Savart Journal
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Analysis of violin combination tones and their
contribution to Tartinis third tone
G. C
1
2
3
4
ASELLI , G. CECCHI , M. MALACARNE AND G. MASETTI
Abstract-- It is widely accepted that the famous Tartini's third tone, i.e., the appearance of an additional third tone of
lower frequency when playing a dyad on the violin, is a subjective phenomenon generated by the listener's cochlear
nonlinearity. However, the recent demonstration that additional tones of audible amplitude can also be generated
by the violin itself during playing of a dyad (violin combination tones), raises the question if these tones might have
influenced Tartini's third sound perception. The experiments reported here were made to ascertain this possibility.
To this end, following Tartini experiments, several dyads played by either one violin or two violins playing one note of
the dyad each, were recorded. The analysis of the spectra shows that violin combination tones are present in all the
dyads investigated, but exclusively when the dyad is played by a single violin and not when the same dyad is played
by two violins. Tartini found the third tones to be the same in both conditions, which means that violin combination
tones in his experiments were either absent or too small to affect the perception of the subjective third tones arising
from cochlear distortion.
I. INTRODUCTION
In 1714 in Ancona (Italy) the famous violinist, composer and musical theorist Giuseppe Tartini discovered that when
playing two simultaneous notes (a dyad) on the violin he could clearly hear a third tone called by him terzo suono [1].
This tone was weaker than the main notes and it was lower than the lower note of the dyad. The origin, characteristics
and musical applications of the third tone have been thoroughly investigated since the eighteenth century. These
extensive studies showed the existence of several tones, rather than one, and the term Combination Tones was
introduced to describe the complexity of the phenomenon better [2].
A great step forward in understanding the nature of the combination tones was made by Helmholtz (pp. 411-413 of
[3]) with the formal demonstration that a non-linear response of the auditory system, in presence of two different
notes of frequency f1 and f2 , can lead to the formation of additional tones. These occur mainly at frequencies
f2 - f1 and f1 + f2 (f2 > f1) and constitute most of the combination tones normally heard but not present in the
air (distortion theory). Helmholtz called them, subjective combination tones to distinguish them from the objective
combination tones generated by some musical instrument where the two sound sources are mechanically coupled
as occurs in the Harmonium [3]. In contrast to subjective tones, objective tones are present and detectable in the
air. More recent studies with sinusoidal tones, confirmed the substantial correctness of Helmholtz's hypothesis, and
revealed the presence of additional subjective tones mainly 2f1 - f2 and 3f1 - 2f2 (see [4], [5]).
The auditory non-linearity was located in the amplification mechanism of the cochlea (for a review, see [6]). Recently
it has been shown that objective combination tones are also produced by some bowed instruments, especially by the
violin [7], [8]. Listening experiments, suggested that these combination tones, hereinafter called Violin Combination
Tones (VCTs), are perceivable [7]. Hence we can assume that they add to the subjective tones to produce the tone
ultimately perceived by the listener. Thus, we may ask if combination tones produced by the violin, could have had
a significant impact on the perception of the third tones described by Tartini.
In fact, Tartini describes two experimental settings for studying third tones, which can be considered quite different
from the physical point of view. The first one involves a violin playing the dyad, whereas in the second one, two violins
five or six steps away, play at the same time one note of the dyad each. In the first setting the two sound sources
are strongly coupled similarly to the Helmholtz harmonium, VCTs are therefore expected. In the second settings, the
only coupling possible between the two violins occurs through the mechanism of the sympathetic resonance. Given
the distance, the coupling between the two violins is expected to be very small or absent and the same is true for
1 Gabriele Caselli, Musical Academy, Pontedera, Italy.
2 Giovanni Cecchi, University of Florence, Italy.
3 Mirko Malacarne, Musical Academy, Pontedera, Italy.
4 Giulio Masetti, ISTI-CNR, Pisa, Italy.
Manuscript received September 24, 2020
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VCTs. Tartini found the third tones to be the same with both settings, which means that either the VCTs were the
same in both cases or they were absent or too small to influence the subjective tones due to the cochlear distortion.
