pitchJND
plain-language theorem explainer
The definition pitchJND k sets the just-noticeable pitch ratio at rung k to 1 + φ^{-k}. Auditory perception researchers and Recognition Science modelers cite this when deriving the φ-ladder for frequency discrimination in the cortical resonance network. It is introduced as a direct abbreviation from the self-similar recognition lattice without further lemmas.
Claim. The rung-$k$ just-noticeable-difference pitch ratio above unison is $1 + φ^{-k}$.
background
In the Pitch Perception from the φ-Ladder module, pitch perception is modeled as a recognition operation on the auditory-cortex φ-rung ladder. Two pitches are perceived as distinct iff their J-cost separation exceeds the per-rung J-cost quantum Jcost(1 + 1/φ^k) for the resolved rung k. Lower rungs correspond to coarse perception; higher rungs to fine perception for trained listeners in the most-sensitive band. The module derives the JND structure as a φ-rung ladder forced by the Recognition Composition Law and the φ-self-similar fixed point.
proof idea
This is a direct definition that encodes the rung-k JND as 1 + 1/φ^k. No lemmas are applied; it serves as the base abbreviation for subsequent theorems on positivity and monotonicity.
why it matters
This definition anchors the music theory track by providing the explicit form of the JND ladder. It feeds the parent theorems pitchJND_gt_one, pitchJND_pos, pitchJND_increment_geometric, and pitchJND_jcost_pos, which establish strict inequality above 1, positivity of J-cost, and geometric decrement by 1/φ at each step. It realizes the φ-ladder forced by T5 J-uniqueness and T6 phi fixed point in the forcing chain, with the eight-tick octave implicit in the recognition lattice. It touches the open question of matching empirical JND data at higher rungs.
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