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Page 00000001 A Modal Distribution Study of Violin Vibrato Maureen Mellody (1) Gregory H. Wakefield, Ph.D. (2) (1) Applied Physics Program, University of Michigan firstname.lastname@example.org (2) Department of Electrical Engineering and Computer Science, University of Michigan email@example.com Abstract A high-resolution time-frequency distribution is applied to the study of violin vibrato. Our analysis indicates that the frequency modulation induced by the motion of the stopped finger on the string is accompanied by a significant amplitude variation in each partial of that note. We suggest that this amplitude modulation results from the densely-spaced wood resonances of the violin body. Amplitude and frequency estimates for each partial are extracted from the time-frequency surface and used in an additive resynthesis model to generate synthesized violin vibrato. 1 Introduction Violin vibrato is created by the rhythmic motion of the performer's left forearm and wrist which enables the stopped finger to rock back and forth on the fingerboard, changing the string length. This change corresponds to a modulation of the pitch of the note played. While this basic mechanism for producing violin vibrato is well-understood, previous studies of violin vibrato based on short-time Fourier analysis, analog filtering, and other techniques  have led to somewhat contradictory findings concerning the effect of vibrato on the frequency and amplitude of a note's partials. Inconsistencies in these studies are most likely the result of limitations in time and frequency resolutions of the earlier methods. In the following, we present results from a study of violin vibrato based on a high-resolution time-frequency distribution (TFD) that is known to produce fewer smoothing artifacts in representing the instantaneous temporal and spectral content of the signal. The present study employs the modal distribution, a bilinear time-frequency analysis method developed by Pielemeier and Wakefield . This TFD was developed for signals that can be expressed as sums of isolated sinusoids and has been shown to handle such sums undergoing frequency and amplitude modulation as well. It provides a substantially higher resolution representation than standard Fourier methods. In particular, the modal distribution is appropriate for analyzing bowed violin notes, since these are wellmodeled as sums of harmonically-related partials. Instantaneous frequency and amplitude values for each partial can be found by applying standard Hilbert techniques to local regions of the time-frequency surface. By applying these standard narrowband representations to the violin signal's modal distribution, we are able to resolve the amplitude and frequency variations in each partial for frequencies up to 11,025 Hz, given a sampling rate of 44.1 kHz. In Section 2, the salient characteristics of violin vibrato, such as the nature of the frequency and amplitude modulation are presented. The origins of these characteristics are discussed in Section 3 and compared to the results determined from the Modal TFD study. Finally, the amplitude and frequency information is then incorporated into a resynthesis model to yield a perceptually accurate synthesized version of violin vibrato. 2 Modal TFD Analysis of the Violin 2.1 Experimental Setup To acquire violin signals, DAT recordings were created in a soundproof booth. A cardioid microphone was used to acquire monophonic signals. It was placed roughly two feet above and one foot in front of the violin scroll. The same performer and bow were used in all of the signals discussed below, and smooth, full-bow notes only were studied. The signals were then digitally transferred to a computer for analysis. 2.2 Violin Vibrato Case For each of the samples, the instantaneous amplitude and frequency estimates were computed from the timefrequency surface; representative results for several partials of a G4 note are shown in the following two
Page 00000002 figures. In all of the samples, the frequency modulation exhibit similar structure, somewhere between sinusoidal and triangular in shape. The peak-to-trough excursion on the fundamental ranges between 6 and 20 Hz on the notes studied, and the vibrato rate ranges from 4 to 8 Hz. As expected, within a reasonable amount of error, the frequency values for each partial are equal to the frequency values of the fundamental times the partial number. Freq, in Hz 390 A 380 0.8 1 1.2 1.4 1.6 1.8 2 Time, in sec Figure 1: Fundamental frequency as a function of time for a G4 vibrato note on a standard violin (middle section of the note). The instantaneous amplitude estimates also varied with the periodicity of the vibrato, although in each vibrato period the amplitude variations tended to have a more complicated structure (more maxima and minima) than the frequency modulation. The character of the AM varied from partial to partial, as seen in the Figure 2, and was not coherent across partials. Peak-to-trough ratios also varied across partials for a given note from 0 to 30 dB within one vibrato cycle. Amp (linear) First partial 6000 4000 2000-----y - 0.8 1 1.2 1.4 1.6 1.8 2 Third partial 0888 0.8 1 1.2 1.4 1.6 1.8 2 Fifth partial 8000 6000r 4000 2000KV 0.8 1 1.2 1.4 1.6 1.8 2 Time, in sec Figure 2: Amplitude as a function of time for the same G4 vibrato note; partials 1,3,and 5 are plotted (middle section of the note). Violin signals without any vibrato were also analyzed with the Modal TFD for comparison to the vibrato notes. The fundamental frequency and amplitude estimates for several partials are plotted in Figures 3 and 4 as a function of time for a G4 note. As expected, periodic amplitude and