COLS Sound Lab — multi‑voice (p^e) Dark • loop off • v24

Explanations — how to “hear” integers

1. Factorisation. We write $M={\prod }_i^{e_i}{p}_i$. Each prime‑power $p^e$ becomes one voice.
2. Geometric wave. $\frac{A}{\pi /2}$ is a normalised triangular wave on the class $modM$ that modulates the audible energy.
3. Arithmetic windows. For $t=n/2$ we measure how close $a^t$ is to $\pm 1$ modulo each $p^e$. The distance is $h_{p^e}$.
4. Per‑voice energy. $E_i=\frac{A}{\pi /2}·(1-h_{p^e})$. Peaks of $E_i$ reveal audible structure.
5. Aggregators. $E_{\min }=max₍i₎E_i$ (at least one factor resonates). $E_{\mathrm{k=2}}$ is the second‑largest among {$E_i$} to emphasise coincidences.
6. Periods → frequencies. We estimate each voice period on the arithmetic signal $S_i=1-h_{p^e}$, then map $f_i=\frac{K}{T_i}$ (Auto/A/B scales), with transpose $\alpha$ and octave factor $\times 2^\mathrm{oct}$.
7. Render. Chord: all voices at once. Arpeggio: voices one after another. Tempo‑sync: step derived from the envelope autocorrelation.
8. Seamless looping. Continuous oscillator phase, buffer length as an integer multiple of the step (for arpeggio), Hann micro‑fade per step, and short end→start crossfade.