Savart Journal

Stradivari violins exhibit formant frequencies

resembling vowels produced by females
HWAN-CHING1 TAI AND DAI-TING CHUNG2
AbstractOver the past two centuries, violins made by Antonio Stradivari (1644-1737) have been more favorably
received by concert violinists and instrument collectors than instruments by any other maker. Some suggest that
Stradivari's success can be attributed to unique tonal characteristics, generally described as brilliance, and this opinion
is still widely expressed by leading violinists today. Others believe that the perceived tonal distinction of Stradivari
violins may be attributed to psychological bias instead of physical differences, influenced by historical reputation and
market evaluation. Furthermore, modern research has yet to clearly identify acoustic differences between Stradivari
violins and other professional quality instruments. Since both violin tones and spoken vowels are perceived through
steady-state spectral features, we hypothesized that voice analysis techniques may help elucidate the tonal properties
of violins. Using linear predictive coding (LPC), a common speech analysis technique, we examined the recorded
scales of four Stradivari violins and ten other professional quality instruments, both old and new. On most violin notes,
there are typically four or five resonance peaks (formants) below 5.5 kHz. Generally, professional quality violins exhibit
formant frequencies (F1-F4) which are equidistant from the formants of male and female voices. But Stradivari violins
tend to produce higher formants which are closer to female voices. Stradivari violins also show greater probabilities to
emulate the formants of bright-sounding front vowels spoken by females, a tendency shared by other violins judged by
concert violinists as having Strad-like tonal characteristics. Our results suggest that, within the sample group being
studied, there are measurable and statistically significant differences between Stradivari violins and other professional
quality violins in terms of formant features. Having higher formants or having formants that resemble female vowels
may be acoustic correlates of the tonal qualities which concert violinists frequently associate with Stradivari violins.

I. INTRODUCTION
Antonio Stradivari (1644-1737) of Cremona, Italy is the most famous violin maker in human history. Over the past
two centuries, more violin virtuosos have preferred to play instruments made by Stradivari than by any other maker,
followed closely only by his neighbor Giuseppe Guarneri "del Ges" (DG) (1698-1744) [1-2]. Leading concert violinists
and instrument collectors usually suggest that they are attracted by the unique tonal qualities of Stradivari and DG
violins, and today a famous specimen by either maker can reach auction prices over $10M USD. Stradivari's tone is
often described as "sweet" or "brilliant" by experts familiar with their sound [1, 3]. However, not everyone agrees that
Stradivari's unparalleled success is primarily due to acoustic factors. Some people suggest that there is no special
acoustic distinction between Stradivari violins (hereafter, Strad violins or Strads) and other high quality violins made by
hundreds of master makers throughout history. They also suggest that subjective tonal evaluation can be easily biased
by psychological factors such as the historical reputation and the exuberant price of Strad violins [4-5].
Many acoustic and physical studies have been conducted to examine if Stradivari violins produce sound differently.
These studies have yet to identify consistent and measurable differences between Stradivari violin and other
professional quality violins (for reviews and discussions, see ref. [5-9]). To reconcile the apparent discrepancy between
the subjective opinion of leading violinists and the lack of objective, confirmatory evidence, some proposed that
Stradivari's uniqueness may originate from "hard-to-define" acoustic properties beyond our analytical capacity, which
are only apparent to our ear-brain system but not to our measuring equipment [10]. However, clear evidence is also
lacking that players or listeners can readily distinguish between Stradivari violins and other master instruments when
played side-by-side. For instance, in a recent blind test, a panel of violinists failed to make such distinctions [11].
1
2

Hwan-Ching Tai. Department of Chemistry, National Taiwan University, Taipei, Taiwan.

Dai-Ting Chung, Chi Mei Museum, Tainan, Taiwan.

Manuscript received April, 2012.

