Physics of Music - Notes

Equations for the Frequency Table


The basic formula for the frequencies of the notes of the equal tempered scale is given by
fn = f0 * (a)n
where
f0 = the frequency of one fixed note which must be defined. A common choice is setting the A above middle C (A4) at f0 = 440 Hz.
n = the number of half steps away from the fixed note you are. If you are at a higher note, n is positive. If you are on a lower note, n is negative.
fn = the frequency of the note n half steps away.
a = (2)1/12 = the twelth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...

The wavelength of the sound for the notes is found from
Wn = c/fn
where W is the wavelength and c is the speed of sound. The speed of sound depends on temperature, but is approximately 345 m/s at "room temperature."

Examples using A4 = 440 Hz:


C5 = the C an octave above middle C. This is 3 half steps above A4 and so the frequency is
f3 = 440 * (1.059463..)3 = 523.3 Hz
If your calculator does not have the ability to raise to powers, then use the fact that
(1.059463..)3 = (1.059463..)*(1.059463..)*(1.059463..)
That is, you multiply it by itself 3 times.

Middle C is 9 half steps below A4 and the frequency is:
f -9 = 440 * (1.059463..)-9 = 261.6 Hz
If you don't have powers on your calculator, remember that the negative sign on the power means you divide instead of multiply. For this example, you divide by (1.059463..) 9 times.

Questions/Comments to: suits@mtu.edu


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