Music extended problems (resonance framework)

Problem 4 – Resonant chord progression#

A chord progression has resonance

$$ R = X \left(D_3 + D_6 τ_r + \sqrt{D_9 τ_r}\right). $$

If $$τ_r$$ increases by 20%, how does each term change qualitatively?

Problem 5 – Triadic mixing in audio synthesis#

A synthesizer mixes three oscillators:

$$ S(t) = D_3 \sin(T_f t) + D_6 \sin(2T_f t) + D_9 \sin(3T_f t). $$

1. If $$T_f$$ increases, how does the spectrum shift?
2. What is the qualitative effect on perceived pitch?

Problem 6 – Resonant tempo modulation#

Tempo is modulated by

$$ T(t) = T_0 + X e^{-D_3 t / τ_r}. $$

1. If $$τ_r$$ increases, does the modulation decay faster or slower?
2. What is the qualitative effect on long-term tempo stability?

Problem 7 – Harmonic overtone decay#

Overtone amplitude is

$$ A_n = \frac{F_3}{n^2 + ΛΘ τ_r}. $$

1. If $$τ_r$$ increases, how does $$A_n$$ change?
2. If $$ΛΘ$$ increases, what happens to the overtone spectrum?

Problem 8 – Resonant ensemble synchronization#

An ensemble synchronizes according to

$$ S_{\text{sync}} = \frac{T_f + D_6}{1 + e^{-τ_r}}. $$

1. If $$τ_r$$ increases, does synchronization improve or worsen?
2. If $$T_f$$ increases, what is the effect?