Frequency Doubling

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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 is often adjusted to match the natural range of a vocalist (in opera, for instance). However, one commonly used standard tone is A above middle C at 440Hz.

Since the frequency of the reference note A 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 the note C? What frequency is it?

To calculate the value of C 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. E.g.

	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.

It is also common to base this measurement system upon the note C instead of A. In this case we will use C1, which is three octaves below middle C, or C4. (C1 is also the first, and thus lowest and left-most, C that appears on a typical piano.) Since C1 is the basis of our scale, its value in cents is 0. Now, starting with C1 we will measure the number of cents until we reach A above middle C, or A4. The distance is 4500 cents.

Now that we have a value for A, we can plug it and the remaining values into our function.

Since the distance of our reference tone is 4500 cents away from our root, we want to calculate the difference between our reference tone and the note we're trying to calculate. Thus, we plug the following for i into our equation:

	f(x) = 2^((x-4500)/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-4500)/1200)

Now, to calculate the frequency of a different note, simply plug the note's value in cents in for x in our equation. E.g., to find the frequency of G below middle C (i.e. G3, or 3100 cents), we perform the following:

	f(3100) = 440 * 2^((3100-4500)/1200) = 195.9977Hz

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.

Download the worksheet here.

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This page © Copyright 2009 Michael Horvath. Last modified: February 21 2010 20:33:45.