IndisputableMonolith.MusicTheory.TuningSystemJCostRanking
This module ranks tuning systems by per-step J-cost on interval ratios. It defines stepCost as J applied to ratios, jiCost for just intonation, tetStep for 12-TET, and certifies ji beats tet via TuningRankingCert. Recognition Science researchers would cite it when extending the cost framework to acoustics and comparing scale efficiencies. The module consists of definitions plus short lemmas on non-negativity and direct comparisons.
claimThe module defines the per-step cost function $stepCost(r) := J(r)$ on interval ratio $r$, the just-intonation cost $jiCost$, the equal-temperament step $tetStep$, and the ranking certificate $TuningRankingCert$ asserting that just intonation has strictly lower total J-cost than 12-TET.
background
Recognition Science uses the J-uniqueness function $J(x) = (x + x^{-1})/2 - 1$ (equivalently $cosh(log x) - 1$) that obeys the Recognition Composition Law. This module imports the RS time quantum $τ_0 = 1$ tick from Constants and the general cost primitives from Cost, then specializes them to frequency ratios treated as multiplicative steps. The local theoretical setting is the MusicTheory domain, where J-cost measures deviation from unity on each interval and enables ranking of tuning systems.
proof idea
This is a definition module with supporting lemmas. It introduces stepCost directly from the imported J, proves stepCost_zero_at_just and stepCost_nonneg by substitution into J-properties, defines tetStep and its band, shows tetStep_jcost_pos, and closes with ji_beats_tet to support the TuningRankingCert. All steps are algebraic reductions or direct inequality checks.
why it matters in Recognition Science
The module supplies the J-cost ranking for tuning systems and thereby feeds future parent theorems in the MusicTheory extension of Recognition Science. It instantiates T5 J-uniqueness and T7 eight-tick octave structure on musical intervals, providing a concrete comparison between just intonation and equal temperament that aligns with the phi-ladder optimality claims. No downstream theorems are recorded yet.
scope and limits
- Does not derive the J-function itself.
- Does not compute numerical costs for concrete scales.
- Does not treat microtonal or non-Western tunings.
- Does not incorporate perceptual or auditory models.