Suppose the fundamental frequency of a string is 880 Hz. Which of the following frequency pairs is both the harmonics of the fundamental frequency, whose frequency ratio forms a fifth? (I) 1760 Hz (II) 2640 Hz (III) 3520 Hz (IV) 3960 HzQuestion 13Select one:A.(I) and (III) onlyB.(II) and (IV) onlyC.(I) and (IV) onlyD.(I) and (II) onlyE.(III) and (IV) only
Question
Suppose the fundamental frequency of a string is 880 Hz. Which of the following frequency pairs is both the harmonics of the fundamental frequency, whose frequency ratio forms a fifth? (I) 1760 Hz (II) 2640 Hz (III) 3520 Hz (IV) 3960 HzQuestion 13Select one:A.(I) and (III) onlyB.(II) and (IV) onlyC.(I) and (IV) onlyD.(I) and (II) onlyE.(III) and (IV) only
Solution
In music, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2.
Given the fundamental frequency of a string is 880 Hz, we are looking for a pair of frequencies that are both harmonics of this fundamental frequency and whose frequency ratio is 3:2.
Let's examine each of the given pairs:
A. (I) 1760 Hz and (III) 3520 Hz Both of these frequencies are multiples of the fundamental frequency (1760 = 8802, 3520 = 8804), so they are harmonics. However, the ratio of these frequencies is 3520/1760 = 2, not 3:2. So, this pair does not form a perfect fifth.
B. (II) 2640 Hz and (IV) 3960 Hz Both of these frequencies are multiples of the fundamental frequency (2640 = 8803, 3960 = 8804.5), so they are harmonics. The ratio of these frequencies is 3960/2640 = 1.5, which is the same as 3:2. So, this pair does form a perfect fifth.
C. (I) 1760 Hz and (IV) 3960 Hz The ratio of these frequencies is 3960/1760 = 2.25, not 3:2. So, this pair does not form a perfect fifth.
D. (I) 1760 Hz and (II) 2640 Hz The ratio of these frequencies is 2640/1760 = 1.5, which is the same as 3:2. So, this pair does form a perfect fifth.
E. (III) 3520 Hz and (IV) 3960 Hz The ratio of these frequencies is 3960/3520 = 1.125, not 3:2. So, this pair does not form a perfect fifth.
So, the correct answer is B. (II) 2640 Hz and (IV) 3960 Hz and D. (I) 1760 Hz and (II) 2640 Hz.
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