Just to add in to what others have said, these notes are in fact used in many styles of music rather frequently, it is just that the piano became the dominant instrument for teaching music and does not allow them to be played.
However one can hear blues guitarists bend strings not entirely up a full semitone. Many brass and woodwind instruments are also capable of this as well as the human voice.
Also worth noting is the music of Jacob Collier, who incorporates micro tonality (the notes between) in a much bigger way. Famously, he recorded the Christmas carol “in the bleak midwinter” and modulated from the key of E to the key of G half sharp.
(At around 4:15 is the key change)
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