Trumpet spectrum transformation systematics
A. H. Benade, Peter L. Hoekje · The Journal of the Acoustical Society of America · 1980
The spectrum transformation function Tmr(f) converts the mouthpiece spectrum into the spectrum averaged over the room's sound field, Tmr is extremely irregular, and is best represented as the product of other functions. Tmc transforms from the mouthpiece to the standing wave amplitude in the cylindrical main bore. Tcb converts this to the on-axis spectrum just outside the bell. Tbr modifies the latter to give the room average spectrum. Tcr = Tcb ⋅Tbr = [1 + (1343/f)9.9]−0.25, and Tbr = [1 − (f/104)] for f ⩽ 4000 Hz. For a tonal component whose frequency coincides with an input impedance peak, Tmc ≃ 1 at low frequencies, ≃ 0.3 at 800 Hz (near the mouthpiece popping frequency Fp), and climbs past 1.0 at 2000 Hz. At high frequencies Tmc → (arim/acyl), the ratio of mouthpiece radius to that of the cylindrical bore. Harmonics of a played tone may not coincide with the input impedance peaks. If the harmonic moves above such a peak, Tmc rises overall and its dip smoothly changes into a peak. If the harmonic lies below an air column resonance, Tmc is below the resonance Tmc for f < fp increasing to very large values above fp. [Work supported by NSF.]