It is clear that investigation of the VCTs under both conditions become essential to establish the possible impact of
VCTs on the third tones perceived by Tartini. To this end we investigated the properties of Violin Combination Tones
with experiments reproducing as much as possible those described by Tartini.
II. METHODS AND EXPERIMENTAL SETTING
In his treatise [1], Tartini states that the third tone is heard equally well either when a player plays the dyad on a
single violin or when two players, five or six steps away from each other, play one note of the dyad each with the
listener placed midway between them. To reproduce both these conditions, the Violin Combination Tones (VCTs)
were investigated with the following protocol: two professional violinists, players 1 and 2, playing two contemporary
Italian violins, were asked to play forte and without vibrato several intervals among those shown by Tartini (pp. 14-17
of [1]), standing at about 2.70m apart (five or six steps), following a precise execution program. Two microphones
(mod.CM3, Line Audio Design, Sweden) were placed near the ears of each player (mics 1, 2 and 3, 4, respectively)
at about 40 cm from the violin. To simulate the experience of the central listener, two further mics (mod. SE 4, SE
Electronics, USA) were placed in the middle between the two players in opposition to each other looking towards
the players (mics 5 and 6). Finally, one mic (mod. KM 184, Neumann, Germany) was placed in front of each player
(mics 7 and 8) at about 50 cm from the violin. Signals were fed to a low-noise, flat-response preamplifier (Line Audio
8MP, Sweden) and from there to a sound card (Motu Traveler mk3, Motu, USA). Signals were recorded individually
at 24bit and 192kHz with the Logic Pro software (Apple, USA) without any compression.
Processing of the audio signals and the Fourier Trans-
forms, calculated with the Fast Fourier Transform algo-
rithm (FFT), were performed on a period of 2s selected
from recording periods of about 5s each. The selection
was made by choosing for the most stable period of the
note recorded. Analysis, data elaboration and statistics
were made with Sigview (USA) and OriginPro software
(USA). The spectra are expressed in dB relative to
the greatest peak. All the plotted FFT records were
filtered with the moving average method (5 points). The
recording protocol, applied to all the dyads analyzed,
was split into sections of about 5s each. Each player
played alone and separately: (1) the dyad, (2) the note
1 (whose fundamental is f1), (3) the note 2 (f2) of the
dyad. Further (4), player 1 and 2 played simultaneously
Figure 1 - FFT spectra of the interval of perfect fourth C5 - F5.
one note of the dyad each, then (5) switched the note
Summed Notes Dyad (SMD) trace, summed note dyad, Single
played. Each section was repeated three or more times.
Player Dyad (SPD) trace, single player dyad. Peaks marked by the
asterisks in SPD, are the VCTs. Note that both traces have a very
By selecting appropriately the microphone recordings,
similar background, fundamental and partial amplitudes, the only
we obtained the following conditions for VCTs examina-
difference being the VCTs. Trace SPD, originally superimposed to
tion: 1) dyads performed separately by player 1 and by
SMD, was shifted upward and the abscissa was limited to 5kHz,
for clarity.
player 2, mics 1, 2 and 3, 4, both termed SPD; 2) dyads
performed by the two players playing simultaneously
one note of the dyad each, mics 5 and 6, termed Two Players Dyad (TPD); 3) dyads obtained by electronically
adding the separated records of note 1 and note 2 of player 1, mic 7, and the same for player 2, mic 8, both termed
SMD.
Conditions SPD and TPD emulate Tartini's experimental settings. Condition TPD was also used by Tartini to inves-
tigate the third tone produced by the oboe, which he found to be even stronger that in violin [1]. Clearly, SMD was
not possible at Tartini's time, because it requires electronic recording techniques.
