More specifically, going up a fifth augments the fourth by a semitone to become the seventh and going down a fifth diminishes the seventh by a semitone to becom

More specifically, going up a fifth augments the fourth by a semitone to become the seventh and going down a fifth diminishes the seventh by a semitone to become the fourth. In your example, F the fourth became F# the seventh and B the seventh became Bb the fourth. A fifth away _IS_ a semitone away. Defining the following eight functions: SEMITONE = +1 mod 12; ANTI_SEMITONE = SEVENTH; FOURTH = +5 mod 12; ANTI_FOURTH = FIFTH; FIFTH = +7 mod 12; ANTI_FIFTH = FOURTH; SEVENTH = +11 mod 12; ANTI_SEVENTH = SEMITONE; We ask ourselves why the following holds: FOURTH(X) = ANTI_SEMITONE(SEVENTH(FIFTH(X))) SEVENTH(X) = SEMITONE(FOURTH(ANTI_FIFTH(X))) Substituting, the answer is clear: X + 5 = X + 7 + 11 - 1 X + 11 = X - 7 + 5 + 1 Keys one fifth apart differ by one note because: 1 = 11 + 7 - 5 mod 12 Algebraically it seems to work out...