatise Gradus ad Parnassum, Johann Joseph Fux organized counterpoint instruction through a pedagogical method known as species counterpoint.12

This system was designed to teach the principles of 16th-century polyphonic writing by abstracting them into five increasingly demanding types, or "species," of two-voice composition. Each species is written against a preexisting melody called a cantus firmus. The progression of the five species is as follows:13

• First species: Note-against-note writing.
• Second species: Two notes against one.
• Third species: Four notes against one.
• Fourth species: Syncopated writing (ligatures).
• Fifth species: Florid counterpoint, which combines the techniques of the previous four species.1

By systematizing these contrapuntal practices, Fux provided a structured tool for composers that remained a standard textbook for generations, studied by notable figures such as Joseph Haydn and Wolfgang A. Mozart.12