How to Science [Part 5: Mathematics]

The reasonable initial guess—that tension and frequency are directly proportional—is quickly rejected when data shows doubling tension (1000g to 2000g) only increases frequency by factor 1.4, not 2.

Pattern analysis reveals frequency doubles when tension quadruples (×4 → ×2) and triples when tension increases ninefold (×9 → ×3), identifying square root as the connecting operation.

Experimentally validated relationship—frequency of vibrating string is directly proportional to square root of tension (f ∝ √T)—known as Mersenne’s Second Law, another hidden connection between mathematics and physical reality.

When unable to generate new test data (limited to six guitar string types), the method tests proposed rules by making “predictions” for already-observed data points—if predictions match observations, the rule is probably correct.

Rather than “staring down data and hoping for magical insights” when patterns aren’t obvious, mathematics provides systematic tools for testing hypotheses and revealing hidden relationships.

The initial guess—frequency inversely proportional to mass per unit length (f = K/M)—produces predictions up to 70% off from observations, far exceeding the ~5% error tolerance from previous laws.

Rather than random guessing among “thousands of possible rules,” systematic refinement identifies what works (inverse relationship), what fails (too-rapid increase), and modifies accordingly—applying square root to slow growth rate.

The refined hypothesis—frequency inversely proportional to square root of mass per unit length (f = K/√M)—dramatically improves predictions, reducing maximum error from 70% down to 4.7%.

The experimentally validated relationship—frequency inversely proportional to square root of mass per unit length (f ∝ 1/√M)—completes Mersenne’s three laws governing vibrating strings.

The three Mersenne laws combine into a single comprehensive equation: f = K/(L√M) × √T, or more elegantly f = (1/L)√(T/M), providing complete mathematical description of string vibration frequency.