Simultaneous string crossings and bow changes require coordination of bow velocity and bow inclination.
Simplified model of the coordination between string crossings and bow changes in fast repetitive bowing patterns. The panels show (a) bow inclination, (b) bow velocity, (c) bow force per string, and (d) bow-change categories associated with the transition. The string-crossing range parameter (0.7 in this example) is normalized with respect to the inclination extent (unity in the model). The bow change (zero bow velocity) is indicated by a thick vertical dashed line (and a × intersecting the signals). The unshifted bow velocity ( ) is indicated in panel (b) by a gray line. The delay of the bow change at results from a phase shift of bow velocity relative to bow inclination (30° in this example). Time is normalized with respect to period, so that time and phase can be used interchangeably.
The two musical patterns used in the perceptual experiment.
Dependence of relative phase on range in the combined 2D slider conditions. (a) Means and 95% confidence intervals per range subdivision. The dash-dotted curve [see Eq. (1) ] indicates the border between cat. 2 (shaded) and cat. 3 (blank) areas. The vertical dashed lines indicate the median values of range in the “optimum” and “minimum” conditions, respectively. The fixed-range conditions at and 0.54 are shown for comparison. (b) Standard deviation of phase responses per range subdivision.
Relative bow force (percent of total bow force) transferred to the new string during the bow change in the 2D conditions (C3 and C4). The error bars indicate the 15th and 85th percentiles, i.e., 70% of the responses fall within the indicated intervals.
False attack features for false attacks of types I and II as a function of range: (top) time durations (in ms), and (bottom) relative bow displacement (in percent of total displacement per bow stroke). The error bars indicate standard deviations. In the top panel the total duration of the string crossing (arcsine relationship with range) is indicated by a dash-dotted line.
Bow speed (mean and standard dev. per range interval) at time when leaving the string-crossing range, as an indicator of the amount of ringing of the “old” string during the new note. In cat. 3 ringing is due to remaining string vibrations in the old bow stroke; in cat. 2 ringing is due to vibrations associated with a restart (false attack of type II).
String parameters used in the violin model: tension , length , string diameter , linear density , Young's modulus , and decay time of the fundamental. The last row indicates the fundamental frequencies of the strings.
Overview of the eight stimuli tested in the perceptual experiment. The used bowing parameters [bow velocity amplitude ( ), total bow force ( ), and normalized bow-bridge distance ( )] were based on real measurements and slightly adapted for the different string combinations to optimize the sound quality.
Distribution of responses in the phase-range conditions C3 and C4 across string-crossing range ( ), subdivided in 10 equal intervals of width 0.1.
Distribution of responses per bow-change category across string-crossing range ( ), subdivided in 10 equal intervals of width 0.1.
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