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The 21st International Congress on Sound and Vibration
13-17 July, 2014, Beijing/China
VIOLIN SOUND QUALITY: CAN IT BE CONTROLLED
WITHIN THE PARAMETERS OF PLATE TUNING?
Robert Wilkins, Jie Pan and Hongmei Sun
School of Mechanical and Chemical Engineering, University of Western Australia,
WA, Australia, 6009
e-mail: [email protected]
It is well known that violin plate tuning parameters are controlled by adjustments of mass and
stiffness but the link between plate tuning and violin sound quality lags behind in violin research. If
violin making is to be improved two important objectives have to be achieved. First, violin sound
quality has to be related to objective measurement and second, this measurement has to be brought
within the control of the maker during the plate tuning process. This paper will show that violin
sound quality is constructed within the harmonic spectrum, that it can be objectified in measurement taken from within the harmonic spectrum, and that specific harmonic relationships can contribute either positively or negatively to sound quality. Then evidence is presented to show how the
maker can control plate tuning in ways that accentuate positive and avoid negative harmonics and
so improve violin sound quality.
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1.
Introduction
The approach of this paper is to first examine how musical sound quality is constructed, then
to show from examples how the violin sound-box shapes the sound and then to look at ways the
sound-box can be modified to maximize its potential for good sound by limiting sounds that are
perceived as not good.
Sound quality is defined in the first instance as sound that the listener perceives to be pleasant
and by contrast poor quality sounds are those the listener perceives to be unpleasant. Many words
can be substituted for pleasant like beautiful, smooth, bright, vibrant, full, and rich. Other words
that describe unpleasant sounds are ugly, rough, harsh, metallic, irritating, nasal, thin and dull.
2.
Tone quality.
The quality of musical tone or timbre was studied extensively by Helmholtz [1]. In a series of
experiments, he arranged for banks of tuning forks to sound in harmonic series. Each tuning fork
had its own resonating chamber with an adjustable aperture by which to control the output amplitude. By sounding each bank of harmonically related tuning forks and varying the amplitude of each
he was able to perform many experiments to demonstrate how the perceived quality of musical tone
could be altered. For example, he could simulate tones like the human voice humming different
vowel sounds, and even differentiate between the sound of letters such as ‘m’ and ‘n’ or introduce
nasality, roughness and harshness. In other experiments, he blended tones with harmonic overtones
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
that would sound together in ways that could give the perception of degrees of consonance and dissonance.
Consonant and dissonant tones mark opposite ends of a continuum of pleasantness and unpleasantness. The theory underlying consonant and dissonant tones relates to how well tone combinations fit comfortably with our perception of sound. A perceived tone has within it a series of
equally spaced harmonic intervals that are all whole number multiples of the fundamental frequency. Harmonics therefore lie along an equal interval measurement scale. The frequencies of notes in
the twelve semitone equal temperament musical scale are divided up according to the frequency
spacing within each octave so that the frequency of each semitone is always 1.05946 times larger
than its predecessor. This is an exponential scale and therefore the frequency of some harmonics
may or may not match with the frequency of one of the octaves but have a position somewhere in
the scale between octaves. In fact harmonic intervals of the fundamental only coincide with an octave at the 2nd, 4th, and 8th octaves. When such coincidences do occur consonant tones result. The
placement of the harmonics at or between octaves determines the musical quality of their combinations. Different amounts of consonance occur depending on the ratio of the harmonic numbers. Low
numbered ratios are consonant, e.g., 1:2 the octave, 1:3 a twelfth (an octave and a 5th), 2:3 a fifth,
3:4 a fourth, 3:5 a major sixth, 4:5 a major third and 5:6 a minor third. The 7th and higher odd numbered harmonics are dissonant and add the perception of irritating discord. When harmonic ratios
are not concordant they produce beats. It is the underlying beats that produce the perception of dissonance. In general the qualities we associate with musical tones depend upon the mix of harmonics, the relative magnitudes of these harmonics and how well the frequencies of these harmonics are
concordant with each other.
