Acoustics Room Acoustics From Phi Ladder

Room acoustics and the golden ratio

Room acoustics is the study of how sound behaves inside an enclosed space. The standard measure of a room's acoustic character is the reverberation time, the number of seconds it takes for a sound to decay by 60 decibels after the source stops. A small, heavily furnished room might have a reverberation time of a few tenths of a second, while a large concert hall can exceed two seconds. The golden ratio, approximately 1.618, is the number that satisfies the equation r² = r + 1.

In Recognition Science, the framework models room acoustics as a discrete ladder of five canonical regimes: anechoic, heavily damped, semi-reverberant, reverberant, and echoic. The framework proves that the ratio of reverberation times between adjacent regimes is exactly the golden ratio. This is a machine-checked theorem in the framework's library of formal theorems, with no unproved assumptions. The framework also identifies a canonical band for speech intelligibility, defined by the cost function J(φ), which falls in the range 0.11 to 0.13.

This structure is not arbitrary. The golden ratio emerges from the framework's forcing chain, which derives fundamental constants from a single cost function. The same ratio that appears in the framework's derivation of the fine-structure constant and particle masses also appears here, organizing the acoustic regimes. The framework treats this as a sign that the same underlying structure governs physical phenomena across scales.

The practical consequence is a way to think about acoustic design: moving from one regime to the next, say from a reverberant hall to an echoic one, is not a continuous slide but a discrete step of a fixed ratio. This gives designers a concrete target for the reverberation time needed to shift a room from one acoustic character to another.

inductive RoomAcousticRegime where
  | anechoic
  | heavilyDamped
  | semiReverberant
  | reverberant
  | echoic
  deriving DecidableEq, Repr, BEq, Fintype

theorem rt60_ratio (k : ℕ) : rt60 (k + 1) / rt60 k = phi := by
  unfold rt60
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

structure RoomAcousticsCert where
  five_regimes : Fintype.card RoomAcousticRegime = 5
  phi_ratio : ∀ k, rt60 (k + 1) / rt60 k = phi
  rt60_always_pos : ∀ k, 0 < rt60 k

What this page does not claim

This module does not derive the golden ratio from first principles; it applies the ratio already derived elsewhere in the framework. The module does not provide a physical mechanism for why reverberation times should scale by the golden ratio. It does not claim that real rooms fall exactly into these five discrete categories.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Acoustics/RoomAcousticsFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: