Music core problem solutions

Solution to Problem 1 – Harmonic resonance energy#

$$ H = D_6 T_f^2. $$

If $$T_f$$ doubles:

$$ T_f' = 2T_f, \quad H' = D_6 (2T_f)^2 = 4H. $$

Answer: Harmonic energy quadruples.

Solution to Problem 2 – Tempo drift#

$$ ΔT = \frac{F_3}{ΛΘ}. $$

If $$Λ$$ increases by 25%:

$$ Λ' = 1.25Λ, \quad ΔT' = \frac{F_3}{1.25ΛΘ} = \frac{ΔT}{1.25}. $$

Answer: Tempo drift decreases by 20%.

Solution to Problem 3 – Beat synchronization time#

$$ t_s = τ_r D_3. $$

If $$τ_r$$ decreases by 15%:

$$ τ_r' = 0.85τ_r, \quad t_s' = 0.85 t_s. $$

Answer: Synchronization time decreases by 15%.