The interface of this with Music Theory is equal temperament, as in [1]. The musical hardware provides white piano keys giving intervals with rational frequency

The interface of this with Music Theory is equal temperament, as in [1]. The musical hardware provides white piano keys giving intervals with rational frequency ratio, so harmonious (called just intonation). With that you can play in only one key (or relative minor). The problem is what to do with the other keys. The solution was equal temperament, that suddenly alowed to play all keys at once without retuning the instrument. That new technology was showcased by Bach in his 'Well tempered clavier', that was a set of etudes each one in a different key, ranging over all possible keys. One could sit down and play them all in a row. In equal temp. one can, even inside the piece, do modulations (local key changes) that one couldn't in just intonation. But the price was to abandon the rational frequency ratios of intervals, turning to a system based on multiplying by 2^(1/12) (irrational, except at the octave, 12 semitones/multiplications away, that is at doubling frequency), hence dropping some of the good properties of just intonation. One can see that frequencies of notes in both systems are similar, but end up not really matching. [1] https://pages.mtu.edu/~suits/scales.html https://pages.mtu.edu/~suits/scales.html