4.4 The place map

The traveling-wave model in 4.3 told us that a tone of frequency $f$ peaks at a particular location on the basilar membrane — the one whose local natural frequency matches the input. What it did not tell us is which location, in millimeters from the stapes. For that we need an empirical fit to data, because the cochlea’s stiffness profile is not given to us a priori; it has to be measured.

The classical fit is the Greenwood function. Donald Greenwood proposed it in 1961 and refined it in 1990 once direct cochlear measurements had become available. For the human cochlea, it is

f(x) = A\,\bigl(10^{\,\alpha\,(1 - x/L)} - K\bigr),

where $x$ is measured from the base (the stapes end), $L = 35$ mm is the total length of the unrolled membrane, and the constants are

A = 165.4 \text{ Hz}, \qquad \alpha = 2.1, \qquad K = 0.88.

▶ Derivation: the log-frequency form from uniform critical-band spacing

The Greenwood function is empirical — its constants are fit to measured cochlear data. But its functional form has a clean derivation from a single perceptual assumption.

Assumption: equal lengths along the basilar membrane correspond to equal log-frequency intervals (octave-uniform tonotopy). This is observed experimentally and is consistent with what we know about pitch perception — pitches an octave apart sound equally far apart regardless of absolute frequency.

Mathematically, this says

\frac{d(\log f)}{dx} = -c

for some positive constant $c$ (the negative sign because $f$ decreases as $x$ increases from base to apex). Integrating:

\log f(x) - \log f(0) = -c x,

so $f(x) = f(0)\, e^{-cx}$ , or equivalently $f(x) = f(0)\cdot 10^{-cx/\ln 10}$ . Writing $\alpha = c L /\ln 10$ to make the exponent dimensionless,

f(x) = f(0) \cdot 10^{-\alpha\, x/L}.

With $f(0)$ corresponding to the basal end at the high-frequency limit, we can rewrite this as

f(x) = f_\text{base} \cdot 10^{-\alpha\, x/L} = f_\text{base}\,\cdot 10^{\,\alpha\,(1 - x/L)}\,\cdot 10^{-\alpha}.

Absorb $f_\text{base} \cdot 10^{-\alpha}$ into a single constant $A$ , and we have

f(x) = A \cdot 10^{\,\alpha\,(1 - x/L)}.

This is the leading behavior of the Greenwood function — the simple log law. The correction term $-AK$ is added empirically to account for deviations near the apex of the cochlea, where the simple log law no longer fits the data; specifically, the cochlear apex has nontrivial hydrodynamics (the helicotrema affects boundary conditions, and the membrane is thicker and more loosely supported). The empirical $K = 0.88$ pushes the apex’s predicted frequency down toward 20 Hz rather than the few-Hz floor the pure log would give.

Putting it all together:

f(x) = A\,\bigl(10^{\,\alpha(1 - x/L)} - K\bigr).

The constants $A = 165.4$ Hz, $\alpha = 2.1$ , $K = 0.88$ are Greenwood’s best fits for human cochleas.

This is the Greenwood function. It is the bridge between cochlear anatomy (a position $x$ ) and cochlear physiology (a characteristic frequency $f$ ).

Plug in the endpoints. At $x = 0$ (base, near the stapes), $f = 165.4 \cdot (10^{2.1} - 0.88) \approx 20\,700$ Hz — close to the high-frequency limit of human hearing. At $x = L$ (apex, helicotrema), $f = 165.4 \cdot (1 - 0.88) \approx 20$ Hz — the low end. The Greenwood function smoothly interpolates between these limits over the 35 mm in between.

For most of the membrane, the $K$ term is a small correction, and the function reduces to its leading behavior $f(x) \approx A \cdot 10^{\,\alpha\,(1 - x/L)}$ . This is exponential in $x$ — which means $\log_{10} f$ is linear in $x$ . Equal distances along the basilar membrane correspond to equal log-frequency intervals. Every millimeter is worth the same fraction of a decade — $\alpha/L = 0.06$ decades per mm, or about 0.2 octaves per mm. The whole 35 mm spans roughly $2\alpha = 4.2$ decades, somewhat more than 10 octaves.

The interactive below renders the place map. Drag the frequency slider, or drag the place slider — they are wired together. Press play tone to hear a pure sine at the current frequency, so you can map what you hear to a literal place along your basilar membrane.

frequency

1 kHz

place along the cochlea

21.0 mm

octave above 20 Hz: 5.6
distance from apex: 14.0 mm

Music and speech on the cochlea

Musical pitches, with their octave-based intervals, are mapped uniformly along the cochlea. An octave is an octave anywhere on the membrane. The 88 keys of a piano span about seven and a half octaves, and they would lay out from roughly 30 mm down the cochlea (the lowest A0, 27.5 Hz) to roughly 5 mm from the stapes (the highest C8, 4186 Hz) — almost the full length of the basilar membrane.

Speech, by contrast, is not log-uniform. The phonemes of “Hey Dr. Miles!” cluster into a relatively narrow band. The vowel formants (F1 around 500–1000 Hz, F2 around 1500–2500 Hz) live in the middle, the broadband /h/ aspiration and the /m/ nasal resonance straddle them, and the fricative energy of /s/, /z/ extends into the basal third above 4 kHz. Most of speech is decoded in the basal half of the cochlea. High-frequency hair cells are the first to die under noise exposure and with age, and so the fricative consonants and the unvoiced parts of speech are the first to fade in mild hearing loss. The sentence “Hey Dr. Miles!” becomes audible but unintelligible — every vowel is heard, every consonant is smeared.

The classical audiology picture of this is the speech banana: the phonemes of conversational speech plotted in the frequency-intensity plane. The banana names the shape; the phonemes inside it are what we hear when someone talks at conversational level. Switch the listener to mild or moderate and watch which phonemes fall off the chart first: the vowels persist into severe loss while the fricatives go early.

listener: all phonemes audible

We now have what the cochlea sends downstream, at least at the level of where. The Greenwood function plus the traveling wave tells us, for any sound: which places on the basilar membrane will move, when, and how much. We know the map.

What we do not yet have is an explanation for how sharp that map is. In a dead cochlea, the peaks of the traveling wave are broad and unimpressive, with effective $Q$ values around 3 to 5. In a living cochlea, at low stimulus levels, the peaks are dramatically sharper: $Q$ values of 30, 50, sometimes past 100. The cochlea is between 10 and 100 times sharper than the passive physics we have written down so far. Something is adding energy to the traveling wave on every cycle, compensating for the damping and amplifying the resonance.

That something is the outer hair cells. The story of how they do it — and the way it reshapes everything we have built up so far — is the next section.