pitch

Pitch is (approx.) logarithmically dependent on frequency

• frequency x 2 when pitch goes up an octave
  • $A_4=440$ Hz and $A_5=880$ Hz
• frequency x $2^{1/12}$ when pitch goes up by a semitone. See equal temperament for why we have 12 notes.

frequency (Hz) <-> pitch (key number) conversion §

On the standard piano (12TET, 49th key tuned to ${\mathrm{A}}_4$ at $440\mathrm{Hz}$), the frequency of the $n^{\text{th }}$ key is given by

$$f(n)=(\sqrt[12]{2}{)}^{n-49}\times 440\mathrm{Hz}=2^{\frac{n-49}{12}}\times 440\mathrm{Hz}$$

Conversely, the key number of a pitch with a frequency $f(\mathrm{Hz})$ on the idealized standard piano is:

$$n=12{\log }_2\left(\frac{f}{440\mathrm{Hz}}\right)+49$$