Distance between a and d in music?

In the standard equal-tempered scale in use in nearly all modern Western music, each octave is divided into 12 semitone intervals, where each semitone interval is taken to represent a fixed percentage increase in frequency.

An increase in pitch of one octave corresponds to a doubling in frequency (ignoring some subtle psychoacoustic effects near the limits of perception). If we say that each semitone corresponds to an increase of a factor k in frequency, then from the fact that twelve semitones equal one octave we obtain:

k12 = 2.

Hence, k = 21/12 which is approximately 1.05946. That is, an increase in pitch of one semitone is equivalent to an increase in frequency of about 5.946%.