optimalT60
Optimal reverberation time in room acoustics is set to the golden ratio φ in Recognition Science units. Acoustics modelers cite it when mapping J-cost thresholds to empirical Beranek bands for concert halls. The declaration is a direct assignment to the constant phi from the constants module.
claimThe optimal reverberation time $T_{60}$ equals the golden ratio $φ$.
background
The module expresses the Sabine formula through J-cost on the dimensionless absorption ratio r. Optimal music performance occurs at the golden-section threshold J(φ), which the module identifies with critical damping for concert halls. This produces the structural prediction that optimal T60 equals φ seconds, placing measured halls such as Carnegie (1.89 s) inside the interval (φ, φ²).
proof idea
Direct definition: optimalT60 is assigned exactly the constant phi with no computation or lemmas applied.
why it matters in Recognition Science
It supplies the numerical value used by the band theorem proving 1.61 < optimalT60 < 1.62, the over-damped bound optimalT60 > 1, and the RoomAcousticsCert structure. The definition realizes the module claim that optimal concert-hall T60 equals φ, connecting to the self-similar fixed point from the unified forcing chain.
scope and limits
- Does not derive the Sabine formula from first principles.
- Does not compute T60 from explicit room volume or absorption area.
- Does not address frequency dependence or non-diffuse sound fields.
- Does not extend the model beyond the single optimal value.
formal statement (Lean)
36def optimalT60 : ℝ := phi
proof body
Definition body.
37
38/-- Optimal T60 is in the empirical Beranek band for concert halls. -/