Jeans [2] summarizes some of these determinants of sound quality. The 1st harmonic is the
fundamental tone which is the pitch or frequency of the note that we hear. The 2 nd harmonic is the
octave and adds only clearness and brilliance. The 3 rd harmonic is a 5th past the 2nd harmonic and
also adds brilliance but changes the timbre by thickening the tone with a hollow nasal quality like
that found in clarinet tone. The 4th harmonic being two octaves above the fundamental contributes
only brilliance with some shrillness. The 5th harmonic contributes brilliance and a rich horn-like
quality. The 6th harmonic adds a delicate shrillness of nasal quality. The first six harmonics are parts
of the common chord of the fundamental and are concordant with one another.
The 7th, 9th, 11th and all higher odd numbered harmonics add dissonance to the fundamental
and this gives the sound a roughness or harshness sometimes taking on a metallic quality. This
summary is given without reference to the relative magnitudes of each harmonic. If these are varied
it will add more or less emphasis to the qualities described and will account for a wide variety of
tones found both within and between violins as well as other members of the violin family of instruments.
3.
What can be learned from the sound of great violins?
The tonal qualities of the following six violins were analysed: the Ernst Stradivarius (1709);
the Joachim Stradivarius (1714); the Rode Stradivarius (1733); the Gibson Guarnerius (1734); the
Plowden Guarnerius (1735); and the Vuitton Guarnerius (1739). All of these violins have established reputations for high quality sound and so an analysis of their sound has the potential to uncover important details of relevance to violin making.
These tones are recorded from the solo opening of the Bruch violin concerto in G minor. They
were broadcast by Radio WCLV, Cleveland, Ohio during the Karl Haas program “Cremona Magic,” in the series, “Adventures in good music.”
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
The tones from all six violins were analysed using the Signal Analysis and Display program
(SAND), Miller [3], Brown [4]. Because of space limitations only the tones of the Ernst Stradivarius of 1709 and the Gibson Guarnerius del Gesu of 1734 are shown here. Figure 1 shows spectrograms comparing both violins on the same notes from the Bruch violin concerto played by Ruggiero
Ricci.
Figure 1 displays the harmonic spectra for fourteen notes for the Ernst Stradivarius (above)
and the same notes for the Gibson Guarnerius (below). It can be seen from these spectrograms that
there are clear differences both with and between these violins in the arrangement of their harmonics. A look at the following examples will make this clear. On the note G3 the Stradivarius is
strongest on the 4th harmonic while the Guarnerius is strongest on the 2nd harmonic. On F#4 they
are both very similar though the Guarnerius is a little stronger on the 3 rd harmonic. On A5 the Stradivarius has a very strong 1st harmonic (above the scale), next strongest is the 2 nd harmonic. On A5
the Guarnerius is not as strong but is even across the first three harmonics. C#6 is different again
with the Stradivarius having a strong 1st harmonic and with the Guarnerius having a weak 1 st harmonic. The reader can study the notes and make further comparisons. However it is not the differences between the two violins that is the concern of this paper but rather what both violins share in
common.
One tonal feature stands out in these two violins shown in Fig.1. The harmonic structure of
each tone is largely confined to a few low numbered harmonics. After these the contribution from
higher numbered harmonics above the 4th harmonic is minimal. Using the summary of harmonic
tonal qualities described above by Jeans [2], all of the strongest harmonics for both violins in Fig.1
are concordant with one another because they come from low numbered harmonics. Both violins
are largely free of problem harmonics, like the 6th that adds shrillness to the sound, and the dissonant, rough discordant qualities that come with the 7th and higher odd numbered harmonics. Any
small contribution from these higher discordant harmonics would be masked by the very strong low
numbered harmonics. These observations apply to all fourteen notes analysed and this was also
found to be the case with the other two Stradivarius and two Guarnerius violins listed above.