Article published: June 14, 2012

url: http://SavartJournal.org/index.php/sj/article/view/16/pdf

Savart Journal

In this study, we investigated the tonal properties of violins by analyzing their resonance peaks (formants) during
actual playing. Generally speaking, the tone quality (or timbre) of a musical note is a set of properties that help
distinguish different types of sound production, independent of pitch and loudness. The tone quality of violins is most
clearly perceived by listeners in the sustained part of a note, suggesting a strong association with steady-state spectral
cues instead of transient ones [5-6]. The analogy in human speech perception would be that vowels are determined by
steady-state cues, while consonants are determined by transient cues [12]. The most important spectral features of
vowel sounds are the resonance frequencies (formants) of the vocal tract. Listeners rely on formant frequencies to
determine both vowel identity and speaker gender. The steady-state spectra of violins also display characteristic
formants similar to those of human voices [13-14]. Therefore, we investigated whether formant analysis techniques
commonly employed in voice research may be useful for studying violin tonal quality.
Through formant analysis based on the linear predictive coding (LPC) algorithm [15-17], we found statistically
significant differences between Strad violins and other professional quality violins in terms of formant properties. LPC
analysis revealed that Strad violins generally produce higher formants. The first four formants of Stradivari violins
coincide rather closely with those of female voices. Other violins generally exhibit formant frequencies centrally located
between those of male and female voices. Moreover, Strad violins show greater tendencies to emulate the formants of
bright sounding front vowels spoken by females. Our data imply that there may be a physical difference that underlies
the perceived tonal distinction of Stradivari violins, and that there may be a correlation between higher formant
frequencies and the brilliant qualities of Stradivari violins often described by concert violinists.

II. EXPERIMENT
A. Background considerations about tone analysis
The steady-state spectra of both the violin and human
speech can be explained by a source-filter model (Fig.
1A) [18-20]. In both, the vibrating source (the string or the
vocal cord) produces a harmonic series that decreases in
amplitude with rising frequency. The filter (the violin body
or the vocal tract) then attenuates certain frequency
bands, generating a pattern of peaks and valleys called
the spectral envelope. The tone quality of the violin note
and the identity of the spoken vowel are primarily
determined by the spectral envelope shape, which in turn
is largely controlled by the frequency response function
of the filter [5, 18, 21]. These resonance peaks in the
spectral envelope are called formants (F1, F2, F3, F4,
and so on).
The first four formants of the human voice convey
important information about speaker gender, vowel
identity, and vowel quality [12, 22]. Formants are
basically resonance frequencies of the vocal tract at
Figure 1. (A) A source-filter model for the human voice and
the violin. The vibrating source produces a harmonic series
which standing waves are formed. Therefore, the shorter
(multiples of F0 in frequency), the amplitudes of which are
vocal tracts of females compared to males lead to shorter
modulated by filter resonance. Resonance peaks in the output
standing waves and higher formant frequencies [23]. In
spectral envelope are called formants (F1-F4). (B) An
speech, listeners use both fundamental frequencies (F0)
example of the fast Fourier transform (FFT) and LPC spectra
and formants to determine speaker gender [22], but
of a violin note, and its formant frequencies calculated by
LPC.
during singing F0 is matched to the musical pitch and
hence formants become the primary gender cue [24-25].
On the other hand, vowel identity is primarily differentiated by F1 and F2 values, which in turn are determined by
different tongue positions and mouth shapes [26].

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Intriguingly, it has been observed that people can consciously match different violin notes with different vowels when
instructed to do so, and the chosen vowel may vary from one semitone to the next on the same violin [14]. The
perceptual capacity to match violin notes to vowels has also been demonstrated in children [27]. Therefore, the brain
may have the capacity to associate the spectral features of violin notes with vowel formants. This also implies that
vowel formants and violin formants may be analyzed similarly. One of the most commonly used computational
approaches for vowel formant analysis is LPC [16, 22], and recently some researchers have also applied LPC to
analyze string instruments [28-29]. In this study, we applied LPC formant analysis to study the spectral envelope of
violins, focusing on steady-state regions of long, sustained notes without vibrato.
B. Violin recording
We recorded the scales of 14 violins, including four Strads (Table 1), during a one-day session at the recital hall of
Chi Mei Museum (Tainan, Taiwan). Except for the Altavilla and the Nagyvary, all violins were generously loaned by the
Chi Mei Museum. The instruments were set up by D.T.C. and tuned to A4 441 Hz. A Zoom Q3 HD digital recorder
(Tokyo, Japan), equipped with two small cardioid condenser microphones in X-Y configuration, was placed 180 cm
above the stage floor to make uncompressed 24 bit/48 kHz stereo recordings. Chu-Hsuan Feng, a professional violinist
who graduated from the Paris Conservatory, used a French bow by Joseph Henry to play the C major scale (G3-C7) on
each violin, once at a distance of 60 cm from the microphone (measured from the bridge), and once at 120 cm. Each
note in the scale was played twice consecutively with down bows (1.5-2.0 s) at forte loudness without vibrato. The
violinist was informed about the maker before recording each instrument, and was asked to maintain a consistent
bowing style while playing different violins.
Average value of all notes

Group 1
"Strad"

Maker

Year

Name

F1
(Hz)

F2

F3

F4

ETL
(cm)

