# Calculating the Frequency of Pitches

Assuming you know the pitches of the scale you want expressed in in the commonly used alphabetic system, eg., D#, then we can calculate the frequency.
### Case #1. Using only the pitches in 12-tone equal temperment.

The relationship between any two adjacent pitches in 12tet is the 12th root of 2, that is, that number which, when multiplied by itself 12 time, yields 2.
Using a scientific calculator (including the one supplied with Windows 3.1), to arrive at the 12 root of 2, we raise the number 2 to the exponent of 1/12. On the Windows calculator, one must supply the decimal equivalent of 1/12: 0.08333.

Then one must figure the number of half-steps up from "A" the desired pitch lies. For our case, let us say that we want to find the frequency of "C". This lies 3 half-steps up from "A". We then raise our result for the 12 root of 2 to the exponent 3 (3 half-steps). Finally we multiply by a know frequency for "A", 440 Hz, or 220 Hz. We make octave transpositions, if neccesary, by multipying or dividing by 2.

## Example

Here's the basic formula:
F = {[(2)^1/12]^n} * 220 Hz
Where:
F = the frequency of the pitch
n = the number of half-steps up from "A" the pitch lies
So, in order to calculate the frequency of "C", we start with
the basic formula:
F = {[(2)^1/12]^n} * 220 Hz
F = {[(2)^1/12]^3} * 220 Hz substitute the number of half-steps
F = {[ 1.059460646483]^3} * 220 Hz calculate the 12 root of 2
F = { 1.189198872076} * 220 Hz raise it to the third power
F = 261.62 Hz multiply by the frequency of "A"

And that's a close enough approximation as we'll ever need (I rounded off just a bit).

### Case #2: Calculatting Frequencies in a Non-12-tone System

For those of us calculating more exotic scales, we need to find those frequencies that lie between the notes of the tempered scale. The procedure is much like above, only with a finer division of the octave.
Usually these things are expressed in terms of the 12tet scale degree plus or minus a certain number of cents. A cent is a 1/100 division of the half-step, or 1/1200 of an octave. So instead of calculating with the 12th root of 2, we need to be using the 1200th root of 2.

## Example

Here's the basic formula:
F = {[(2)^1/1200]^n} * 220 Hz
Where:
F = the frequency of the pitch
n = the number of cents up from "A" the pitch lies
So, in order to calculate the frequency of C# +47 (447 cents, or
4 half-steps at 100 cents each, plus an additional 47 cents), we
start with the basic formula:
F = {[(2)^1/1200]^n} * 220 Hz
F = {[(2)^1/1200]^447} * 220 Hz substitute the number of cents
F = {[1.000577789275]^447} * 220 Hz calculate the 1200 root of 2
F = {1.294594115215} * 220 Hz raise it to the 447th power
F = 284.81 Hz multiply by the frequency of "A"

And, again, I rounded a little bit.

That's all there is to it. Once you know what pitches you want for your scale, then construct a tuning table, from which you can then figure bar and resonator lengths.