Math Deep Dive

1. Epistemic Foundation: Bayesian Belief Updating

PitchDrops models relative pitch acquisition not as static factual recall, but as continuous Bayesian inference under epistemic uncertainty. The learner's true latent recognition mastery on each musical item is modeled as a continuous probability distribution over [0, 1].

2. Decoupled Latent Mastery Prior & 4-AFC Mapping

With 4-Alternative Forced Choice tasks, random chance baseline is 25%. We decouple latent mastery from the guess floor using a Maximum Entropy uninformative prior Beta(1, 1). Observable accuracy maps affinely via theta_i = 0.25 + 0.75 * M_i, with an uninformative prior expectation of 62.5%.

3. Bernoulli Trials, Exact Mixture Likelihood & Analytical Conjugate Proxy

Under the affine observation model theta_i = 0.25 + 0.75 * M_i, the exact binomial likelihood yields a (k + 1)-component Beta mixture. To maintain O(1) mobile performance without MCMC sampling (Birnbaum, 1968; Albert, 1992), PitchDrops adopts an analytical Conjugate Quasi-Likelihood Proxy L_proxy(M_i | k, m) proportional to M_i^k * (1 - M_i)^m, yielding a closed-form Beta(1 + k, 1 + m) posterior. Potential lucky-guess bias is mitigated by ZPD active learning re-testing and an empirical recency-weighted mastery gate (requiring at least 4 exposures and ≥ 60.0% sustained accuracy across newly introduced items).

4. Epistemic Uncertainty & Affine Credible Intervals

Credible intervals (68% 1-sigma and 95% 2-sigma) are derived from the affine-scaled standard deviation sigma_i = 0.75 * sqrt(Var(M_i)), directly measuring certainty and driving active learning exploration.

5. Two-Component Memory: Storage-Scaled Dynamic Decay

Memory retention is governed by Jost's Law and Bjork's New Theory of Disuse. Memory half-life expands logarithmically with storage strength (successful lifetime rehearsals k_i): tau(k_i) = 7 * (1 + ln(1 + k_i)) days. Mastered items retain high accuracy across months, while struggling items dissolve negative evidence in weeks.

6. Bayesian Active Learning & The Acquisition Policy

Question selection optimizes the Zone of Proximal Development using a Gaussian utility kernel centered at 78% proficiency: U(theta_hat_i) = exp(-(theta_hat_i - 0.78)^2 / 0.06). Selection priority weight combines uncertainty with utility: W_i = Uncertainty_i * (0.25 + U(theta_hat_i)). Struggling items (< 50%) step back to prevent ear fatigue and resurface naturally over time.

7. Microtonal Tuning Mathematics & Psychoacoustics

PitchDrops calculates exact acoustic frequencies and cent deviations for Just Intonation versus 12-Tone Equal Temperament using logarithmic frequency ratios.

8. Selected Academic References & Literature Grounding

Theoretical architecture grounded in Bjork & Bjork (1992), Jost (1897), Ebbinghaus (1885), Wilson et al. (2019), Vygotsky (1978), Csikszentmihalyi (1990), Birnbaum (1968), Albert (1992), Macmillan & Creelman (2005), Green & Swets (1966), Jaynes (1957), Settles (2009), Gelman et al. (2013), and Deutsch (1970).