We present Hα spectroscopic observations and detailed modeling of the Balmer filaments in the supernova remnant (SNR) Tycho (SN 1572). We used GH α FaS (Galaxy Hα Fabry–Pérot Spectrometer) on the William Herschel Telescope with a 3.‧4 × 3.‧4 field of view, 0.″2 pixel scale, and {σ }{instr}=8.1 km s‑1 resolution at 1″ seeing for ∼10 hr, resulting in 82 spatial‑spectral bins that resolve the narrow Hα line in the entire SN 1572 northeastern rim. For the first time, we can therefore mitigate artificial line broadening from unresolved differential motion and probe Hα emission parameters in varying shock and ambient medium conditions. Broad Hα line remains unresolved within spectral coverage of 392 km s‑1. We employed Bayesian inference to obtain reliable parameter confidence intervals and to quantify the evidence for models with multiple line components. The median Hα narrow-line (NL) FWHM of all bins and models is {W}{NL}=(54.8+/- 1.8) km s‑1 at the 95% confidence level, varying within [35, 72] km s‑1 between bins and clearly broadened compared to the intrinsic (thermal) ≈20 km s‑1. Possible line splits are accounted for, significant in ≈ 18 % of the filament, and presumably due to remaining projection effects. We also find widespread evidence for intermediate-line emission of a broad-neutral precursor, with a median {W}{IL}=(180+/- 14) km s‑1 (95% confidence). Finally, we present a measurement of the remnant’s systemic velocity, {V}{LSR}=-34 km s‑1, and map differential line-of-sight motions. Our results confirm the existence and interplay of shock precursors in Tycho’s remnant. In particular, we show that suprathermal NL emission is near-universal in SN 1572, and that, in the absence of an alternative explanation, collisionless SNR shocks constitute a viable acceleration source for Galactic TeV cosmic-ray protons.