Dm

Df

Antonio Stradivari

1707

Dushkin

562

1557

2795

3818

16.13

0.122

0.098

Antonio Stradivari

1709

Viotti-Marie Hall

533

1648

2748

4042

16.02

0.152

0.079

Antonio Stradivari

1713

Wirth

512

1701

2822

3888

15.94

0.124

0.104

Antonio Stradivari

1722

Elman-Joachim

580

1731

2803

3809

15.61

0.144

0.098

547

1659

2792

3889

15.92

0.135

0.095

Giuseppe Guarneri "del Ges"

1733

Lafont-Siskovsky

511

1562

2604

3713

16.83

0.111

0.097

Giuseppe Guarneri "del Ges"

1744

Ole Bull

532

1638

2718

3798

16.53

0.138

0.112

Ansaldo Poggi

1974

564

1734

2729

3859

15.83

0.140

0.114

535

1645

2684

3790

16.40

0.130

0.108

543

1515

2528

3739

16.95

0.109

0.102

446

1598

2689

3705

17.13

0.102

0.110

493

1481

2725

3910

17.13

0.136

0.123

484

1581

2842

3696

16.85

0.138

0.128

491

1544

2696

3763

17.02

0.121

0.116

Group 1 average
Group 2
"Strad-like"

Group 2 average

Group 3
"Old"

Gasparo da Sal

1560

Andrea Amati

1570

Nicol Amati

1624

Giuseppe Guarneri "filius Andrea"

1706

Group 3 average
Group 4
"Modern"

Ross

Paganini

Jean-Baptiste Vuillaume

1860

508

1588

2544

3704

16.82

0.088

0.110

Armando Altavilla

1922

477

1757

2535

3848

16.58

0.089

0.132

Joseph Nagyvary & Guang-Yue Chen

2006

504

1493

2516

3680

17.25

0.114

0.108

Group 4 average

497

1613

2532

3744

16.88

0.097

0.117

Average of 14 violins

518

1613

2685

3801

16.49

0.122

0.108

Table 1. Recording and analysis of 14 professional-quality violins. Calculations of formants (F1-F4), equivalent tube length
(ETL), and distances to male and female vowels (Dm and Df) are explained in the main text.

C. Pre-categorization of violins
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As the curator of the Chi Mei collection of over 800 master violins,
D.T.C. has interacted with many concert violinists who have borrowed
and played the instruments being studied. To see if subjective
evaluations by musicians may correlate with objective, measurable
acoustic attributes, we separated the non-Strad violins into three
categories before the recording and analysis. The categorization
criterion was simple: if a violin had been considered by many concert
violinists as having Strad-like tonal characteristics, it belonged to
Group 2 [Strad-like]. Violins considered to lack Strad-like tonal
characteristics were further classified by age: those over 300 years old
belonged to Group 3 [old], and those made after 1850 belonged to
Group 4 [modern].
Concert violinists often express the opinion that the tonal character of
top-tier Stradivari violins consists of two major factors: a generally aged
sound and a unique quality often described as brilliance. While most
violinists agree that well-made and well-preserved violins would
generally acquire an aged sound over two or three centuries of playing,
there is much debate about the nature and the existence of the
brilliance factor. Furthermore, neither factor has been successfully
characterized and explained through modern acoustic research.

Figure 2. Genealogy of famous violin-making

families in Cremona, Italy. The names in bold
are included in this study.

The four Strads included in this study are generally considered by concert violinists as fine examples of
well-preserved Strads with brilliant and beautiful tones, and, as a matter of fact, all of them were formerly owned by
world-class soloists. Also, according to many concert violinists, it is very rare to encounter other violins which possess
Strad-like characteristics, and those rare exceptions are mostly made by DG, the famed neighbor of Stradivari. Group
2 [Strad-like] violins in this study included two historically renowned DG instruments and a 1974 modern Italian violin by
Ansaldo Poggi, which, according to some concert violinists, apparently lacked the aged sound of antique instruments
but exhibited exemplary tonal brilliance reminiscent of the Strads.
Group 3 [old] consisted of four antique Italian violins that predated the Strads studied. These included the works of
Giuseppe Guarneri "filius Andrea" (DG's father), Nicol Amati (the teacher of Stradivari and DG's grandfather), and two
founding pioneers of Italian violin making, Andrea Amati of Cremona and Gasparo da Sal of Brescia (see Fig. 2).
Although the tonal qualities of Group 3 violins are also highly appreciated by concert violinists, they are not considered
to possess the brilliant characteristics of Strads. Group 4 [modern] included a violin made by J. B. Vuillaume, one of the
most famous copyists of Stradivari in the 19th century, and two professional quality violins made in the 20th and 21st
century.
D. Choosing an LPC method for violins
In speech research, LPC algorithms have been refined over decades to produce reliable formant identification.
These optimizations are based on known features of human speech and tested against speech transcripts annotated
by listeners. To our knowledge, no study has yet optimized LPC algorithms for violins. Hence, we took three popular
speech analysis programs (Praat, SFS and WaveSurfer) and empirically tested their performances on violin
recordings, trying different combinations of built-in analysis parameters.
After testing these three software programs, we found that Praat's LPC analysis (based on Burg's maximum entropy
method [30]) with the appropriate parameters produced the most consistent results with violin spectra. We judged the
quality of LPC prediction by the stability of formants (least shift within the same note) and by visually comparing LPC
formant predictions to FFT spectra. It turned out that default LPC parameters for analyzing female voices in Praat (5
formants under 5.5 kHz) also worked best for violin analysis. Because the length of recorded violin notes was >1500
ms, the analysis window was increased from 25 ms for speech to 150 ms for violins for better frequency accuracy.
Pre-emphasis was not applied to violins because they have considerably less high-frequency roll-off than human