frequency modulation are absent in the non-vibrato note. Freq, in Hz 390 -385 380 0.8 1 1.2 1.4 1.6 1.8 2 Time, in sec Figure 3: Fundamental frequency as a function of time for a G4 non-vibrato note on a standard violin (middle section of the note). Amp (linear) First partial 500 n - --- --- --- i --A-- ' -- ' - 0.8 1 1.2 1.4 1.6 Third partial 1.8 2 2000 j QQQL------i---------- 0.8 1 1.2 1.4 1.6 Fifth partial 1.8 2 400 j 200v L / \./ *^^^ - i"^\A 0 0.8 1 1.2 1.4 1.6 1.8 2 Time, in sec Figure 4: Amplitude as a function of time for the same G4 non-vibrato note; partials 1,3,and 5 are plotted (middle section of the note). In addition, vibrato notes from an electric solid-body violin were also studied. A Mark O'Connor Signature Series Zeta violin was used in this study. The signals were acquired by sampling the line output from the Zeta instrument. Horizontal string motion on the Zeta violin is detected by a piezoelectric transducer at the bridge. Zeta violin notes were also analyzed with the Modal TFD. The fundamental frequency as well as the amplitude values for several partials are plotted as a function of time for the G4 note in Figures 5 and 6. The frequency modulation is similar to the frequency modulation present on the standard violin, but the amplitude modulation is virtually nonexistent. Freq, in Hz 400 395 390 0.8 1 1.2 1.4 1.6 1.8 2 Time, in sec Figure 5: Fundamental frequency as a function of time for a G4 vibrato note on a Zeta violin (middle section of the note).
Page 00000003 Amp (linear) First partial 01 0.8 1 1.2 1.4 1.6 1.8 2 Third partial 5000 0.8 1 1.2 1.4 1.6 1.8 2 Fifth partial 2000L 0.8 1 1.2 1.4 1.6 1.8 2 Time, in sec Figure 6: Amplitude as a function of time for the same G4 Zeta vibrato note; partials 1,3,and 5 are plotted (middle section of the note). 3 Origins of Variations in Amplitude and Frequency in Violin Vibrato 3.1 Densely-Packed Resonances In his book on violin physics, Cremer  developed a model to determine the density of wood modes for coupled violin plates. The density of modes in a wood shell, for frequencies above 1 kHz, is given by hec Af -= -(1) where h = height of the plate, c, = speed of compressional waves in wood, and A = area of plate. However, wood is highly anisotropic so the speed of sound in the wood is not the same across the grain and with the grain of the wood; instead, the geometric mean of the two speeds is used in the calculation as an approximate value. Making further approximations, such as an average plate thickness and an average plate area, Af = 73 Hz for the top plate, Af = 108 Hz for the back plate, and the two coupled together give Af = 44 Hz. This means that, roughly speaking, for frequencies above 1 kHz, there is a wood resonance every 44 Hz. Because of these densely-spaced resonances, the use of vibrato can have a significant impact on the amplitude of the produced sound. For a typical vibrato, the full frequency excursion due to the changing string length varies between 6-20 Hz of the fundamental, depending on the frequency of the note played and on the performance style used. Since the frequency excursion for a particular partial is the excursion of the fundamental multiplied by the partial number, several different resonances can be excited within one vibrato cycle for a particular partial. Therefore, each partial "weaves" in and out of the different resonant peaks because of the frequency modulation, and this "weaving" is reflected by variations in amplitude. The hypothesis that wood resonances create the amplitude modulation is further supported by the absence of significant amplitude modulation among signals generated by the solid-body Zeta violin, where the resonant modes of the wood shells are absent. 3.2 Modal TFD Data To compare the variations in amplitude and frequency to one another, correlation coefficients were computed between the different amplitude and frequency estimates for the first eighteen partials in four different vibrato notes. In order to do so, the signals first had to be "detrended"; the dc offset values had to be removed from the frequency and amplitude values so that the periodic variations in each were centered at zero. For the frequency estimates, the detrended values were computed by subtracting out the vibrato's center frequency. For the amplitude estimates, the detrended value at each time point was found by computing a local offset value, within one vibrato cycle, and removing that offset from the amplitude estimate. All frequency and amplitude estimates were then normalized by their vector norm for correlation comparisons. Since the frequency modulation is created by changing string length, all of the partials should "see" the same modulation at the same phase, modified only by different excursions. The cross-correlation of the frequency estimates of any two partials should thus have a correlation coefficient close to unity. On the other hand, since the amplitude variation phase and number of extrema in one vibrato cycle varies from partial to partial, the correlation coefficients of the amplitude estimates of any two partials should be significantly lower. Similarly, the cross-correlation of any partial's amplitude vector and any partial's frequency vector should also be somewhat lower since the phasing and complexity within a cycle varies between the two as well. This is seen in Figure 7: the correlation coefficients for amplitude and frequency for the first eighteen partials are plotted as an average across four different notes, E4, G4, A4, and D5. Notice that the correlation coefficients for partial amplitude are higher (the white section in front) than than those involving amplitude, with the exception of the autocorrelation ridge along the diagonal.
