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Analyzing musical signals to obtain the time-varying magnitudes and frequencies of instruments’ partial frequency components is important for resynthesis, transcription, and instrument physics. Windowing techniques, including Fourier series extensions, short-time Fourier transforms, and constant-Q transforms, generate bias in time and frequency dictated by the uncertainty principle. This is significant to analysis requirements of such properties as attack, which involve changes over millisecond time ranges and require frequency accuracy on the order of cents. Alternatives such as the Wigner distribution avoid the uncertainty principle restriction and associated bias, but nonlinear cross products of magnitude and frequency computations are not smoothed as with windowing methods, increasing those sources of bias. All these techniques belong to Cohen’s class, a framework where this paper develops the modal distribution, exhibiting decreased total bias. Computation of the modal distribution and a constant-Q version are detailed. Comparative examples to windowing methods are provided. Further research on modal distribution magnitude and phase estimation verifies the advantage of this distribution over others.

Just-noticeable differences in bandlimited vertical vibration were studied for subjects sitting on automotive seats. Stimuli consisted of octave band frozen Gaussian noise centered at 4, 8, and 16 Hz. Two-interval forced-choice paired comparisons were used, with a reference level of 8 mg. The level of the alternative stimulus in the pairs varied from 8.25 to 10 mg. Stimulus intensity was measured with a seat pad accelerometer. Stimulus durations of 4 s plus 1/2-s tapers were used for all frequency bands, with 1/2 s between stimuli. Sets of trials with 2-s durations at 16 Hz were done as a test of duration effects. Three subjects were trained with feedback until performance stabilized. Then 200–250 trials were performed in blocks of 25 or 50 for each of four alternatives at each frequency and each subject. Thresholds determined from the psychometric functions ranged from 0.6 to 1.8 mg, with most between 0.6 and 1.2. Little frequency dependence was evidenced over all frequency bands, and little duration effect was seen between 2- and 4-s stimuli at 16 Hz.

Time resolution, frequency resolution, and superposition requirements present a difficult trade-off in time-frequency representations for musical signals, both for individual instrument and ensemble analysis. Existing methods have significant limitations, particularly at low frequencies. A representation is developed in Cohen's class which improves on existing methods for a large class of musical signals. The pseudo-Wigner distribution is extended to a constant-Q form well suited to the frequency resolution required for musical signals and pitch analysis. This has excellent resolution, but fails to provide even limited superposition. A kernel is designed for Cohen's class, based on a musical signal model, providing a limited superposition property while maintaining improved resolution when applied to the constant-Q pseudo-Wigner distribution. Results are presented comparing these techniques to short time Fourier methods.