Condition SPD is characterized by the strongest mechanical and acoustic coupling between the two played notes,
similarly to the Harmonium. The primary sound sources (the strings) are acoustically and mechanically coupled
mainly through the bridge and the violin's top plate (secondary sources) and the bow. Combination tones are therefore
expected. Mechanical interaction between the two sound sources when the dyad is played by two players 2.7m apart
(TPD) is clearly absent. In principle, however, some acoustic interaction cannot be excluded: in fact, air vibration
induced by the note played by one violin might put into vibration the top plate of the other violin (sympathetic
resonance) giving rise to some interaction between the two notes which in turn might induce VCTs generation. Due
to the weak interaction between the two violins, it seems obvious to predict that VCTs, if present, are very weak,
however to our knowledge this point has not been verified experimentally. The importance of the conditions in which
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the dyad is obtained by two notes recorded individually and summed electronically later (SMD), is that no acoustic
and no mechanical interaction at all can be present. Thus the FFT spectra in SMD dyads are expected to contain
only fundamentals and partial tones with no combination tones. Comparing SMD spectra with those recorded in SPD
and TPD allowed us to highlight the possible presence of violin combination tones and their examination. Finally, the
switching between players allowed us to draw some preliminary qualitative observations related to a given violin or
player.
III. EXPERIMENTAL RESULTS AND DISCUSSION
The results presented in this paper show that violin
combination tones are present exclusively when the
dyad is played by a single player whereas records
with two players and summed notes show no signs
of them at all. Violin combination tones occurred in
all the following intervals analyzed (scientific pitch):
perfect fifth (C5 - G5), perfect fourth (C5 - F5), major
#
third (C5 - E5), minor third (C
-
5
E5), major sixth
#
(E5 - C6 ), minor sixth (E5 - C6), tuono maggiore
(G5-A5) and tuono minore (A5-B5), all played in the
key of C major except the last two, played in the key of
G major. However, VCTs amplitude varied considerably
among the various intervals and also between the two
players. The analysis of this variability, which likely
depends on several factors, was not pursued being not
presently our main interest.
Figure 2 - Comparison of FFT spectra of SPD and SMD traces
Figure 1 shows the comparison between the FFT spec-
for the interval of perfect fifth C5 - G5. SPD and SMD are almost
perfectly superimposed, except for the VCTs on the SPD trace (red
tra of single player and summed note dyads for the
line) identified by the asterisks. The upper trace shows the relative
interval of perfect fourth C5 - F5. The summed note
spectra 20log(SPD/SMD) which highlights the difference between
spectrum (SMD) shows only peaks corresponding to
the SPD and SMD traces. C4 corresponds to the VCT at the lower
frequency.
the fundamentals and partial tones of the dyad, as
expected, whereas the single player spectrum (SPD)
shows several smaller peaks in between the fundamentals and partials. These peaks, marked with the asterisk, are
the violin combination tones which are regularly distributed all along the spectrum.
If the frequencies f1 and f2 (with f1 < f2) are in a rational relation with f2 = f1 m/n the pitch of all VCTs,
expected from the distortion theory [3] is hf1 + kf2 = (hn + km)f1/n, where h and k are integer numbers. Thus,
the VCTs' peaks occur at frequencies that are integer multiples of f1/n or the greatest common divisor (GCD) of f1
and f2 in accordance with [7].This can be seen in Figure 1 where VCTs repeat themselves every f2 - f1 interval
corresponding to f1/n ratio, named Fundamental Combination Tone (FCT), and equal in this case to 178Hz. Note
that if f1 and f2 are not exactly in a rational relation, f2 is equal to f1 m/ne, where the real number e is the small
intonation error unintentionally caused by the players. The pitch of VCTs is then hf1 +kf2 = (hn+km)f1/nke
and it does not coincide with GCD. In a few of the studied dyads the value of e was not negligible; however, this did
not affect the presence of VCTs. The appearance of combination tones at regular and precise intervals shows that
they occur as if they were the harmonics of a fundamental virtual note corresponding to the FCT plus or minus the
amplified error.
