The Gradus Ad Parnassum (Step or Ascent to Mount Parnassus) is a theoretical and pedagogical work written in Latin, which Fux dedicated to Emperor Charles VI in 1725.
It is divided in two major parts. In the first part, Fux presents a summary of the theory on Musica Speculativa, or the analysis of intervals as proportions between numbers. This section is in a simple lecture style, and looks at music from a purely mathematical angle, in a theoretical tradition that goes back, through the works of Renaissance theoreticians, to the Ancient Greeks. The works of Mersenne, Cicero and Aristotle are among the references quoted by Fux in this section.
The second part, on Musica Pratica, is the section of this treatise where the author presents his instruction on counterpoint, fugue, double counterpoint, a brief essay on musical taste, and his ideas on composing sacred music, writing in the a cappella and in the recitativo style. This part is in the form of a dialog, between a master (Aloysius, Latin for Luigi, who is meant to represent Palestrina's ideas) and a student, Josephus, who represents Fux himself, a self-admitted admirer of Palestrina. At the outset, Fux states his purpose: "to invent a simple method by which a student can progress, step by step, to the heights of compositional mastery..." and gives his opinion of contemporary practice: "I will not be deterred by the most passionate haters of study, nor by the depravity of the present time." He also states that theory without practice is useless. Thus, his book stresses practice over theory.
While Gradus ad Parnassum is famous as the origin of the term "species counterpoint", Fux was not the first one to invent the idea. In 1610, Girolamo Diruta, a composer of the Venetian school, published Il Transilvano, which presented the Renaissance polyphonic style as a series of types: one note against one note, two notes against one note, suspensions, and so forth. Fux's work repeated some of Diruta's, possibly coincidentally, since he is not known to have had a copy. In any event, Fux presented the idea with a clarity and focus which made it famous as a teaching method.
In species counterpoint, as given in Fux, the student is to master writing counterpoint in each species before moving on to the next. The species are, in order, note against note; two notes against one; four notes against one; ligature or suspensions (one note against one, but offset by half of the note value); and florid counterpoint, in which the other species are combined freely. Once all the species are mastered in two voices, the species are gone through again in three voices, and then in four voices. (Occasionally, in modern counterpoint textbooks, the third and fourth species are reversed with suspensions being taught before four notes against one.) Fux expressed the intention of adding sections on how to write counterpoint for more than four parts, indicating that rules in this area were to be "less rigorously observed". However, citing his poor health as a result of gout and age, he chose to conclude the book as it stood.
Even though Fux made a number of errors, particularly in his description of third species (four notes against one) in which he allowed for idioms that do not belong to the 16th century, but rather to the 18th, modern counterpoint education is greatly indebted to Gradus ad Parnassum as the codex of the five species. Most subsequent counterpoint textbooks have taken Fux as their starting point, from the book by Albrechtsberger (Gründliche Anweisung zur Komposition) to 20th century examples such as the book by Knud Jeppesen (Counterpoint: The Polyphonic Vocal Style of the Sixteenth Century).
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