The 4H-pyran-4-one (4PN) molecule is a cyclic conjugated enone with spectroscopically accessible singlet and triplet (n,π*)excited states. Vibronic spectra of 4PN provide a stringent test of electronic-structure calculations, through comparison of predicted vs measured vibrational frequencies in the excited state. We report here the T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, recorded under the cooling conditions of a supersonic free-jet expansion. The jet cooling has eliminated congestion appearing in previous room-temperature measurements of the T1 ← S0 band system and has enabled us to determine precise fundamental frequencies for seven vibrational modes of the molecule in its T1(n,π*) state. We have also analyzed the rotational contour of the 000 band, obtaining experimental values for spin-spin and spin-rotation constants of the T1(n,π*) state. We used the experimental results to test predictions from two commonly used computational methods, equation-of-motion excitation energies with dynamical correlation incorporated at the level of coupled cluster singles doubles (EOM-EE-CCSD) and time-dependent density functional theory (TDDFT). We find that each method predicts harmonic frequencies within a few percent of observed fundamentals, for in-plane vibrational modes. However, for out-of-plane modes, each method has specific liabilities that result in frequency errors on the order of 20-30%. The calculations have helped to identify a perturbation from the T2(π,π*) state that leads to unexpected features observed in the T1(n,π*) ← S0 origin band rotational contour.