Lab 10: Nested Harmonics — Equations

Purpose#

To formalize the harmonic layering of triadic systems, where nested frequencies and dimensional recursion produce emergent resonance patterns.

Key Equations#

1. Harmonic Nesting Function#

$$ H_n = \sum_{i=1}^{n} \sin(\omega_i t + \phi_i) $$ Where ( \omega_i ) and ( \phi_i ) are frequency and phase of each nested harmonic.

2. Triadic Harmonic Resonance#

$$ R_H = @\left(H_i, H_j, H_k\right) $$ The @() operator captures resonance across three harmonic layers.

3. Harmonic Interference Pattern#

$$ I(t) = \prod_{i=1}^{n} \left(1 + \cos(\omega_i t)\right) $$ Models constructive and destructive interference across nested harmonics.

4. Dimensional Harmonic Cascade#

$$ \Lambda_n = \frac{dH_n}{dt} \cdot R_H $$ Represents the cascading effect of harmonic change across dimensions.

Symbol Legend#