The harmonic mean of two numbers, a and b, equals StartStartFraction 2 over StartFraction 1 over a EndFraction plus StartFraction 1 over b EndFraction EndEndFraction . As you vary the length of a violin or guitar string, its pitch changes. If a full-length string is 1 unit long, then many lengths that are simple fractions produce pitches that harmonize, or sound pleasing together. The harmonic mean relates two lengths that produce harmonious sounds. Find the harmonic mean for the string lengths of 1 and one-half .
First, we need to find the reciprocals of the two lengths:
1: 1
1/2: 2
Now, we use the formula for the harmonic mean:
HM = 2 / (1/1 + 1/2)
= 2 / (1 + 0.5)
= 2 / 1.5
= 1.333
Therefore, the harmonic mean for the string lengths of 1 and one-half is 1.333.