Figure 2 shows the comparison between one player dyad and summed note dyad spectra, for the perfect fifth interval
(C5 - G5). Traces are almost perfectly superimposed, except for the VCTs on the SPD trace (red line) identified
by the asterisks, which emerge from the background. This aspect is even clearer in the relative spectrum trace
(20 log(SPD/SMD)) also shown in the figure, which highlights differences between the two traces by eliminating
all the common features. Six VCTs peaks stand out clearly from the flat baseline.
#
The comparison between two players and summed note dyads spectra for the major sixth interval (E5 - C6 ) is
shown in Figure 3. It can be seen that TPD trace has no VCTs at all and it is very similar to SMD trace. The absence
of VCTs is also clearly shown by the relative spectrum. This means that, as far as VCTs are concerned, the condition
SPD (one player) and TPD (two plyers), both used by Tartini for investigating the third tone, are not equivalent.
The combination tones are present when a single player plays the dyad, but they disappear from the spectrum when
the same dyad is played (one note each) by two violin players. Tartini stated that the third tone was heard equally well
in both conditions, which means that combination tones generated by the violin were not relevant to his observations.
Thus, the third tones he described were unaffected by VCTs being exclusively subjective combination tones due to
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cochlear distortion. A possible, although unlikely exception, is shown in Figure 2. The lowest pitched combination
tones at C4 is in Tartini's third tone range. Although we did not perform any proper listening experiments, for this
particular dyad we noticed a clear difference between the SPD dyad played as it was recorded and after filtering out
the narrow region of the VCT at C4, which indicates that this VCT was perceivable.
Thus it may be asked why Tartini in his 1754 treatise did
not mention this combination tone but rather indicated
for the perfect fifth interval a third tone at unison with C5.
Tartini did not explain how the experient was performed,
but if he used the configuration with two players, the
third tone at C4 was not mentioned simply because
it was not there. Another possibility is connected with
the players and violins he used since in our experience
VCTs' amplitude depends on both these factors. Finally,
it is interesting to note that after and even before the
treatise of 1754, Tartini assigned the third tone not
to C5 but to C4 [9] in perfect coincidence with the
lowest pitched VCT shown in Figure 2. Thus, we cannot
exclude that, at least for the interval of perfect fifth, the
violin combination tone at C4 might have influenced
Figure 3 - Comparison of FFT spectra in configuration TPD (two
Tartini's listening. It is tempting to speculate that VCTs,
violins, 2.7m apart) with SMD for the major six E5 - C#. No
6
VCTs peaks are present on the TPD trace which is almost perfectly
with their variability, might have been at least partially
superimposable to SMD. Small glitches in coincidence with some
responsible for the difficulties experienced by Tartini in
partial tones, are caused by differences in intonation and amplitude
attributing the right octave to some of the third tones
of the notes played in TPD and SMD. TPD and relative spectrum
traces are shifted upward for clarity.
perceived.
IV. STATISTICAL ANALYSIS
All the figures of the paper show clearly that VCTs exists only when the dyads are played by a single instrument and
not when the same dyad is played by two instruments (one note each). To give more solidity to these observations
we decided to investigate their statistical significance. This was done for those dyads which were played at least
four times in all conditions. We report here the results for the perfect fourth (C5 - F5) and perfect fifth (C5 - G5)
dyads. Spectra amplitude was measured on linear spectra ( performed directly on the output of the sound card ),
in coincidence with the first six violin combination tones on TPD, SMD and TPD traces. An ANOVA test was then
performed on the results. Means and SEM of the data obtained are shown graphically in Figure 4.
