Beyond the beat: Modeling metric structure in music and performance
Overview of the model’s processing stages.
Signal plots corresponding to (a) ANI channels 4, 10, and 30 with center frequencies 252, 507, and 3266 Hz. (b) The same ANI channels after performing a rms, downsampling, and lowpass filtering. The calculation of is discussed in Sec. II C. (c) The resulting five bands after onset detection and summation of every eight adjacent channels. Equation (2) is used to calculate .
Comparison of comb filter and reson filter responses. The reson filter has a resonant frequency of 2 Hz, and the comb filter’s lowest resonance is at 2 Hz. Both filters are passed four different series of Dirac impulses for comparison: (a) a series at 2 Hz ending at 4 s, (b) a series at 2 Hz, followed by a single impulse at 4.25 s which is antiphase with both of the filter outputs, (c) a series at 2 Hz, phase shifted at 4.25 s, and (d) a series at 2 Hz which immediately doubles in frequency to 4 Hz after 4 s.
Surface plots of the resonator outputs from each of the five bands are shown on the upper left. Grayscale values range from negative (black) to positive (white) values. Surface plots of the rms of the resonator outputs, referred to as periodicity surfaces, of the same five bands are shown on the upper right. Grayscale values range from zero (white) to positive (black). The APS, which resulted from averaging the five periodicity surfaces, is shown on the bottom surface plot. The MPP is on the right adjacent side of the APS. The black line of the MPP plots the mean of the APS over time, while the gray bars plot the standard deviation of the APS over time.
The four sets of resonator outputs, periodicity surfaces, and MPPs illustrate how a resonator bank behaves when Gaussian noise is added to the timing of Dirac impulses in an otherwise isochronous series. (a) No deviation . Deviations in onset times are drawn from Gaussian distributions with (b) , (c) , and (d) .
The four sets of resonator outputs, periodicity surfaces, and MPPs illustrate how a resonator bank behaves when presented with a series of Dirac impulses that has an accent structure. Nonaccented impulses are assigned half the maximum amplitude, while accented impulses are assigned the maximum amplitude. (a) The input isochronous series has no accented impulses. (b) Every other impulse is accented. (c) Every third impulse is accented. (d) Accents are alternated every second and third impulses.
The same accent patterns of Dirac impulses in Fig. 6 were convolved with a Gaussian distribution function and used as inputs for comparing periodicity surfaces produced by reson filters [top plots in (a)–(d)] with periodicity surfaces produced by comb filters [bottom plots in (a)–(d)]. A constant gain for the reson filters was used for a more equivalent comparison. The comb filters were tuned according to Scheirer (1998). The reson filters clearly illustrate the frequencies present in the accent patterns while the peaks in the periodicity surfaces produced by comb filters do not significantly change with the introduction of accents.
APSs and MPPs produced from stimuli. (a) A pattern played by a single high hat. The score above the plot notates a 2.67 s long pattern that is iterated 12 times, resulting in a 32.0 s excerpt. The excerpt is used as an input to the model to produce the APS and MPP. (b) A stimulus with two instruments: snap and high hat for which every note from the excerpt in (a) is played by one or both of the instruments. The duration of the score is also 2.67 s long and iterated 12 times. [(c) and (d)] APSs and MPPs of excerpts from commercial musical recordings retrieved from the iTunes store. Each row in the raster plots above each APS corresponds to a different participant, and shows the times at which he/she tapped when instructed to tap along with the perceived beat in the music.
Periodicity surfaces and MPPs of response tapping data recorded as MIDI. The MIDI note ons are converted to Dirac impulses that are subsequently amplitude scaled by MIDI velocity, and then passed to a single resonator bank. The vertical lines at the top plot the amplitude scaled Dirac impulses produced by the left and right hand taps. The trials shown are from (a) an isochronous tapping task and (b) a free-form tapping task. The stimulus being presented during both trials was the musical excerpt for which the APS and MPP were illustrated in Fig. 8(c).
(a) Accent patterns that were used as exemplars for four different metric classes. Four measures of each pattern are illustrated here. The amplitudes of the accented onsets (except the highest onset) and the tempi were randomized, while maintaining the accent hierarchy to produce 100 exemplars for each class. (b) Mean peak height values of the MPPs of the four sets of exemplars and their standard errors. The peaks were binned by the approximate frequency ratio relative to the frequency of the peak with the highest amplitude of each MPP.
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