Additional Remarks
Micro-modulations
Why are synchronized micro-modulations of the fundemantal frequency and the low-pass filter cutoff so essential for believable synthesis of wind instrument tones? Their intricate relationship was investigated in-depth by Horner and Beauchamp in [5] and can also be explained by the physics of the tone production in the instrument [2,3,4,17]. Regardless if we look at brass, single-reeds or double-reeds, these instruments usually consist of a tube resonator and some kind of excitation (e.g., the player's lips) that makes the air within the tube vibrate. Wind instrument tones exhibit clearly harmonic spectra whose brightness changes with dynamic level, i.e. the relative amplitudes of higher harmonics increase with intensity. If the fundamental frequency of a particular tone is modulated over time (e.g., by vibrato) then the resulting brightness changes proportially, since the harmonics of the excitation signal shift their relationship to the tube's resonant peaks. Those resonant peaks are determined by the tube length, which in turn is determined by the fingering the wind player applies to the instrument. This results in modulated attenuation of higher harmonics, which can be effectively emulated by a time-variant low-pass filter. In the plots below, we show the fundamental frequency trajectory and low-pass filter cutoff trajectory of selected tones with vibrato. For visualization purposes, we shift the filter cutoff down to the fundamental. We can observe that often the modulation of fundamental frequency is inversely proportional to the brightness. For some recordings, the opposite relationship can be observed. This is probably due to different execution of the vibrato by the musician.
Pulsetable oscillator
What is the advantage of the pulsetable oscillator in comparison to a wavetable oscillator? The combination of wavetables with time-variant filters was already proposed 30 years ago by Horner and Beauchamp [5]. They also raised the question how many entries (i.e., single-period waveforms) the wavetable oscillator should be loaded with. They reported that their system performed well with 5 waveforms corresponding to the trumpet tones Bb3, F4, Bb4, F5, and Bb5. Audio examples on the accompanying website for the paper by Derenyi and Dannenberg [6] illustrate that using only one waveform for the complete pitch-range is insufficient. The reason is that the resampling of the waveforms stretches or shrinks their spectral envelopes, creating unnatural timbres. A pulsetable oscillator does not have this issue as the pulseshape is not affected by pitch changes. Especially for the brass family, the characteristic pulseshapes are also almost invariant to the note pitch. Therefore, all tuba synthesis examples shown here only used a single pulse representing the lowest note playable on the instrument (C1). In the audio examples below, we show that for the trumpet, a single pulseshape can be sufficient, while a single waveform as used in a wavetable oscillator is not sufficient.
