Frequency Doubling
- Author:
- Michael Horvath
In music, the frequency of a note or tone doubles with each octave. This means that if the frequency of a reference tone is 40Hz, then the frequency of the same tone but one octave higher is 80Hz. The frequency of the tone one octave higher still is 160Hz. And so on.
In musical performance, instruments are tuned to a reference note so that all the instruments in an orchestra sound in the same key. The exact note and frequency of this tone varies from orchestra to orchestra, and from performance to performance, and is often adjusted to match the natural range of a vocalist (in opera, for instance). However, one commonly used reference tone is "A above middle C", or A4, tuned to 440Hz.
Since the frequency of the reference note A4 is 440Hz, the frequency of A one octave lower is 220Hz, and the frequency of A one octave lower still is 110Hz. But what about middle C, or C3? What frequency is it?
To calculate the value of C3 we use an inverse logarithmic function. An inverse logarithmic function, also called an "exponential" function, takes the following form:
f(x) = n^i
where the number n (the base) is raised to the i'th power. And, since the frequency is doubling at each octave, the value of the base must be 2. I.e.
f(x) = 2^i
So far so good. But what about the other values?
First of all, one common way of measuring notes is using a unit called "cents". In this system, the distance between each half step (the distance between the notes B and C, and the notes E and F for instance) is 100 cents. Since there are twelve half steps in an octave, the size of an octave is 1200 cents. This remains true regardless of the note or scale used.
We are also going to have to set a root note for our measurements. In this case we will use A1, which is three octaves below A4. Since A1 is the root of our scale, its value in cents is 0. Now, starting with A1 we will measure the number of cents until we reach "A above middle C", or A4. The distance is 3600 cents. Let us also measure the number of cents between A1 and C3. In this case, the distance is 2700 cents.
Now we can plug all our values into our function.
Since the distance of our reference tone is 3600 cents away from our root, we want to calculate the difference between our reference tone and the note we're trying to measure. Thus, we plug the following for i into our equation:
f(x) = 2^((x-3600)/1200)
We also want to divide by 1200, since this is the basic unit of our scale in cents. Since our reference tone is 440Hz, we multiply the entire equation by 440. E.g.
f(x) = 440 * 2^((x-3600)/1200)
Now, to calculate the frequency of middle C, simply plug the note's value in cents in for x in our equation. I.e., to find the frequency of middle C (C3, or 2700 cents), we perform the following:
f(2700) = 440 * 2^((2700-3600)/1200) = 261.625565300599Hz
Hopefully you have found this tutorial useful toward understanding how frequency doubling in music works, as well as what steps are needed to calculate the frequency of any wanted note with regard to a single reference note.