Greenwood place map

The place map is how the cochlea turns a sound wave into a spatial code. The local resonance ω = √(S/m) on a logarithmically graded stiffness forces an exponential map, reproducing Greenwood's CF(x) = A(10^{a·x} − k) to machine precision.

Read off the inherited map, the apex is 19.848 Hz and the base 20677.07 Hz, a span of 10.024823 oct. Because CF(x) is strictly monotone, place → frequency is an order-isomorphism — the property §7 uses to turn “where a failure sits” into “which band is lost.”