Chebyshev Polyphony: Distributional Rhythms of ψ(eᵗ)

Audio Ready Time t = 0.000 (x = 1.000)

Blue Spikes: Powers of 2 ($2^k$, $\Delta t = \ln 2 \approx 0.693$)

Green Spikes: Powers of 3 ($3^k$, $\Delta t = \ln 3 \approx 1.098$)

Amber / Rose Spikes: Powers of $p$ ($p^k$, $\Delta t = \ln p$)

Active Impulse Hit: None

View Range: [0.00s, 6.00s] Continuous Endless Mode Active

Active Prime Configuration

1 Prime Selected (p = 2)

Number of Primes ($N$): 1 Primes: [ 2 ]

Instrument Timbre

All powers of prime $p$ ($p, p^2, p^3\dots$) share the exact same pitch sound signature.

Playback Speed 1.00x

Prime Inspector

Toggle & Preview

Each prime $p$ creates an endless pulse train with period $\Delta t = \ln(p)$ seconds (at 1x speed).

Mathematical Foundations: Chebyshev's $\psi(x)$

1. The Second Chebyshev Function

The Chebyshev function $\psi(x)$ is defined as the sum of the von Mangoldt function $\Lambda(n)$:

\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \ln(p)

It increases by a step of height $\ln(p)$ whenever $x$ reaches a prime power $x = p, p^2, p^3, \dots$.

2. Logarithmic Transformation $t = \ln(x)$

In original variable $x$, prime power events get exponentially spread out. By taking $t = \ln(x)$ (or $x = e^t$), prime powers occur at $t = k \cdot \ln(p)$.

\frac{d}{dt} \psi(e^t) = \sum_{p} \sum_{k=1}^{\infty} \ln(p) \cdot \delta\left(t - k\ln(p)\right)

Taking the distributional derivative yields a sum of Dirac delta impulses. Notice that for each fixed prime $p$, the impulses occur at equal intervals $\Delta t = \ln(p)$, forming a perfect periodic percussion train infinitely into $t \to \infty$!