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Wikipedia defines a musical invention as a short composition (usually for a keyboard instrument) with two-part counterpoint and goes on to say inventions are mainly associated with Bach.
OK. So what, if any, relation does this have to the modern use of the word "invention" as in "The light bulb was an invention of Thomas Edison"? Was there something particularly novel about Bach's inventions?
Inventions is a title that Bach uses in his foreword to the collection of this work. It is adapted from the theory of rhetoric. The inventions used by Bach means clearly that he first shows an invented motif and develops of this simple figure an entire (but short) composition:
"Straightforward Instruction, in which amateurs of the keyboard, and especially the eager ones, are shown a clear way not only (1) of learning to play cleanly in two voices, but also, after further progress, (2) of dealing correctly and satisfactorily with three obbligato parts; at the same time not only getting good inventiones, but developing the same satisfactorily, and above all arriving at a cantabile manner in playing, all the while acquiring a strong foretaste of composition."
Title and historical background of inventiones:
http://www.music.qub.ac.uk/~tomita/essay/inventions.html#1
Traditionally, this concept of 'invention' denoted an important stage in composition; it originated from a famous Roman orator, Marcus T. Cicero's rhetoric, which was still widely studied in the 18th-century Germany: In his De Inventione, Cicero listed five stages in creating an oration, namely invention (inventio), arrangement (dispositio), style (elocutio), memory (memoria) and delivery (pronuntiatio). He explains, 'one must first hit upon what to say; then manage and marshal his discoveries, not merely in orderly fashion, but with a discriminating eye for the exact weight, as it were, of each argument; next go on to array them in the adornments of style; after that keep them guarded in his memory; and in the end deliver them with effect and charm.'
Bach did not call them inventions. He first called the two-part ones "praambulum" and the three-part ones "fantasia", then switched to "inventio" and "sinfonia". Those are Latin (or Italian) terms, not English ones. "inventio" comes from "invenire", "going inside" and is basically used similarly to "introduction". It has become a fixed expression in music for introducing and developing a short theme in contrapoint.
Independently, the term "invention" has become synonymous to, well, invention in English.
According to Honegger-Massenkeil reference the first musical piece called invention was by Claude Janequin and dates from 1555. This weakens the strong Bach connection.
There the title already described a piece not restricted by a specific scheme or formalism. From this kind of originality it is not far to the phrase that each piece is invented.
There is a lot that is special about the work of Bach but the fact that some of it is called Invention has nothing to do with it.
Invention comes from latin and originally means 'discovery'. All the way through the Middle Ages, that was what it meant. Today still, the technical legal term for acquiring ownership of a lost treasure is 'invention'.
