For standard tuning, concert A is defined to have a frequency of 440 Hz. On a piano, A is five white keys above C, but 9 half steps above C counting both the white and black keys. A full octave...

For standard tuning, concert A is defined to have a frequency of 440 Hz. On a piano, A is five white keys above C, but 9 half steps above C counting both the white and black keys. A full octave consists of 12 half steps (semitones). In equally-tempered tuning, each half step has the ratio of 1.0595 above the preceding step. (This ratio is the 12th root of 2.0.)

a. What is the frequency of A-flat, one half step below A for equal temperament?

b. Working down, find the frequency of each succeeding half step until you get down to C. (Carry your computations to four figures, to avoid rounding errors. For each half step, divide by 1.0595.)

c. In just tuning, middle C has a frequency of 264 Hz. How does your result in part b compare to this value?

d. Working up, find the frequency of C above concert A in equal temperament. Is this frequency twice that obtained in part b for middle C?