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voices. When analyzed using these empirically determined LPC parameters, each violin note typically showed four or
five stable formants (Fig. 1B). In this study, we focused on the analysis of the first four formants.
E . Formant analysis by LPC
LPC analysis was performed using Praat Software (version 5.2.28) [15], which implemented the maximum entropy
spectra proposed by Burg [30]. We used the following LPC parameters: maximum formant frequency=5500 Hz,
maximum formants=5, time window=0.15 s, dynamic range=70 dB, and no pre-emphasis. The analysis was performed
on the stereo recording, without separating left and right channels. Each pair of repeated notes was analyzed using
Praat's spectrogram display, with formants labeled by colored dots to help visualization. We tried to identify a region
within each note pair with at least four formants that showed the greatest stability. The LPC analysis window was then
placed at the center of that steady-state region. Generally, a violin note showed four to five formants, although some
regions might exhibit only three formants or unstable, shifting formants. Hence, it was advantageous to play each pitch
twice to ensure that a region with four or five steady-state formants was available for analysis. For each violin, 14 LPC
spectra were extracted (G3, A3, B3, C4, E4, F4, and G4, each recorded at two distances, omitting D4 due to technical
errors during recording). Publicly available anechoic recordings [31] of a violin and a Bb trumpet (G3-G4 at ff and mf
loudness) were downloaded from the University of Iowa website and analyzed similarly. Statistical analyses
(Mann-Whitney test, Kruskal-Wallis test, and Fisher's exact test) were performed with GraphPad Prism 5 software (La
Jolla, CA).

III. RESULTS AND DISCUSSION

A. Strad violins exhibit higher formants
Empirically, we observed that Praat's LPC formant analysis is best suited for analyzing the spectral envelope when
F0 is smaller than 400 Hz. If F0 is too high, the harmonic partials are too far apart to reflect the peaks and valleys in the
spectral envelope, and formants are much harder to define. For this same reason, when people sing a note above 400
Hz, vowel intelligibility starts to decrease drastically with rising pitch as formants become harder to recognize, and also
because F0 starts to exceed F1 [32]. Thus, from a psychoacoustic perspective, it is also less meaningful to consider
formants when F0 is too high and the harmonic gap is too large.

Figure 3. Stradivari violins have higher formant frequencies compared to other violins. Four Stradivari violins were
compared against 10 other violins (Group 2+3+4) in (A) and against only Group 3+4 violins in (B). Statistical comparisons
were made by Mann-Whitney test (*p<0.05, **p<0.01). The box represents 25%, median, and 75% values, and the whiskers
represent 5% and 95% values.

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In this study, we applied LPC formant analysis to the lowest

octave (G3-G4, F0=196-392 Hz) of the recorded violin scale.
Even at the lowest notes, a violin can still produce clear
formants above 4 kHz, which is an octave above the highest
playable note (for example, see Fig. 1B). Hence, the spectral
envelope of the lowest notes still provides much information
about the full-range response of the violin. The radiativity
profile of the violin drops rather quickly above 4 kHz [33],
although the useful output range may extend to 7-8 kHz.
Typically, we found that each violin note exhibited 4-5
formants below 5.5 kHz, and we focused on understanding
the first four formants. The average F1-F4 values of the 14
recorded violins are listed in Table 1. Comparing Strad violins
against all other violins (Group 2+3+4), we found that Strads
tended to produce higher formant frequencies (Fig. 3A). The
differences in F1, F3, and F4 were statistically significant
(p<0.05), and there was also a trend toward significance in F2
(p=0.090). When we excluded Group 2 violins from the
comparison, which were specially selected for their Strad-like
tonal characteristics, the statistical difference between Strads
and other violins (Group 3+4, old and modern) became even
more pronounced. In this case, Strads exhibited significantly
higher frequencies in all four formants (Fig. 3B).

