Question

Problem 12.82: Musical scale. The frequency ratio of a semitone interval on the equally tempered scale is 21/12.

Part A Show that this ratio is 1.059. SubmitMy AnswersGive Up Part B Find the speed of an automobile passing a listener at rest in still air if the pitch of the car's horn drops a semitone between the times when the car is coming directly toward him and when it is moving directly away from him. v = 18.38   m/s   SubmitMy AnswersGive Up Incorrect; Try Again; 3 attempts remaining

Answer 1

here,

Problem 1 :
This problem is related to equal temperament, which state that the distance between each step of the scale is the same interval.

Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications.

Specifically, the smallest interval in an equal-tempered scale is the ratio:

r^n = p
where the ratio r divides the ratio p (typically the octave, which is 2/1) into n equal parts.

so, In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:

r^12 = 2
r = 2^(1/12)
r = 1.059

Problem 2 :
fl = largeer frequency
fs = smaller frequency

While approaching ,
Ration of frequency = C / (C-V)
where,
C = speed of sound , 340 m/s
v = velocity of car

fl/fs = 1.059

C/(C-V) = 1.059

C = (C-V)* 1.059
340 = (340 - v)*1.059
1.059 * v = 360.06 -340

v = (360.06 -340) /1.059

V = 18.94 m/s or 19 m/s (rounded off)