[linux-audio-user] transposing frequency

[Pete Leigh]

On 11/12/05, Atte André Jensen <atte.jensen at gmail.com> wrote:
> Hi
>
> Suppose I have an A with a frequency of 440 hz. Which
> formula should I use for transposing x semitones up/down?
> So which frequency would the Bb a half step higher for
> instance have? What about cents (1/100's of half steps)?

If A is 440 then the A an octave up (call it A') is 2 x 440 = 880.

To go from A to A' in 12 equal steps (we're assuming equal
temperament), we need an interval, call it I, so that multiplying
by I will give the frequency a semitone higher, and doing that
12 times will go up one octave:

   A x I x I x I x I x I x I x I x I x I x I x I x I = A',

so, rearranging:

  I^12 = A' / A

but A' / A = 440/880 = 2.

So I^12 = 2.

So I is the twelfth root of 2. Multiply the frequency of a note by
that and you get the frequency a semitone higher. The twelfth
root of two is approximately 1.05946309436, or, as kcalc has
it: 1.059463094359295264523454505045663154306

( http://en.wikipedia.org/wiki/Twelfth_root_of_two )

To go up one cent, the same logic indicates you'd multiply by
the 1,200th root of two, since there are 100 cents in a semitone.

kcalc tells me it's:

1.000577789506554859250142541782224725466

( http://en.wikipedia.org/wiki/Cent_%28music%29 )

- Pete.