What is the frequency interval length of k semi-tones? (k = 1…11)

Question

Go up 12 semi-tones from frequency f on a guitar, and the frequency interval difference in terms of f is f = 2f - f. What is the formula for the intermediate 11 non-trivial intervals? So for k=2, that's two semi-tones. If I start at frequency f on guitar, and go up two frets, what is the frequency interval length?

Answer 1

With g being the new frequency, f being the frequency of reference, and k being the interval in number of semitones:

g = f * 2ˆ(k/12)

With k = 12 (one octave), you get g = 2 * f.

With k = 0 (unison), you get g = f.

In between, you get an exponential curve (power of two).

For the frequency difference between g and f:

g - f = f * [2^(k/12) - 1]

Note also that k needs not be positive, and may also have decimals. If the interval you are looking for is 3 semitones and 14 cents down, take k = -3.14.