Transcript Analysis of Violin and Viola Vibrato Waveforms - Physics

Analysis of Viola Vibrato Waveforms
Katherine Hagen1,3, Kathleen Horvath2, and Gary Chottiner1
1Department
of Physics, 2Department of Music, Case Western Reserve University; 3Department of Viola, Cleveland Institute of Music
Vibrato is a series of expressive pitch oscillations produced by left hand and arm
movements on the viola. In this project, the characteristics of viola vibrato
waveforms in various technical and musical contexts were investigated. Several
viola majors at the Cleveland Institute of Music were audio- and video-recorded;
plots of selected notes’ peak frequencies vs. time near their fundamentals and
harmonics were generated and characterized. After additional recording of
selected subjects playing on each other’s instruments, it was determined that
vibrato waveforms are affected by resonance patterns in particular instruments
as well as subjects’ physical vibrato motions.
INTRODUCTION
Vibrato is one of the least
standardized techniques in string
playing. Neglected for centuries by
pedagogues who felt it must come
straight from the soul, vibrato is
taught through exercises today, but
teachers’ ideas regarding its execution
and musical role vary widely.
Scientists have discovered that viola
vibrato has an amplitude of about one
quarter tone and a frequency of about
5-7 Hz, but other characteristics
remain ambiguous.
Magnitude of PSD (Volts2)
ABSTRACT
INITIAL PLOTTING. A three-dimensional plot of Subject 4 playing B4 “with vibrato.” The
first four partials are shown. The frequencies associated with power spectral density (PSD)
peaks at each time were saved, scaled, and plotted on two-dimensional plots.
CATEGORY 1. The peak-frequency waveform near the fundamental resembled a
symmetric sine or triangle wave. Left plot: scaled peak frequencies near the first four
partials (in black, red, green, and blue, respectively) of Subject 14 playing B4 “with vibrato”
follow a single waveform closely. Right plot: the peak frequencies near the fundamental
with a sine wave fit. The waveform’s slopes generally match the sine wave’s; small
inconsistencies at extrema varied with window size and were attributed to noise.
CATEGORY 2. The peak-frequency waveform near the fundamental had large spikes in
the positive and/or negative direction. Left plot: scaled peak frequencies near the first four
partials (black, red, green, and blue, respectively) of Subject 6 playing A4 “with narrow,
slow vibrato.” Right plot: Subject 5 playing A4 “with vibrato” on Subject 6’s viola. The
spikes in both plots originate near 440 and 445 Hz, indicating that they are a result of the
resonance characteristics of Subject 6’s viola body and/or strings.
CATEGORY 3. The peak-frequency waveform near the fundamental was asymmetric. Some
samples also had resonance effects (see Category 2). Left plot: the first four partials (black,
red, green, and blue, respectively) of Subject 9 playing B4 “with vibrato.” The vibrato
upswings (pitch-raising motions) appear to be faster than the downswings (pitch-lowering
motions). Right plot: the first four partials of Subject 4 playing B4 “with vibrato;” the
upswings are slower than the downswings.
METHODS
RESULTS AND CONCLUSIONS
Several (n = 14) CIM viola students were recorded using a Marantz
CDR300 digital recorder, two Rode microphones, and three Canon video
cameras in CIM’s Kulas Center for International Music Education studio,
which is soundproof and nonresonant (S/N  50 dB). Each subject played a
seven-note sequence nine times with prescribed fingerings and bowings.
Instructions were given each time regarding vibrato speed (slow or fast) and
width (wide or narrow) or an emotion to convey (happiness, sadness,
tenderness, or anger). Each subject also completed a survey about his or her
musical history, instrument make and size, and vibrato ideals.
Later, audio recordings were made of selected subjects (n = 4) playing
vibrato with selected speeds/widths and emotions on each other’s instruments
and bows; for this session, an Edirol R-09 digital recorder was used in a
nonresonant CIM practice studio (S/N  25 dB).
Single-note vibrato samples, each about 2-3 seconds long, were analyzed in
MATLAB. A Discrete Fourier Transform algorithm with frequency
interpolation was used to calculate and plot the power spectral density (PSD)
vs. frequency vs. time of each sample (see three-dimensional plot above). The
frequencies associated with PSD peaks near the fundamental and first few
harmonics were then determined and plotted vs. time in Origin.
