Sigh... I won't go into full explanation here, but try to give you a lead what to search for:
In the beginning, there was so-called
just intonation, sounding beautiful for physical reasons (
overtones), but being sentenced to be played alone: one instrument tuned in a just G scale and another tuned in just F sounded awful when played together, because the frequencies wouldn't exactly fit over each other. In just intonation, A# and Bb were two different notes. Stringed instruments with just scale frets were impossible to build (
though some try to sometimes), so frets were given up on altogether (that's how the violin family came to be).
Then these rebellious scales were being tempered, e.g. the unfitting notes of different just scales were slightly shifted so they would fit, until
equal temperament was reached which was basically an octave divided into 12 equal half steps (each one raising the frequency by the 12th root of 2). Suddenly, different instruments could play together, fretted instruments had simple frets and A# and Bb became the same note. The names of the notes, however, remained. I don't understand it either (nor do I understand why Washington DC is outside Washington state) - just accept that the world is weird.
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