IndisputableMonolith.Acoustics.MusicPitchJNDFromJCost
This module derives the musical pitch just noticeable difference fraction of the octave as 1/φ^8 from the J-cost function. Acoustics researchers applying Recognition Science to perception cite it when scaling auditory thresholds to the phi-ladder. The module supplies the core definition together with positivity and bound lemmas.
claimThe pitch JND fraction of the octave equals $\phi^{-8}$, where $\phi$ is the golden-ratio fixed point of the forcing chain.
background
Recognition Science builds perceptual scales from the J-cost function satisfying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The imported Constants module supplies the base time quantum: "The fundamental RS time quantum (RS-native). τ₀ = 1 tick." The Cost module supplies the underlying defect and cost structures. This acoustics module applies those objects to define pitchCost and pitchJNDFraction on the phi-ladder.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the pitch JND fraction that anchors acoustic models inside the Recognition framework. It directly implements the eight-tick octave (T7) and the phi-ladder rung structure for auditory JND, feeding the sibling certificates PitchJNDCert and cert.
scope and limits
- Does not convert the fraction into absolute frequency units.
- Does not compare the value against measured psychoacoustic thresholds.
- Does not treat non-octave musical intervals.
- Does not address other sensory JNDs such as loudness.