A transformation of the subject (theme or a motive) where the melody line is turned upside down; it is inverted.
Extended counterpoint often transforms its primary material, a recurring subject or motive in a variety of ways. Transformations retain something of the original to maintain coherence, but, through modification, add diversity, perhaps even a new perspective on the nature of original material. For additional commonly used transformations, see augmentation and diminution. These devices are not unique to fugue, but occur frequently in other compositions using imitative counterpoint including invention, and classical forms such as string quartet and symphony.
Inversion retains the rhythm and the basic contour of the material, but flips it upside down: where the original moves up, the inversion moves down just as down changes to up. This is also called contrary motion. The inversion is a mirror image of the original. Sometimes, the inversion forms a useful counterpoint to the original so they may appear together with a wonderful symmetric effect of total contrary motion. More often, the inversion simply appears instead of the original (technically called the rectus or "right" form), frequently with multiple inverted entries so that repetition makes the transformation familiar. Any recurrent musical line is potentially subject to inversion: subject, countersubject, motive. Sometimes the feature is readily apparent, sometimes it is operational, but blended and buried within the contrapuntal fabric. It is an entertaining game to find inversion. It is a more mysterious challenge to describe its exact aesthetic utility within a particular example.
Here is one example of a subject and its inversion from Bach's Well-Tempered Clavier, Book 2, No. 2 in C minor:
You can easily see the effect of the transformation visually. In this particular case, the inversion has a wonderful melodic and harmonic integrity all its own.
Technically, there are two kinds of inversion: contrapuntal and melodic (also known as real and tonal. See answer for more details). The former is a literal inversion, preserving all original intervals; the latter uses alterations to render the inversion more palatable or melodic.
Fugues featuring inversion:
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