Figure 4. Formant frequencies of different violin groups.

Statistical comparisons were made by Kruskal-Wallis
one-way ANOVA, followed by Dunnett's post test to
compare all pairs (*p<0.05, **p<0.01, ***p<0.001). The
box represents 25%, median, and 75% values, and
whiskers represent 5% and 95% values.

Comparing each group separately, we observed that Strad

violins had the highest averages in F1 through F4, and that
Group 2 [Strad-like] had the second highest averages in F1,
F2, and F4 (Table 1). By one-way ANOVA (Fig. 4), there were
significant differences in F1 (p=0.016) and F3 (p=0.0006)
between the groups, and a trend toward significant differences
in F4 (p=0.074). Strads had higher F1 than Group 3 [old]
(p<0.05), and higher F4 than Group 4 [modern] (p<0.001).
Altogether, our data suggest that the four Stradivari violins
examined generally produced higher formant frequencies,
especially in F1 and F3.
B. Violins and vowels share several formant features
Recent studies by Bissinger have shown that, in addition to
the cavity (air) mode around 280 Hz, the radiativity profile of
the violin is dominated by three corpus modes: the corpus
bending mode around 500 Hz, the BH (originally called
"bridge-hill") mode around 2.4 kHz, and the bridge-filter mode
around 3.5 kHz [33]. These vibration modes were measured
with an impact hammer striking the bridge [34], which was
rather different from our study in which violins were naturally
bowed and recorded in a concert hall. Nevertheless, we still
expected to see LPC-estimated formants to closely relate to
the major corpus modes measured by impact response.
The F1-F4 averages of our 14 violins were 518, 1613,
2685, and 3801 Hz, respectively (Table 1), and therefore F1,
F3, and F4 appeared closely related to the dominant corpus

Article published: June 14, 2012

Figure 5. (A) 10 basic female and male vowels shown in

the F1-F2 formant space, based on published
measurements [21]. (B) Violin notes (G3-G4, 196-392
Hz) have greater F1 and F2 variations than
corresponding trumpet notes. Recordings taken from a
publicly available instrument sound library [30].

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modes mentioned above. The major resonance frequencies are slightly higher in our recordings than Bissinger's force
hammer measurements, which may be due to complex experimental factors such as the act of bowing, the violin hold,
microphone placement, and room acoustics. This also validates that our LPC parameters were chosen properly. We
can tentatively assign F1 to the corpus bending mode (B1- and B1+), F3 to the BH mode, and F4 to the bridge-filter
mode. Upon closer examination, there is also a minor peak in the violin corpus mobility curve around 1625 Hz [33],
which may potentially relate to violin F2.
Interestingly, the major resonance modes of the violin corpus (0.5, 2.4, 3.5 kHz) follow a 1:5:7 ratio starting at 0.5
kHz, while male voice formants (511, 1411, 2370, 3428 Hz, Table 2) also follow a 1:3:5:7 ratio starting at 0.5 kHz. This
may explain why LPC analysis commonly used in voice research can be successfully applied to violin spectra. The
average F1-F4 of our 14 violins (518, 1613, 2685, 3801 Hz) fell squarely between the formants of the male voice (511,
1411, 2370, 3428 Hz) and the formants of the female voice (619, 1686, 2842, 4053 Hz). Whether by coincidence or by
design, the strongest resonance modes of the violin above 400 Hz can emulate the formants of the human vocal tract.
Moreover, violins and voices both exhibit large variations in formant frequencies, especially in F1 and F2. In human
speech, variations in F1 and F2 are caused by variable tongue positions and mouth shapes associated with different
vowels, and each vowel occupies a distinct region in the F1-F2 plane (Fig. 5A). Unlike the voice organ, there are no
moving parts in the violin, and therefore F1 and F2 variability are possibly due to the existence of relatively sharp
resonance peaks, which would only be excited strongly if a harmonic partial is very close to their resonance frequencies.
To rule out the possibility that F1 and F2 variability is a computational artifact due to applying LPC to our violin
recordings, we also tested digital recordings from other sources. Every violin we have analyzed consistently displayed
large variations in F1 and F2 on successive notes, while some other instruments such as the trumpet displayed very
stable F1 and F2 (Fig. 5B). The much larger F1 and F2 variance of the violin compared to the trumpet (p<0.0001 by
F-test) are apparently related to the inherent properties of the instrument, not computational artifacts.
Interestingly, the formant frequencies of violin notes show no correlations with pitch (F0), which is again analogous
to human voice. Raising the pitch in speech or singing has little effect on formant frequencies, except in some highly
trained singing styles or when the musical pitch (F0) approaches or exceeds F1 in normal speech [25, 35]. The fact that
both violin notes and vowels show large F1 and F2 variations in similar frequency ranges may explain why people
frequently hear different corresponding vowels in adjacent semitones played on the same violin [14].

