# Tuning

The basic formula for the frequencies
of the notes of the
equal tempered scale is given by

*f*_{n} = f_{0} * (a)^{n}

where

*f*_{0} = the frequency of one fixed note which must
be defined. A common choice is setting the A above middle C (A_{4})
at *f*_{0} = 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.

*f*_{n} = 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

*W*_{n} = c/f_{n}

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 A_{4} = 440 Hz:

C_{5} = the C an octave above middle C. This is 3 half steps
above A_{4} and so the frequency is

f_{3} = 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 A_{4} 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.

This calculation is easily accomplished using a
Spread Sheet (excel spreadsheet).

If your browswer is blocking xls downloads, use
this version and replace the ".txt"
extension with ".xls" before using.

Questions/Comments to: suits@mtu.edu

There are no pop-ups or ads of any kind on these pages.
If you are seeing them, they are being added by a third
party without the consent of the author.

To table of frequencies

To Physics of Music Notes

To MTU Physics Home

Copyright Info