How to calculate the ratio between Semitones in the diatonic scale?

Question

There are twelve semitones in the diatonic scale, so the frequency doubles every twelve steps.

Knowing this, how do you calculate the ratio between the frequency of one semitone and the next?

Answer 1

Assuming that the ratio between the frequencies of consecutive notes (one semitone apart) is constant, denote this ratio by $x$. Then $x^n$ gives the ratio of the frequencies of notes that are $n$ semitones apart. So for notes one octave (=12 semitones) apart, the ratio of the frequencies is $x^{12}$. On the other hand, we are told that the ratio of these frequencies is 2. Therefore $$x^{12}=2,$$ giving $$x=2^{\frac{1}{12}};$$ in other words the ratio of the frequencies of consecutive semitones is the 12th root of 2.