Vowel

IY

AE

OW

OO

UH

ER

Example

beet

bit

bet

bat

Bob

bought

book

boot

but

Bert

back

central

central
openmid

Avg

IPA

Backness

front

front

front

front

back

back

back

close

nearclose

openmid

nearopen

nearopen

openmid

nearclose

close

openmid

131
302
2172
2851
3572
18.07

130
438
1837
2482
3533
17.01

124
541
1690
2456
3511
16.45

122
645
1621
2357
3463
16.22

120
673
1097
2457
3463
17.80

119
614
990
2465
3408
18.79

125
486
1168
2307
3359
19.09

129
341
1067
2219
3342
21.70

120
590
1194
2401
3423
17.92

121
477
1276
1707
3201
20.56

124
511
1411
2370
3428
18.36

231
378
2586
3286
4127
15.10

227
512
2196
2995
4265
14.25

219
661
2013
2955
4219
13.66

215
841
1932
2981
4146
13.13

213
837
1245
2945
3957
15.20

216
745
1190
2853
3922
15.91

220
522
1386
2791
3976
16.41

222
409
1361
2729
3976
17.72

215
723
1445
2862
4052
14.91

217
558
1503
2024
3888
17.33

220
619
1686
2842
4053
15.36

Height
Male
F0 (Hz)
F1
F2
F3
F4
ETL (cm)
Female
F0 (Hz)
F1
F2
F3
F4
ETL (cm)

Table 2. Formant data of 10 basic vowels in General American English spoken by males and females. Fundamental (F0) and
formant (F1-F4) frequencies are compiled from published data in ref. [21]. Average formant values and equivalent tube lengths
(ETL) were calculated in this study. IPA: international phonetic alphabet.

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Figure 6. Mean formant values of individual violins plotted in the F1-F2 plane and the F3-F4 plane, compared with male
and female means. Stradivari violins (red) tend to be clustered near female means.

C .Strad violins exhibit formants similar to the female voice

Since professional quality violins generally have formant frequencies similar to those of human vowels, and since
people can often match violin notes to different vowels, it seems likely that the proximity of a violin note to a vowel or a
voice type in the formant space may affect its perception. When we compared the average formants of individual violins
to those of female and male voices, we noticed that all four Stradivari violins displayed a proximity to the female voice
both in the F1-F2 plane and the F3-F4 plane (Fig. 6). The only other violin to show a comparable tendency was the
1974 Poggi violin, which was pre-selected into Group 2 for its supposed tonal resemblance to Strad violins.
Because voice formants arise from standing waves in a tube with an open end (the vocal tract), they are in fact not
independent variables but jointly affected by vocal tract length [23]. A simple estimation of vocal tract length from
formant frequencies is given by this formula (c is the speed of sound at 344 m/s, ETL is equivalent tube length):

4,

(1)

This equates to 15.36 cm for females and 18.36 cm for males

(average of 10 vowels, Table 2). While the actual vocal tract is not a
straight tube and appears slightly shorter than ETL estimates [36], it
has been shown that relative ETL differences are excellent predictors
in voice gender differentiation [37]. Thus, it is very likely that voice
formants are not just perceived as independent and continuous
variables in our brain, but also as conjoined attributes that fall into
different gender-vowel categories. Since good quality violins also
display vowel-like formants, it seems reasonable to apply ETL
calculations to violins as a method to jointly analyze all four formants.
Compared to analyzing each formant individually, the differences
Figure 7. Stradivari violins have lower ETL
between Strads and other violin groups became even more
values than old (Group 3) and modern (Group
pronounced in ETL analysis (Fig. 7). By one-way ANOVA, the
4) violins. Red lines represent average male
difference between groups was highly significant (p=0.0043). The
and female ETL values. Statistical significance
was determined by Kruskal-Wallis one-way
Strads had the lowest mean ETL, followed by Group 2 [Strad-like]. By
ANOVA, followed by Dunnett's post test to
Dunnett's post-test, we found that the average ETL of Strads
compare all pairs (*p<0.05, **p<0.01). The box
(15.920.22 cm, meanSEM) was significantly smaller than those of
represents 25%, median, and 75% values, and
Group 3 [old] (17.020.22 cm, p<0.01) and Group 4 [modern]
whiskers represent 5% and 95% values.
(16.880.21 cm, p<0.05). The Strad average was also much closer to
the female vowel average (15.36 cm) than the male vowel average
(18.36 cm). Since instruments in Group 3 are older than the Strads tested, the higher formants and lower ETL of the
latter cannot be simply be attributed to aging. In fact, the Poggi violin from 1974 has the second lowest ETL despite its
young age, suggesting that it is possible to produce a modern instrument with very high formants.
Article published: June 14, 2012

