§6 · The audible band: a geometry-carved bandpass

The audible band: a geometry-carved bandpass

Why we hear ~20 Hz–20 kHz is derived as a geometry-carved bandpass on the inherited √-law: the √-law halves the stiffness decades into octaves, so N_oct = ½·log₂(S_base/S_apex). The shape and every edge sign are forced; the absolute edges are the irreducible measured-geometry [O].

Read off the inherited map, the apex is 19.848 Hz and the base 20677.07 Hz, a span of 10.024823 oct — and inverting the √-law gives a stiffness ratio 1.0853×10⁶, with ½·log₂(S_ratio) closing the keystone to |Δ|=<1×10⁻⁹. The low edge is a helicotrema high-pass, the high edge an ossicular-mass low-pass at −2 (−12 dB/oct).

The keystone is exact

The √-law halves stiffness decades into octaves. The apex CF 19.848 Hz and base CF 20677.07 Hz give a 10.024823 oct span; inverting CF ∝ √S gives a stiffness ratio 1.0853×10⁶, and ½·log₂(S_ratio) returns the span to |Δ|=<1×10⁻⁹ [V].

Why ~10 octaves is forced once the (measured) ratio is known; why a ratio ~10⁶ is the basilar membrane’s graded geometry [L], never a fit.

The span decomposes; the ½ is the forced part

The span splits cleanly into a forced exponential and a forced apical bend. The bare exponential 10^(a·x) contributes 6.976049 oct and the helicotrema apical bend adds 3.048774 oct, summing to the full 10.024823 oct [V].

The exponential SHAPE (octaves per stiffness decade) is forced by the √-law; the absolute decades (~10⁶) are measured geometry [L].

Both edges are forced by topology and mass

The low edge is a helicotrema high-pass. The Greenwood offset −A·k is a constant subtracted from an exponential, hence low-end-only: its fractional weight ratio apex/base is 125.893 (=10^a), bending the apex down to ~20 Hz, and the apical hole short-circuits slow pressure — a lows-cut robust to filter order [F].

The high edge is a middle-ear low-pass. The ossicular mass forces a −2 (−12 dB/oct) asymptote robust to damping, and the base’s finite stiffness fixes a finite ceiling CF_max = 20677.07 Hz [F].

The band is the product; magnitudes are the obstacle

The audible band is the unimodal product of the rolloffs — a bandpass whose shape and every edge sign are geometry, not new physics [F].

The absolute edges and the three corners are the irreducible measured-geometry [O]; a number for any of them would be tuning. It is the same discipline as §5 — there one scalar Q was open, here the open quantities are the measured geometry.

Honest negatives — what is not claimed

  1. N1. The absolute band edges CF_min / CF_max (Hz) are [O] — they need S_apex, S_base, the ossicular mass, and the helicotrema area. Only the bandpass shape, the ½ exponent, and each edge sign are forced.
  2. N2. The low (helicotrema) corner is [O] — it needs the helicotrema area and cochlear compliance; only the existence and side (apical/low) are forced.
  3. N3. The high (middle-ear) corner / CF_max is [O] — it needs the ossicular mass and S_base; only the existence, the side (high), and the −12 dB/oct asymptote are forced.
  4. N4. The stiffness RATIO (~10⁶) is a measured anatomical input — the membrane widens and thins base→apex; the √-law forces only the factor (½) converting it to octaves.
  5. N5. The band here is the long-wave / place-resonance account (same scope as §5); the full 2-D/3-D fluid transfer and the middle ear’s multi-resonance are the deeper [O].
  6. N6. The absolute BM stiffness gradient law S(x) and the developmental program that builds the graded membrane are anatomy / the DNA volume’s — [O] here.
  7. N7. The felt pitch / loudness RANGE as experience is the mind volume’s (firewall); §6 moves only the physical band edges.