Enharmonic equivalence
Reinterpretation of a note at the same pitch, e.g. C-sharp to D-flat.
What it means
Enharmonic equivalence refers to renaming a note with a different spelling while keeping the same sounding pitch, for instance when C-sharp is renamed D-flat or F-sharp becomes G-flat. In today's common equal temperament, both spellings sound identical, even though in just or meantone temperament they originally denoted slightly different pitches.
In the notation
In the notation, enharmonic equivalence appears as a change of note name and accidental at the same key or fingering position on the instrument, for instance when the same black key on the piano is notated as C-sharp or D-flat depending on the harmonic context.
Musical use
Composers use enharmonic equivalence deliberately to connect harmonically distant keys without the sounding pitch actually changing: a diminished seventh chord, for example, can be resolved into four different keys through the enharmonic reinterpretation of a single note. Enharmonic equivalence also comes into play when moving between sharp and flat keys on the circle of fifths, such as between F-sharp major and G-flat major, since both keys sound identical on tempered instruments.
Enharmonic equivalence should be distinguished from actual modulation, which describes a real harmonic move into a new key, whereas enharmonic equivalence changes only the spelling, not the sounding pitch itself. In non-equal tuning systems, such as just intonation, enharmonic equivalence is strictly speaking not a pure reinterpretation, since the notes involved actually have slightly different frequencies there.
Performance practice
When reading notation, recognise enharmonically equivalent notes by the fact that they require the same key, fret, or fingering combination on your instrument, even though they are named and notated differently. On keyboard instruments, note that enharmonic equivalence changes nothing about playing technique, while on some wind and string instruments with non-equal tuning, subtle intonation differences can be deliberately brought out.