To determine the value of [tex]\( p \)[/tex] for the standard concert pitch, which is 440 cycles per second, we need to use the provided formula for [tex]\( p \)[/tex]:
[tex]\[ p = 60 + 12 \log_2\left(\frac{f}{440}\right) \][/tex]
Let's go through the steps necessary to find [tex]\( p \)[/tex]:
1. Identify the given frequency (f):
The given frequency [tex]\( f \)[/tex] is 440 cps (cycles per second).
2. Substitute the frequency value into the formula:
We substitute [tex]\( f = 440 \)[/tex] into the formula:
[tex]\[
p = 60 + 12 \log_2\left(\frac{440}{440}\right)
\][/tex]
3. Simplify the fraction inside the logarithm:
[tex]\[
p = 60 + 12 \log_2(1)
\][/tex]
4. Evaluate the logarithm: [tex]\( \log_2(1) \)[/tex]:
We know that [tex]\( \log_2(1) = 0 \)[/tex], because any number to the power of 0 equals 1.
5. Calculate the value of [tex]\( p \)[/tex]:
[tex]\[
p = 60 + 12 \cdot 0
\][/tex]
[tex]\[
p = 60
\][/tex]
6. Rounding the value (if needed):
The final value of [tex]\( p \)[/tex] is already an integer, so no rounding is required.
Therefore, the associated value of [tex]\( p \)[/tex] for the standard concert pitch (440 cps) is:
[tex]\[ p = 60 \][/tex]