It is clear that spectra amplitude at VCTs is much
greater in SPD than in TPD and SMD traces. Since
the analysis showed that ANOVA F and p values were
significant for all the VCTs, we performed a post-hoc
Tukey test on the data. The results are summarized
in Table 1. It can be seen that spectrum amplitude at
VCTs of the dyad played by a single instrument (SPD),
is significantly higher than that of the dyad played by
two separated instruments (TPD) and of the summed
note dyad (SMD). The data also show that SMD and
SPD amplitudes are not significantly different from each
other. Since SMD dyads have no VCTs, the same is
true for TPD dyads. Similar results were also found for
the VCTs from 7 to 12 of the perfect fourth interval
of Figure 1, and for all the other intervals in which it
was possible to measure VCTs (not shown here). These
results confirm in a quantitative way the results showed
by all the figures.
Figure 4 - Mean SEM (n = 4) of spectrum amplitude in corre-
spondence of VCT1(low frequency)-VCT6 (high frequency) for the
perfect fourth and perfect fifth intervals for condition SPD, SMD and
V. MODEL AND DATA SIMULATION
TPD. Ordinate represents the numeric value of the 24 bit range
multiplied by 1.6 105. Because of the VCT presence, peaks in
SPD are is significantly greater than in TPD and SMD, whereas
Although the precise mechanism of combination tone
TPD and SMD are statistically not different.
generation by the violin is unknown, following Helmholtz,
we know that any non-linearity in the sound production,
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transmission or detection, produces a distortion that in principle can generate the combination tones. The summation
of two sinusoidal signals at frequency f1 and f2 in a non-linear system leads to the production of additional
frequencies including 2f1, 2f2, etc. (harmonic distortion) and f2 + f1, f2 - f1, etc. (intermodulation distortion).
In general, the frequencies generated are given by
hf1 + kf2 (with h, k integer numbers different from
zero). Based on this information, we investigated the
response of a simple model in which the occurrence
of combination tones was simulated by adding to the
SMD record (containing no VCTs) the product of the
two recorded waves (i.e. the signal of note 1 multiplied
by the signal of note 2) scaled by a factor z, to introduce
some intermodulation distortion in it. We called this
SSPD. The results of the simulation with z = 1/2
for the perfect fifth interval (C5 - G5), are depicted
in Figure 5.
It can be seen that the simulated traces reproduce
rather well the experimental traces. Evaluation of the
experimental and simulated relative spectra shows that
Figure 5 - FFT spectra for the interval of perfect fifth C5 - G5.
their difference in any point of the graph is comparable
Simulated Single Player Dyad (SSPD) represents the simulated
to the noise, demonstrating the quality of this simple
response, traces a and b represent the experimental and the simu-
lated relative spectra, 20log(SPD/SMD) and 20log(SSPD/SMD),
model. More complex intermodulation distortion models,
respectively. Note the good similarity between experimental and
including more harmonics and more orders, did not add
simulated traces. Asterisks on SSPD trace, indicate VCTs peaks.
more useful information to the analysis. The combina-
Traces SPD, SSPD, originally superimposed to SPD trace, a and
b both originally at zero dB, were shifted upwards for clarity.
tion tones generated by the model with a periodicity of
f2 - f1 , encompasses fundamentals and harmonic
tones in accordance with the theory which predicts the presence of VCTs in coincidence with the primary and partial
tone peaks. This seems in contrast with the relative spectra in Figure 5 that shows no sign of coincident VCTs. This
contrast however, is only apparent as the addition, for example, of the VCT at C4 in Figure 5 to the fundamental at
C5 would produce an increment of less than 1dB, well below the noise floor.
The possible effects of the coinci-
Perfect fifth
Perfect fourth
dent VCTs cannot be determined
SMD-SPD
TPD-SPD
TPD-SMD
SMD-SPD
TPD-SPD
TPD-SMD
when they are analyzed on the sim-
ple record SPD, as done by Lohri [7].
