Acoustics Room Acoustics From Phi Ladder Rt60 Ratio

The RT60 ratio

Reverberation time, usually written RT60, is the number of seconds it takes a sound to decay by 60 decibels after the source stops. In a small, heavily padded room the value is short; in a concert hall it can be two seconds or more. Acousticians use RT60 to describe how live or dead a space sounds, and it is one of the standard numbers in room design.

In Recognition Science, a framework that starts from a discrete record of recognition events and derives constants and scales, room acoustics is modeled with five canonical regimes: anechoic, heavily damped, semi-reverberant, reverberant, and echoic. The framework's machine-checked library of formal theorems proves that these five regimes are exactly the ones in the model, and that the RT60 values assigned to them form a geometric sequence. The declaration rt60_ratio states that for any step k, the ratio of the next RT60 to the current one equals phi, the golden ratio, about 1.618. In plain terms: each step up the ladder multiplies the reverberation time by phi. The theorem also proves every RT60 value is positive, so the ratio is always well defined.

The golden ratio appears throughout the framework as the unique self-similar scaling forced by its cost function, so the RT60 ratio is not an independent assumption. It is a consequence of the framework's definitions. The five regimes and the phi ladder are a modeling choice, not a measured law of physics. The framework does not claim that real concert halls follow this ratio, nor that the five named regimes are the only possible acoustic states. The theorem is about the model's internal consistency: given the definitions, the ratio follows.

What the declaration does not claim is broader than what it does. It does not predict a measured RT60 for any real room, and it does not say that the golden ratio is the best or only way to organize acoustic spaces. It is a formal result about a defined object, useful as a building block for further framework results, not as a design rule for architects.

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

theorem roomAcousticRegime_count : Fintype.card RoomAcousticRegime = 5 := by decide

theorem rt60_pos (k : ℕ) : 0 < rt60 k := pow_pos phi_pos k

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

What this page does not claim

The theorem does not predict any measured RT60 for a real room. The golden ratio is not claimed to be the only or best way to organize acoustic spaces. The five named regimes are not claimed to be the only possible acoustic states.

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)

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