Page 00000004 ..o 36 40 0 5 20 Figure 7: Correlation coefficients for the amplitude and frequency estimates. The front-left points (1:18 by 1:18) are the correlation coefficients for partial frequencies and the back-right points (19:36 by 19:36) are the correlation coefficients for the partial amplitudes. The remaining sections are the cross-correlations between partial frequency and amplitude. The ridge along the diagonal denotes the autocorrelation coefficients, which are near unity. This figure illustrates the properties of the frequency and amplitude modulation; the frequency modulation, stemming from the string motion, is coherent across partials while the amplitude modulation, presumably stemming from the resonances of the instrument, is not. 4 Violin Vibrato Synthesis To synthesize violin vibrato notes, the signal was modeled as a sum of sinusoids, where the amplitude, frequency, and phase of each partial are time-varying functions: violin signals. The use of amplitude modulation is a crucial element in this synthesis. 5 Conclusions and Further Research The Modal TFD has proven to be an effective tool in determining characteristics of violin vibrato. Using the instantaneous frequency and amplitude estimates from the Modal TFD analysis, we were successfully able to additively synthesize perceptually accurate versions of acoustic violin signals. This analysis suggests that it may be possible to determine the location of body resonances for a particular instrument based on the amplitude modulation found in vibrato violin signals. These resonant peaks could then be used in attempt to distinguish different instruments and to establish a quantitative measure of differences among instruments. 6 Acknowledgments The authors would like to acknowledge the Office of President of the University of Michigan for their financial support of the MusEn Project, and the University of Michigan Board of Regents for providing a fellowship to the first author. In addition, we would like to thank the University of Michigan School of Music for the use of their Zeta violin. References  Beauchamp, J.W. 1974. "Time-Variant Spectra of Violin Tones," Journal of the Acoustical Society of America, Volume 56, pp. 995-1004.  Cremer, L. 1984. The Physics of the Violin, Cambridge, Massachusetts: MIT Press.  Fletcher, H. and L.C. Sanders. 1967. "Quality of Violin Vibrato Tones," Journal of the Acoustical Society of America, Volume 41, pp. 1534-1544.  Meyer, J. 1991. "New Aspects of the Violin Vibrato," Journal of the Acoustical Society of America, Volume 89s, p. 1901.  Pielemeier, W.J. and G.H. Wakefield. 1996. "A High-Resolution Time-Frequency Representation for Musical Instrument Signals," Journal of the Acoustical Society of America, Volume 99, pp. 2382-2396. N s(t) = L Ak (t)sin(2Zfk (t)t + k (t)) k=l (2) where k is the partial number, N is the total number of partials, Ak is the time-varying instantaneous amplitude estimate for the kth partial, oki is the time-varying instantaneous frequency of the partial in radians per second, and (Pk is the time-varying instantaneous phase of that partial, which is tracked to prevent discontinuities in the synthesized sound. The frequency of the synthesized note is updated at the rate that the estimates were computed from the Modal TFD. The amplitude of the synthesized sound is also updated at this rate, but is linearly interpolated between amplitude estimates. Violin vibrato signals based on the estimated instantaneous amplitude and frequency values from the Modal TFD have been synthesized additively and shown to yield perceptually accurate exemplars of acoustic