The "First Viennese School" is a name mostly used to refer to three classical music composers who wrote in the classical music era late eighteenth century in Vienna - Wolfgang Amadeus Mozart, Franz Joseph Haydn and Ludwig van Beethoven. Franz Schubert is sometimes, but rarely, added to the list.
The term was created retrospectively, and purely as a result of the name Second Viennese School having become attached to the group of composers connected with Arnold Schoenberg in the Vienna of the early twentieth century.
Whilst, Schubert apart, these composers certainly knew each other (with Haydn and Mozart even being occasional chamber-music partners), there is no sense in which they were engaged in a collaborative effort in the sense that one would associate with 20th century 'schools' such as the "Second Viennese School", or Les Six. Nor is there any significant sense in which one composer was 'schooled' by another (in the way that Berg and Webern were taught by Schoenberg), though it is true that Beethoven for a time received lessons from Haydn.
Attempts to extend the "First Viennese School" so as to include such later figures as Anton Bruckner, Johannes Brahms, and Gustav Mahler are merely journalistic, and never encountered in academic musicology. .
