It's not a fixed mapping. It's more like a function that produces a mapping. It depends on the selection of:
- A pitch standard, like A440, which anchors one arbitrarily chosen note to a specific frequency.
- A tuning system, which defines the frequency ratios between notes.
Almost all modern music assumes _A440_ and _12-tone equal temperament_. In 12TET, adjacent flats and sharps map to the same frequency. It's said that they are _enharmonic_. But there are many other systems where this isn't true. In _just intonation_, notes are defined by simple frequency ratios, and there is no enharmonicity at all. Every number of sharps and flats map to different frequencies.
Even this is not quite right. For keyboard instruments, that have a fixed set of pitches that notes map to, you have to choose a temperament, a fixed set of pitches for each note (unless you're on a clavichord, where you can adjust it a bit by how hard you strike the key). Fretted instruments may or may not be tempered. Guitars are, since the frets on it limit the pitch range of a given note, though it is still a range. Viols aren't, because the frets are soft gut and used to brighten the sound, so the pitch range is much wider for a given note.
For string or wind instruments playing without keyboards, there is no temperament. Each note represents a range of possible pitches, and what pitch you choose depends on the notes around it. For example, if you are repeating or holding a note, you don't want it to change pitch. But if the harmony around that note is changing, it may be in tune at one point in time, then out of tune at a later one. String quartet playing involves a great deal of working out compromises about what pitch each note should have.
Good point! I admittedly was writing this off as "performance details" of music written in a modern 12-tone paradigm, but you're definitely right that plenty of music is written specifically for ensembles with continuous pitch with those nuances in mind. See also, a cappella vocal music, particularly barbershop. Even for guitar, some players tune to favor particular chords shapes sounding maximally consonant, even though the frets are designed for 12-TET.
- A pitch standard, like A440, which anchors one arbitrarily chosen note to a specific frequency.
- A tuning system, which defines the frequency ratios between notes.
Almost all modern music assumes _A440_ and _12-tone equal temperament_. In 12TET, adjacent flats and sharps map to the same frequency. It's said that they are _enharmonic_. But there are many other systems where this isn't true. In _just intonation_, notes are defined by simple frequency ratios, and there is no enharmonicity at all. Every number of sharps and flats map to different frequencies.