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Savart Journal

D. Strad violins emulate the formants of female front vowels

Since there is considerable overlap between male and female vowels in the four-dimensional formant space, we
decided to determine the mapping relationship between individual notes and individual vowels by measuring
normalized Euclidean distances. First, we tried to determine the distance, called Dm, of each violin note to its closest
male vowel in the 4D formant space, which is given by this formula:
2
2
2
2
F

F4
F3
F2
1
1 / 4
1 +
1 +
Dm =
1 +
F1m F2 m F3m F4 m

(2)

Computationally, we tested the F1m-F4m values of all ten male vowels in Table 2 to see which vowel produced the
smallest distance and called that distance Dm. Similarly, we also calculated the distance to the closest female vowel
and called it Df, and hence each note has a Dm value and a Df value. Among the four groups, Strad violins exhibited the
highest mean Dm and the lowest mean Df, followed by Group 2 [Strad-like] (Table 1). By one-way ANOVA, the
difference between groups was significant for Dm (p=0.0151, Fig 8A), and highly significant for Df (p=0.0007, Fig. 8B).
Df was significantly higher for Strads versus Group 3 [old] (p<0.01) and Group 4 [modern] (p<0.01). Dm was also higher
for Strads versus Group 4 [modern] (p<0.05). When we plotted average Df vs. Dm for each individual violin in Fig. 8C,
we noticed that Strads and Strad-like violins were located in the lower right region of the graph, characterized by more
feminine and less masculine vowel characters.
While calculating Dm and Df, we also identified
the closet vowel to each note in the 4D formant
space (considering both male and female vowels
together). Then we noticed that there is a much
higher probability for Strad and Strad-like violins
to project closely to female front vowels (Table 3).
Front vowels (including [I], [E], [AE], and [IY]) are
vocalized by placing the tongue forward, which
raises F2-F4. Because of their higher formants,
female front vowels are the brightest sounding
vowel group. Although different languages may
have somewhat different vowels, basic
categorizations such as front vs. back vowels are
universal in all languages because they are
determined by tongue and mouth movements.
The vowels most frequently emulated by Strad
violins are female front vowels [I], [E], and [AE]
(41%), followed by female back vowels [OO] and
[U] (20%). By contrast, about 36% of Strad notes
mapped to the ten male vowels combined.
Overall, the probability to map to a female front
vowel was higher for Strads vs. Group 3 [old]
(p<0.01, Fisher's exact test) and for Strads vs.
Group 4 [modern] (p<0.01). It was also higher for
Group 2 vs. 3 (p<0.01) and for Group 2 vs. 4
(p<0.05). Taken together, the tendency of a violin
to project notes that map closely to female vowels
(smaller Df and larger Dm), especially the female
front vowels, seem to correlate well with their
perceived tonal brilliance.

Article published: June 14, 2012

Figure 8. Stradivari violins are characterized by larger distances to

male vowels (Dm) and smaller distances to female vowels (Df), shown
in (A) and (B), respectively. The box represents 25%, median, and
75% values, and the whiskers represent 5% and 95% values
(Kruskal-Wallis one-way ANOVA, followed by Dunnett's post test to
compare all pairs, *p<0.05, **p<0.01). In (C), mean Dm vs. mean Df is
plotted for 14 violins. Stradivari (red) and Strad-like violins (orange)
tend to cluster in the lower right region.

