Question

Frequency JND is around 0.5% for any two nearby frequencies (e.g., you can tell 100 Hz and 100.5 Hz notes apart, but maybe not 100 and 100.2 Hz), and the LFD is at best 7%. a. (2) What is the % difference between any two adjacent notes on figure A-1 which shows the standard Western 12 tone scale? Be careful not to skip the black notes for this problem! b. (2) But when two notes on a Western scale, even adjacent ones, are played simultaneously, why can musicians tell them apart anyway?

Answer 1

This interval is the tempered semitone.

With the cent defined as 1 /100 semitone and ¢ denoting the ratio for it,

such that ¢100 = (2)1/12, (5)

it follows that ¢ = (2)1/1200.

As an interval of n cents would be given by the ratio ¢n = 2n/1200 ,

finding the number of cents n in any interval of frequency ratio R requires that 2n/1200 = R. (7)

Taking the logarithm of each side gives log (2n/1200) = log R,

and by applying Equation (2) it follows that € n 1200 log 2 = log R.

The number of the cents in the interval is then given by n = 1200 € log R log2 s

o that n = 3986 log R.