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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. (See fig. 15.22.) 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?
(a)
What is the frequency of A-flat.
Answer to Problem 4SP
The frequency of A flat is
Explanation of Solution
Given info: The frequency of A is
Write the formula to calculate the frequency of A flat.
Here,
f is the frequency of A
Substitute
Conclusion:
Therefore, the frequency of A flat is
(b)
Find the each succeeding half up to reach C.
Answer to Problem 4SP
The succeeding frequencies are
Explanation of Solution
Given info: The frequency of A is
Since C is the 9 half steps below A, to find the succeeding frequencies divide the frequency of A by 1.0595 to get the one step below of A and go on up to nine times. The 9th frequency is corresponding to the frequency of C.
The succeeding frequencies are
Conclusion:
Therefore, the succeeding frequencies are
(c)
Compare the given tuned frequency of C with part (b).
Answer to Problem 4SP
The difference in frequency is
Explanation of Solution
Given Info: The tuned frequency of C is
Write the expression to calculate the difference between the tuned frequency of C with calculated.
Here,
Substitute
Conclusion:
Therefore, the difference in frequency is
(d)
The frequency of C above A and will this frequency is twice as compared with part (b).
Answer to Problem 4SP
The frequency of C is
Explanation of Solution
Given info: The frequency of A is
In this case the C is five keys above A including white and black keys and three keys above without including black keys or halves. Therefore to find the frequency of C, the frequency of A is multiplied by the value 1.0595 by three times.
Therefore the frequency of C is
Conclusion:
Therefore, the frequency of C is
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