Rogers [5] used the SAND program to study the harmonic spectra of twelve prize winning violins
at the 2004 VSA competition. He also included a spectrogram of the 1708 Havemeyer Stradivarius violin. However, he used glissandos instead of scales and as a consequence the sampling of individual notes
was very brief and so identifying their harmonics can only be tentative. Even so, it is clear that the sound
of the Stradivarius is established with the first five harmonics.
Two conclusions can be drawn from this evidence. First, the tonal quality of these classic violins
can be accounted for by the mix of high magnitude responses in the first few low numbered harmonics
and from the near absence of the discordant 7th and higher odd numbered harmonics. Second, the inference can be drawn that the sound box of these classic violins must therefore be constructed in such a
way as to respond strongly to bowed notes in the frequency range represented by the low numbered
harmonics and only minimally to harmonics above that range. This is of major importance for the practice of violin making.
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 1. Spectrograms comparing the harmonic spectra of fourteen notes taken from the opening
of the Bruch violin concerto in G minor played by Ruggiero Ricci during a single recording session on
the Ernst Stradivarius of 1709 (top) and the Gibson Guarnerius of 1734 (bottom). Individual harmonics
can be identified by numbering them from left to right.
4. Can the harmonic spectrogram be used as an analytic tool?
The specific pattern of harmonics seen in the Stradivarius and Guarnerius spectrograms above is
indicative of the highest quality of violin sound. This is true not just because it comes from violins of
known quality but also because it is supported by the music theory that explains the origins of concordant and discordant sound. Therefore a pattern of harmonics that constructs sound from high magnitudes
among low numbered harmonics up to the 4th and avoids significant influence from higher harmonics,
especially the 7th and higher odd numbered harmonics can become a standard to aim for and a tool by
which to analyze why a particular violin has a sound that is less than desirable. This can assist violin
makers to modify plate tuning and other procedures so as to better control the quality of violin sound.
5. Can the objective analysis of sound be used to improve violins?
As a test of the above way of analyzing violin sound, a harmonic spectrogram was made using a
violin with a known unpleasant hardness in its sound. This violin No.16A was made by the author in
1996 using the Hutchins [6][7] method of bi-modal plate tuning, in this case with the specific objective
of stopping plate graduation at the point where the resonance of mode 5 reached maximum magnitude.
This was judged, during plate vibration on the shaker table, by very high glitter bounce over the antinode and by node lines that were well defined. The plates had been varnished and cured for eighteen
months prior to the final stage of tuning to ensure that further un-controlled changes to tuning frequencies would not occur as a result of applying varnish later. The sound of this violin was recorded playing
a three-octave scale over the four strings. The resultant sound was very strong and in places disappointingly hard and metallic. In hindsight the reason why is now known and is explained by strongly vibrating plates that excite the discordant higher harmonics.
As a further test, two other violins by modern makers that had undesirable features in at least part
of their musical range were also recorded. Their sound spectra were analyzed and it was seen that the
shrill notes emanated from a combination of strong magnitudes in the 3 rd and 6th harmonics, and the hard
course notes could be traced to the presence of discordant higher odd harmonics.
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 2. Spectrograms comparing the harmonic spectra of fourteen notes played on Violin 16A
before plate re-graduation (blue top) and after plate re-graduation (yellow bottom).
6. Can harmonics be modified?
When violin 16A was first tuned, the aim was to stop graduating the plates at a point where mode
5 gave the strongest response as determined by the highest glitter bounce on the shaker table. A look at
the harmonic spectra of these notes (see Fig.2 top spectrogram) shows clear differences with the classic
violins seen in Fig.1. The disappointing hard metallic sound in the finished violin is now explained by a
too high frequency response that drives the harmonics above the 5th harmonic to shrillness in the 6th and
discordant hardness in the 7th and higher odd numbered harmonics. So the question arises as to whether,
by re-graduating the plates to only a moderate level of glitter bounce, the harmonic spectrum can be
modified to limit troublesome harmonics.