VCT1
0.0018
0.0025
0.97
0.00071
0.00008
0.9
VCT2
0.00025
0.0045
0.26
0.0036
0.0033
0.98
This is because coincident VCTs are
VCT3
0.00098
0.0012
0.99
0.025
0.034
0.65
superimposed on the peaks of funda-
VCT4
0.00033
0.0019
0.38
0.017
0.015
0.99
mentals or partials whose amplitudes
VCT5
0.0011
0.0021
0.89
0.000016
0.000063
0.95
VCT6
0.00007
0.000069
0.99
0.00012
0.00018
0.94
are unknown. Unlike that used by
Lohri [7], our method of recording
Table 1 - p values computed with the post-hoc Tukey test for the comparison between
the SMD dyad in addition to SPD
spectrum amplitudes of the SPD, SMD and TPD traces in coincidence with combination
tones VCT1 (lower frequency) to VCT6 (higher frequency) for the perfect fifth and perfect
allows us, in principle, to measure
fourth intervals. Asterisks indicates statistical significance (p < 0.05).
VCTs easily. In fact, the subtraction
between SPD and SMD records, as
shown in Figure 5, allows us to isolate all the VCTs including the coincident ones, which can then be analyzed.
It should be pointed out, however that to obtain the necessary precision, the amplitudes and frequencies of the
notes in SPD and SMD, records, played necessarily at different times, needs to be the same within less than 1%.
This precision cannot be obtained during normal violin playing but it can be obtained with a bowing machine. As a
possible easier alternative the amplitude of the VCTs coincident with the partials could be calculated with the model
simulation shown in Figure 5.
Listening experiments were conducted by Lohri [7] to ascertain if a difference was perceived between records before
and after the elimination (by a computer program) of the VCTs present. The results were somewhat unsatisfactory as
the difference due to the VCTs presence was not perceived by all the listeners involved. The drawback of this method
is that VCTs coincident with fundamentals and partials obviously cannot be eliminated so reducing the overall effect
of VCTs. It seems reasonable to expect more convincing results by comparing SPD record, containing all the VCTs
including the coincident ones, with TPD which contains no VCTs at all. Future experiments will be planned to verify
this point.
As regarding the nature of the non-linearity inducing the intermodulation distortion, we can hypothesize many reasons
for it, like bridge movement and violin top movement, to mention just two of them. Instead, the coupling of strings
through the bow, seems to be excluded by the results of [7] and by our preliminary experiments (not reported here)
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showing the presence of VCTs even when the dyad is played by stimulating the violin strings with an electromagnetic
system similar to that described by Gough [10]. Clearly much more work is needed to identify the mechanism of the
VCTs production.
Although not investigated in this paper, the variability of VCTs between different violins and players suggest the
interesting possibility, which we have planned to investigate, that VCTs might contribute to violin sound quality.
VI. CONCLUSIONS
When a dyad is played on a violin, or when two notes are played separately on two violins, Combination Tones
can be heard that are harmonics of the frequency difference of the two notes. This has already been reported by
Tartini. Our research shows that when a dyad is played on a single violin, the Combination tones are also created
by the instrument and can be found by FFT of the sound. When the dyad is played on separate instruments, the
Combination Tones can also be heard, as they are created in our cochlea, but are not recorded by the microphone
and do not show up in the FFT. Our work underwrites the observations of Tartini, but enriches it by demonstrating
that the Combination Tones, as heard by our ears, are due in part to the non-linearity (creation of Combination
Tones) of our ears as well as to the non-linearity of the instrument. The observation that Tartini did not find any
difference between these two conditions, suggests that violin combination tones in his experiments were too small
to be perceived. Planned future work will be made to investigate to what extent violin combination tones correlate
with subjective sound quality.
ACKNOWLEDGMENTS
The authors wish to thank the violin players, Roberto Cecchetti and Lorenzo Petrizzo for kindly contributing to the
experiments and Peter Griffiths and Gianpaolo Coro for critical reading of the manuscript.
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