A relatively short melodic or rhythmic fragment that recurs frequently enough in a piece of music to be noteworthy: it often influences the entire nature of the piece.

A motive is shorter than a subject or a theme, most often just a few notes. It occurs frequently throughout the piece, distributed throughout the multiple voices or parts of the ensemble, and often subject to transformations that retain its essential identity nonetheless. The most famous music motive in Western music is likely the 4 notes from the first movement of Beethoven's Fifth symphony:

Many important motives are not quite so prominent: they are often a small portion of a larger entity such as a subject, countersubject or theme. But as motives, they come to saturate the fabric of the music so that they feel to be the most atomic reduction of the meaning of the music. A great deal of musical coherence is due to the frequent use of these atoms in building larger musical structures. While some pieces are fundamentally based on one or more motives (e.g. Beethoven's fifth or Bach's inventions), most music has a discernable pattern of motives. Altshculer calls this "motivicness". There are other terms like "motivic development" which refer to musical drama that rests chiefly on an artful transformation of one or more motives which saturate the music. This characterizes a large majority of classical music and is one of its chief aesthetics. This is particularly clear in the string quartets, literally of the Classical period. (See Inside the Box).

It is fascinating to study just the subjects of master fugues. While a subject is a continuous and complete musical statement, it often deconstructs into a set of motives, the very motives that saturate the fugue whether in the subject in counterpoint. If you examine a fugue subject and imagine it as background rather than foreground, you can begin to imagine the motivic nature of the counterpoint that might fit like lock and key with the subject. Here are subject tables for a total of 72 fugues:

Bach - Well-Tempered Clavier, Book 1
Bach - Well-Tempered Clavier, Book 2
Shostakovich - Op. 87

In fugue, motives are chiefly used to weave strands of connective tissue between subject entries known as episodes. Often, they are fragments of subject or countersubject, more brief to enable continuous and wandering movement, but connected, as portions of the unique signature of each peice always recalling its fundamental character. Like their larger counterparts, subjects and themes, they are often subjected to transformations such as inversion, augmentation and dimimution.

Just about every Bach or Shostakovich prelude or fugue offers extensive examples of motivies. Another great illustration from Beethoven is the first movement of his string quartet in F-major, Op. 18, No. 1.

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