Chapter 8.4, Problem 35PPS

a.

To determine

The number of times notes up the piano keyboard is this.

Answer to Problem 35PPS

  $37$ Times.

Explanation of Solution

Given information:

The first key on a piano keyboard corresponds to a pitch with a frequency of $27.5$ cycles per second. With every successive key, going up the black and white keys, the pitch multiplies by a constant. The formula for the frequency of the pitch sounded when the $nth$ note up the keyboard is played is given by $n=1+12{\log }_2\frac{f}{27.5}$ .

  $(a)$ A note has a frequency of $220$ cycles per second. How many notes up the piano keyboard is this?

Calculation:

To find how many notes up the piano keyboard is this, to substitute $f=220$ in the formula $n=1+12{\log }_2\frac{f}{27.5}:$

  $\begin{array}{l}n=1+12{\log }_2\frac{f}{27.5}\\ =1+12{\log }_2{2}^3\\ =1+12(3)\\ =37\end{array}$

Therefore, the solution is $37$ times.

b.

To determine

The number of times notes up the keyboard.

Answer to Problem 35PPS

  $61$ times.

Explanation of Solution

Given information:

The first key on a piano keyboard corresponds to a pitch with a frequency of $27.5$ cycles per second. With every successive key, going up the black and white keys, the pitch multiplies by a constant. The formula for the frequency of the pitch sounded when the $nth$ note up the keyboard is played is given by $n=1+12{\log }_2\frac{f}{27.5}$ .

  $(b)$ Another pitch on the keyboard has a frequency of $880$ cycles per second. After how many notes up the keyboard will this be found?

Calculation:

To find out after how many notes up the keyboard this will be found, substitute $f=880$ in the formula $n=1+12{\log }_2\frac{f}{27.5}$ :

  $\begin{array}{l}n=1+12{\log }_2\frac{f}{27.5}\\ =1+12{\log }_2{2}^5\\ =1+12(5)\\ =61\end{array}$

Thus, the solution is $61$ times.