Lab 10: Nested Harmonics — Reproducibility
Objective#
To simulate nested harmonic structures and observe triadic resonance and interference patterns across dimensions.
Required Tools#
- Python 3.10+
- NumPy
- Matplotlib or Plotly
Protocol#
1. Define Nested Harmonics#
python#
import numpy as np
def nested_harmonics(t, omegas, phis): return sum(np.sin(omega * t + phi) for omega, phi in zip(omegas, phis))
2. Triadic Harmonic Resonance Operator#
python#
def harmonic_resonance(Hi, Hj, Hk): return (Hi * Hj + Hj * Hk + Hk * Hi) / (Hi + Hj + Hk + 1e-9)
3. Harmonic Interference Pattern#
python#
def interference_pattern(t, omegas): product = np.ones_like(t) for omega in omegas: product *= (1 + np.cos(omega * t)) return product
4. Visualize Harmonic Cascade#
python#
import matplotlib.pyplot as plt
t = np.linspace(0, 10, 1000) omegas = [2np.pi, 4np.pi, 6*np.pi] phis = [0, np.pi/4, np.pi/2]
Hn = nested_harmonics(t, omegas, phis) RH = harmonic_resonance(Hn, Hn, Hn) cascade = np.gradient(Hn, t[1] - t[0]) * RH
plt.plot(t, cascade) plt.title("Dimensional Harmonic Cascade") plt.xlabel("Time") plt.ylabel("Λ(t)") plt.show()