Why can it be deduced that a hypothetical fourth harmonic would have a smaller p

Question

Why can it be deduced that a hypothetical fourth harmonic would have a smaller period?

The timbre, or quality of a musical depends the number and relative strengths of the including the fundamental frequency of the node. Figure 1a illustrates the first three harmonics of a tone. The addition of the first two harmonics is pictured in Figure 1b, and the addition of the first 3 harmonics is shown in Figure 1c. Figure 1 Elements of a complex tone If a fourth harmonic exists for the tone graphed in Figure 1, then, compared to the third harmonic, the fourth harmonic will have: lower amplitude lighter amplitude lower frequency higher frequency A fourth harmonic would have a shorter period than the other three. Since T = 1/f, the fourth harmonic would have a higher frequency than the third harmonic. Therefore, answer choice D is the best answer.

Explanation / Answer

if the frequency of the 1st is f1 and the period is T1. Then,

for 2nd harmonic frequency is f2 = 2*f1, period T2 = T1/2

for 3rd harmonic frequency is f3 = 3*f1, period T3 = T1/3.

this is the definition of the Harmonics and it is also clear from the figure 1a.

now,

for n-th harmonic frequency is fn = n*f1, period Tn = T1/n.

so for 4th harmonics frequency is f4 = 4*f1, period T4 = T1/4.

so T4 < T3.

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