IndisputableMonolith.Acoustics.MusicPitchJNDFromJCost
This module defines the just noticeable difference fraction for musical pitch as 1/φ^8, obtained by applying the J-cost to frequency ratios near unison. Acoustics and music-perception researchers working in RS-native units cite it for threshold modeling. The module supplies the core definition together with elementary positivity and boundedness lemmas.
claimThe pitch JND fraction of the octave is $1/φ^8$, where $φ$ is the golden-ratio fixed point of the J-cost function.
background
Recognition Science obtains all constants from the J-cost functional equation and the self-similar fixed point φ. The module imports the RS time quantum τ₀ = 1 tick from Constants and the J-cost definitions from the Cost module. It introduces pitchJNDFraction as the frequency-ratio deviation at the perceptual threshold, together with the associated pitchCost function.
proof idea
This is a definition module. It states pitchJNDFraction := 1/φ^8 and proves the three elementary properties pitchJNDFraction_pos, pitchJNDFraction_lt_one and the corresponding cost lemmas by direct substitution and the known inequalities for φ.
why it matters in Recognition Science
The module supplies the acoustic scale that enters higher-level Recognition Science statements on auditory perception and the eight-tick octave structure. It closes the link between the J-cost composition law and the numerical JND fraction used in music modeling.
scope and limits
- Does not convert the fraction into an absolute frequency interval in Hz.
- Does not incorporate masking or loudness-dependent effects.
- Does not compare the derived value with empirical psychoacoustic data.
- Does not extend the result to non-octave intervals or tempered scales.