Why Can an Inverted Melody Still Sound Related to the Original?

Melodic inversion does not mean turning the page upside down. A pitch axis is chosen, and intervals that originally rose now fall while falling intervals rise. An ascending third may become a descending third, while rhythm and note order usually remain. This differs from retrograde: retrograde changes sequence; inversion changes direction.

The brief opening idea of Bach's C major Invention, BWV 772, soon appears in inversion. A more systematic example is The Art of Fugue, BWV 1080, where several contrapuntal movements develop rectus and inverted forms of the same subject. We can still recognise it not because the pitches remain identical, but because rhythm, interval sizes, opening gesture and manner of entry preserve a relational shape. The inversion resembles its source while changing its gravity: a line that opened upwards may now close downwards, and its harmonic destination is rewritten.

This is more than an intellectual game. It shows that a subject's identity is not identical with a fixed row of keys; identity can be maintained by relations that survive transformation. At the piano, treating inversion only as an analytical label can leave both versions sounding directionless. It is better to sing the original contour, then hear how the inversion performs a corresponding action through contrary motion while each voice remains independent.

Inversion need not copy every semitone exactly. Tonal inversion may adjust intervals to fit the scale and harmony, so resemblance depends on sufficiently stable relations rather than a mechanical mirror. What matters is hearing how a composer keeps a subject recognisable after reversing its direction, and how that reversal creates new structural possibilities.

https://musictheory.pugetsound.edu/mt21c/MelodicAlteration.html
https://imslp.org/wiki/Die_Kunst_der_Fuge%2C_BWV_1080_%28Bach%2C_Johann_Sebastian%29