Acoustics Room Acoustics Rt60 Rs

Reverberation time and its cost

Reverberation time, usually written RT60, is a standard measure in room acoustics. It is the number of seconds it takes for a sound to decay by 60 decibels after the source stops. Speech studios are typically tuned to about 0.2 to 0.5 seconds, while large concert halls run closer to 1.1 seconds. The classical formula for estimating RT60, the Sabine equation, uses the room's volume and its total sound absorption, and it has guided acoustic design since Wallace Sabine published it in the 1890s.

The Recognition Science framework takes a different starting point. It models a room's ideal reverberation time as a step on a discrete ladder: RT60 = phi^k * 0.1 seconds, where phi is the golden ratio, about 1.618. On that ladder, phi^3 * 0.1 gives 0.424 seconds, which sits inside the studio range, and phi^5 * 0.1 gives 1.1 seconds, the concert hall figure. The framework's library contains a machine-checked definition of a cost function, called Jcost, that measures the gap between an actual value and an expected one. For this acoustics module, that cost is defined as Jcost(m/e), where m is a measured or chosen value and e is the expected value from the ladder.

What the module actually proves is deliberately modest. Three facts are established in the machine-checked library: the cost vanishes when the measured value equals the expected value, the cost is never negative for positive inputs, and the golden ratio minus 1.5 is positive. The module also bundles these three facts into a certificate structure, which is a formal way of saying that the basic sanity checks hold. None of these theorems is specific to sound. The same three facts appear in 2383 sibling modules, because the cost function is defined once and reused everywhere.

In plain language, the module does not prove that a studio should take 0.424 seconds. It proves that the framework's cost function behaves well enough to be used as a tool for comparing values, and it records where the acoustics application was meant to go. The missing piece is a definition of m and e in acoustics' own terms, for example linking m to a measured decay time and e to the ladder value. Until that definition exists, the reverberation time claims remain a research note, not a theorem.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove that any particular RT60 value is optimal for a studio or hall. The module does not establish that the golden-ratio ladder is the correct model for reverberation time. The module does not contain acoustics-specific physics beyond the definition of domainCost.

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/Room_Acoustics_RT60_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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