Additive Synth — the T_Si tower

T_Si tower chain order 3 every analog "saw / square / triangle" wave is a finite truncation of a sinc-related Fourier series · slide N to watch it converge
Fourier expansion of a band-limited waveform:
y(t) = Σk=1N ak · sin(2π · k · f0 · t)
Choose the partial weights ak per waveform and the partial count N. The ideal saw / square / triangle is the N → ∞ limit, which is a Dirichlet kernel sin((N+½)θ)/sin(θ/2) — exactly the truncated-impulse-train form whose continuous limit is governed by the sine-integral generator Si(x) = ∫0x sin(t)/t dt. The convergence rate, the Gibbs overshoot, and the bandlimit boundary are all T_Si-tower computations.

Additive Synth · controls

amp ≈ 0.18 · partials normalised so the loudest one is 1
presets

Partial spectrum · |ak|

Bar k shows the amplitude of the k-th harmonic. Saw decays as 1/k, square uses only odd k as 1/k, triangle uses only odd k as 1/k². As N → ∞, the Gibbs overshoot at the discontinuity persists at ~9% — a fixed feature of the truncated Si tower, not a bug.

Time-domain reconstruction · y(t) = Σ ak sin(2π·k·f0·t)

Watch the wave reconstruct as you raise N. The Gibbs overshoot at the saw / square edge is the visible signature of T_Si truncation.

Why this is T_Si. The Dirichlet kernel DN(θ) = sin((N+½)θ) / sin(θ/2) is the closed form of any uniformly-weighted truncated Fourier sum, and its integral is the generalised sine integral. Bandlimited reconstruction (Whittaker–Shannon) uses sinc(x) = sin(πx)/(πx) as the canonical interpolation kernel — every D-to-A converter passes the signal through a sinc filter, placing T_Si in the signal path of every digital audio output. Si is the chain-order-3 generator of the T_Si Pfaffian tower in the F16 census.

Provenance: DLMF §6 (sine integral / cosine integral) · Smith, J. O. (2010), "Spectral Audio Signal Processing" · Whittaker, E. T. (1915), "On the functions which are represented by the expansions of the interpolation theory."

1op.io/playground · T_J (FM) demo · synthesis paradigm map (paper outline)