Have you ever heard a note and wondered about where it fits into the audio spectrum? Consider the case of the sound technician who hears a feedback tone in a professional sound system and wonders what band on his equalizer will filter it out. Or the case of the music theory student who wants to know just what the frequency range of a piano, or flute, or bass violin is and only knows the tonal range. In this article, I've worked out several formulae that solve these problems, while at the same time, pointing out some of the mathematics of audio in general. These come in handy in the synthesis field, where tonal ranges must be converted to frequency spans. Quickly, what would be the range needed on an oscillator to cover a span from C2 to A6? Well, the sequence below will show how it is done.

First we need a way to convert the alphabetical part of the note to a number that can be used in our formulae. The equivalency line shown in figure 1 works out fine and it is this system that we will use since the most common tones are in the equally tempered scale.

The octave numbers will remain the same, assuming A⁴ = 440 Hz. For reference sake this means that on a typical 88 key piano, the lowest key is A⁰ with each key above it also having the same octave number until the next A comes up, that one being A¹, etc. all the way up to the top key that turns out to be C⁷.

The first formula is the note to frequency one, and it is fairly basic to anyone who has studied music theory. First, some constants must be established. We will be using 27.5 as a base frequency since it is equivalent to A⁰ or 0⁰ This was derived from A-440 in the fourth octave being divided by two over and over four times bringing it down to the 0th octave (any frequency divided by two is equivalent to the same note one octave lower.) The next constant is the infamous 12√2 here being equal to 1.05946309435. This is the basis of the equally tempered scale, and its derivation is found in the PAIA synthesizers user's manual. Any frequency multiplied by this constant is equal to the next note in the scale.

Now, knowing these facts, I've come to this equation:

which works out with all frequency standards. (F freq., O octave, N note)

Now for a few examples: What is the frequency of Bb⁶?

Using the equivalency line, Bb(A#)=1 so:

Frequency is 1864.655 to three places.

What is the frequency of Middle C (C³)? C=3 so:

Freq, is 261.626

This is a nice formula, but not half as nice as when it is used in reverse. In this form you can put in any number, a readout from a frequency counter, the year you were born, or a preset clock frequency and find the nearest note. It's a bit more complicated, and has two steps to it, but is quite useful, especially to electronic designers.

Here it is: first find the octave:

Where F = Selected frequency
and R = Ref. Freq. = 27. 5 Hz.

Note: Truncate octave to Integer value. Do not round!

And then the note:

Round to the nearest integer.

What is the note value for line frequency? (60 Hz.)

= 1.125 or 1 to the first integer

= 1.506 or 2 when rounded, and 2 = B on line in figure 1 so 60 Hz. = B¹

What would be the note associated with a 5 kHz. oscillator tone?

= 7.506, or 7 when cut to the integer

= 6.07 or 6 = D# (from fig. 1)

There are a few odd ball cases where this formula doesn't seem to work, and these are when any frequency is below the reference of 27.5, because the answers are negative. But they do work, it's just a matter of reversing the note/number equivalency line, as below. Whenever an answer turns out negative, as before the octave is chopped to the nearest integer, with any value less than -1 being equal to 0. The note value is still rounded, though, and this value is used in the new tone line.

Example: What is the note of a 20 Hz. tone? (the theoretical limit of hearing)

= .45 or 0 by dropping the decimals.

= -5.513 rounded to -6

In fig. 2 below, 6 = D# so 20 Hz. = D#^-1*

* Note: When counting backwards, this automatically lowers octave by one.

To check this, the frequency of 40 Hz. should be one octave higher, or

= .5405 or 0 to integer

= 6.48 or 6 when rounded.

Using figure 1, 6 = D#, so 40 Hz. = D#⁰ It checks!

I hope these formulae will come in handy for you someday, either in the synthesis field, electronic design, or just for a little math fun. Now if someone says, "Name that tone!" you'll know how to go about doing it.