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?
With g being the new frequency, f being the frequency of reference, and k being the interval in number of semitones:
In between, you get an exponential curve (power of two).
For the frequency difference between g and f:
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