Why is there no black key on a keyboard in between the notes of E and F?
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Notes were not arbitrarily defined, they were discovered. The ancient Greeks experimented a lot with various sets of notes to find out which sound good together, they tried various systems of music, some of which sound downright exotic to the modern ear, but what caught on was the seven diatonic modes, which the Greeks called the Ionian, Lydian, Dorian, Aeolian, Mixolydian, Phrygian and Locrian. These modes were built from notes that are the modern “white keys”. The Greeks found out that these specific notes form nice sounding chords and melodies.
Then, in the Age of Enlightenment, European musicians started to examine the diatonic closer, and found out that not all of the notes are equally spaced: some of them differ by a tone, and some by a half tone. So they invented black keys: new notes to be inserted everywhere where there would be a whole tone difference. If there would be a half tone difference between the diatonic notes inherited from the Greeks, though, there was no space to insert a new note. Then equal temperament was invented: the frequences of notes, both old and new, were slightly adjusted to allow easy transposition of every mode and key to every tonic note. And this is how the modern set of notes was created.
We still use the Greek modes in modern music, especially Ionian, which is more often called the major scale or natural major. We just learned to add some extra notes for a little variety.
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