The Weber-Fechner law describes how human beings perceive differences. Suppose, for example, that a person first hears a tone with a frequency of 400 hertz (cycles per second). He is then tested with higher tones until he can hear the difference. The ratio between these values describes how well this person can hear differences.
a. Suppose the next tone he can distinguish has a frequency of 404 hertz. What is the ratio?
b. According to the Weber-Fechner law, the next higher tone will be greater than 404 by the same ratio. Find this tone.
c. Write the discrete-time dynamical system for this person. Find the fifth tone he can distinguish.
d. Suppose the experiment is repeated on a musician, and she manages to distinguish 400.5 hertz from 400 hertz. What is the fifth tone she can distinguish?