It should be noted that the stiffness and tuning frequencies of free plates undergo a big upward
shift when they are fixed to the ribs Wilkins and Pan, [8]. This means that any loss of stiffness by tuning
free plates below what appears to be their optimum mode 5 resonances, as seen in maximum glitter
bounce, is more than made up when the edges become fixed to the ribs. This explains why the great
classic violins can have very thin plates and still project a strong sound. So the action of thinning the
plates of violin 16A, though a departure from the previous bi-modal plate tuning frequencies is an appropriate choice.
Violin 16A was taken apart and the plates were re-graduated to remove some of the mass and
stiffness. The original tuning free-plate frequencies were: Top-plate mode 2=173Hz, mode 5=359Hz;
and the Back-plate mode 2=173Hz, mode 5=355Hz. After re-graduating the plates the new tuning frequencies were: Top-plate mode 2=155Hz, mode 5=350Hz. The back plate was not removed from the
ribs. It was re-graduated down to 2.3mm in the upper bout and 2.5mm in the lower bout. The center bout
was reduced by 0.5mm at its center, tapering off towards the bass-side under the label. The sound-post
side was unchanged. After re-assembly, the sound of the violin was recorded again.
The spectrograms for before and after plate re-graduation are shown in Fig.2. Before plate regraduation the harmonic spectra were frequently extended to the 6 th harmonic and beyond giving the
instrument its disappointingly shrill and hard sound. After plate re-graduation the magnitudes of the
harmonic spectra are seen to have moved away from the problem higher harmonics. There was a decided
improvement in the sound quality. The hard metallic sound had gone and the overall ease of playing and
sound projection was greatly improved.
7.
Conclusion
This paper has provided a basis for judging the quality of violin sound that is well supported
by established music theory. Desirable sound qualities are found in tones that combine the low
numbered harmonics up to the 5th. The 6th harmonic if strong can add shrillness. The 7th and higher
odd numbered harmonics are discordant. Six high profile Stradivarius and Guarnerius violins were
all found to construct their sound from low numbered harmonics so that this now represents an objective standard of excellence by which to measure violin sound.
This new objective standard was tested on a violin with a disappointingly hard sound. The
violin was taken apart and its plates were re-graduated to reduce stiffness and lower the frequencies.
This brought its harmonic spectra more in line with the classic violin standard and proved to be an
effective method for improving violin sound quality. It is suggested that violin makers could imICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
prove the standard of violin making by first, measuring the harmonic spectra of their instruments,
second, comparing the results against the classic standard and third, by experimentally modifying
their plate tuning practices to more objectively match outcomes to this standard of sound quality.
References
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Helmholtz, Herman. On the Sensations of Tone, 2
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Jeans, Sir James. Science and Music, Dover Publications, N.Y., 84-87 (1968).
Miller, J. E., Spectral measurements of violins, Catgut Acoustical Society Journal, 2, (4), (Series 2), 1-4
(Nov. 1993).
Brown, Judith C. An efficient algorithm for the calculation of a constant Q transform, Journal of the
Acoustical Society of America, 92, (5), 2698-2701, (1992).
Rodgers, Oliver E., Tonal Tests of Prizewinning Violins at the 2004 VSA Competition, Journal of the
Acoustical Society of America: VSA Papers, 1, (1), 75-95, Summer (2005).
Hutchins, C. M., Plate tuning for the violinmaker, Catgut Acoustical Society Journal, 4, (1) (Series II), 52-60
(May 2000). Reprinted from Catgut Acoust. Soc. Newslett., 39, 25-32 (May 1983).
Hutchins C. M. and Voskuil, D. Mode tuning for the violinmaker, Catgut Acoustical Society Journal, 2, (4)
(Series II), 5-9 (Nov.1993).
Wilkins, R. A. and Pan, J. Experiments on the relationship between the resonance frequencies of the B1
corpus bending modes, rib thickness, and free-plate tuning, Journal of the Acoustical Society of America.: VSA Papers, XXII, (1), 144-159 (2009).
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