Peak-frequency plots generated of a representative sample of single-note
tracks fell into one or more of the three categories described above. Category 1
waveforms (and, in general, waveforms free from resonance effects) were often
observed when subjects were instructed to play with narrow vibrato or to
communicate sadness. In these contexts, a small range of frequencies—and
likely a soft dynamic and a bow contact point far from the bridge—ensured
that resonances in subjects’ instruments were not easily excited and that
symmetric finger rolls were directly translated into pitch oscillations.
Category 2’s sharp extrema were attributed in [3] to resonance effects in
instruments which caused sudden phase shifts at specific frequencies near the
fundamental. However, the absence of corresponding resonances near the
harmonics “smoothed” the vibrato sound a bit, as illustrated in Fig. 4 [3]. These
waveforms were seen in subjects’ wide, happy, and angry vibratos—ones with
large frequency ranges, loud dynamics, and bow contact points near the bridge,
all of which provided ample opportunity for resonances to be excited.
Category 3’s asymmetries frequently appeared in combination with
Category 2’s resonance effects and occurred often on naturally resonant notes
(those with a matching open string). Vibrato exercises described by some viola
pedagogues suggest asymmetrical finger movements [5], but the relative effects
of finger movements and instrument resonances are still unclear.
Time (sec)
Frequency (Hz)
Figure 1. The viola (bottom) is slightly larger than
its better-known cousin, the violin (top) [1].
Figure 2. Plot of a cellist playing E3 vibrato and
nonvibrato from Geringer and Allen (2004) [2]. The
vibrato waveform is approximately symmetric and
resembles a sine wave.
Plots of vibrato waveforms in
published studies indicate a wide
Figure 3. A plot of a violist playing G6 with range of shapes are possible: Figs. 2-4
vibrato (dotted line) and without (solid line) from illustrate three of them. None of these
Brown and Vaughn (1996) [4]. The waveform’s studies were explicitly investigating
upswings are slightly slower than its downswings.
the characteristics of waveform shapes, and it is unclear whether the differences in the plots are a result of subjects’
technical skills, instrument characteristics, the acoustics of the room in which
recordings were made, or differences
in pitch-detection algorithms.
Goals of this project are:
- To quantify the differences between
CIM violists’ vibrato waveforms, with
respect
to
instruments,
given
instructions, and vibrato techniques
Figure 4. A plot of a violinist’s vibrato on D5 from
and ideals.
and Wakefield (2000) [3], including peak
- To explain the observed waveforms Mellody
frequencies near the fundamental (dotted line) and
by comparing them to those a scaled average of the harmonics (solid line). The
published in scientific studies and/or fundamental-frequency waveform resembles a sine
wave but has sharp discontinuities at its minima.
described by pedagogues.
References
Acknowledgements
[1] Courtesy commons.wikimedia.org.
[2] Geringer, J.M. and M.L. Allen. “An analysis of vibrato among high school and university violin and cello students.” J. Res. Mus. Ed.
52, 167-178 (2004).
[3] Mellody, M. and G.H. Wakefield. “The Time-Frequency Characteristics of Violin Vibrato: Modal Distribution Analysis and
Synthesis.” J. Acoust. Soc. Am. 107, 598 611 (2000).
[4] Brown, J., and K. Vaughn. “Pitch Center of String Instrument Vibrato Tones.” J. Acoust. Soc. Am. 100, 1728-1735 (1996).
[5] Castleman, H. “Flexibility of Vibrato: Controlling Width and Speed.” J. Am. Viola Soc. 17, 21-27 (2001).
Thanks to my advisors for their expertise and guidance; to Greg Howe and the CIM Distance Learning/Kulas Center for International
Music Education staff for the use of their studio; to CIM Marketing for logo assistance; to my viola teachers, Jeffrey Irvine and Lynne
Ramsey, for their inspiration and patience; and especially to my CIM colleagues who generously donated their time as research subjects.
For Further Information
Please contact Katherine Hagen at [email protected].