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Savart Journal
E. Strad and DG violins project to
unique regions in the formant space
For over 200 years, the majority of
violin virtuosos have preferred to play
either Stradivari or DG instruments. We
therefore investigated if these violins could
produce unique sounds inaccessible by
other violins. By plotting all recorded notes
from the 14 violins in three formant planes
(F1-F2, F2-F3, and F3-F4), we searched
for regions preferentially occupied by Strad
and DG instruments. There turned out to
be several such "hot zones" in the formant
planes examined (Fig. 9). Over 75% of the
notes in these projection zones belonged
to the four Strads and two DGs, with each
instrument contributing at least 8% (the top
three were all Strads). The Poggi also
contributed to 7% of the notes and, by
contrast, the four violins in Group 3 [old]
contributed just 16%. Strikingly, only 1% of
the notes in these zones belonged to
Group 4 [modern] violins. Upon closer
examination,
these
zones
were
characterized by higher F2-F4 values, and
corresponded closely to female front
vowels [I], [E], [AE], and [IY]. This supports
our earlier observation that Strad and
Strad-like violins are distinguishable by
their tendencies to emulate female front
vowels, which happen to be the brightest
vowel group due to higher formant
frequencies.
F. Possible correlations with perceived
tonal quality

10

Violins

Female front vowels

Other vowels

Group 1

23 (41%)

33 (59%)

Group 2

16 (38%)

26 (62%)

Group 3

8 (14%)

48 (86%)

Group 4

5 (12%)

37 (88%)

Table 3. The number of notes in each violin group that map to different
vowel categories in the 4D formant space.

Figure 9. Preferential projection zones of Strad and DG violins in vowel

formant planes. Grey ellipsoids represent regions predominantly occupied by
notes from Strad and DG violins.

The conventional approach to studying

the resonance properties of violins is to
measure the response curve by averaging many notes or by physically exciting the instrument without the player.
However, this is not how we actually listen to violins. We do not perceive violin tone through its long-time average or the
hammer response, but through the harmonic partials of individual notes during very short time intervals. In this study we
demonstrated the feasibility of analyzing violin resonance through the LPC spectra of individual notes recorded during
normal playing. Our approach can examine the natural sound of the violin without the use of special equipment, just
studio microphones. Compared to deriving one response curve per instrument, our method provides much greater
statistical power by considering each note individually.
In this study, we pre-categorized the violins according to age as well as tone qualities subjectively described by
concert violinists. After recording and analysis, we found that formant features (F1-F4, ETL, Dm, and Df) appeared to
correlate better with perceived tonal brilliance than with age. The only potential age-related difference was observed in
F3, where old violins seemed to display higher values (Fig. 4). But it is unclear if this is a coincidence or if it can be
generalized to a larger set of instruments.

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In human voice perception, higher formants can lead to increased brightness. Voices with higher formants (women
and children) are brighter than voices with lower formants (men); vowels with higher formants like front vowels also
sound brighter than back vowels [38-39]. Since good quality violins display vowel-like formants, and since people can
hear vowel-like characters in violin notes [14, 27], it is plausible that higher violin formants would also lead to perceived
brightness. Our data suggest that having higher formants may be an acoustic correlate of what concert violinists
perceive as Strad-like tone qualities, but it will require further psychoacoustic experiments to determine if there is an
underlying causal relationship.
G. Comparison with previous studies
Another potential explanation for the brilliant tone of Strad violins is that they simply produce more high frequency
energy than other violins, which can be measured from the response curve of the instrument. Unfortunately, in this
study we did not have sufficient recording samples to derive frequency response curves from long-time average
spectra. Nevertheless, comprehensive studies by Dnnwald have already demonstrated that Strads do not have
greater output above 4.2 kHz [40-41]. Dnnwald also thought that the favorable tone of old Cremonese violins may be
associated with increased output in the mid-range (1300-4200 Hz) [41], while Meinel thought it was favorable to have
stronger responses around 2-3 kHz [42]. In a recent study, Anders Buen measured the response curves of 15
Stradivari and 18 modern violins [43]. Based on the curves he published, we made our own calculations to find that
Strad violins exhibited stronger resonance around 3073 Hz compared to modern instruments. Studies in electronic
violins also suggested that increasing the output in a broad band centered around 3 kHz improved tone quality [44].
Some researchers suggest that having stronger output around 2.5-3 kHz on a violin may impart an advantage that is
related to a phenomenon called "singer's formant" [6, 8, 44]. It has been shown that classically trained singers have the
ability to merge F3 and F4 (and perhaps even F5) to create a new formant that is stronger and higher than the original
F3. Because human hearing is most sensitive around 3-4 kHz, developing singer's formant helps the audience hear the
opera singer above the orchestra and the chorus [25, 45]. Similarly, having increased output around 2.5-3 kHz may
also contribute to better projection on the soloist's violin (more easily heard above orchestral violins). Among opera
singers, there is an interesting correlation between voice type and the center frequency of singer's formant. The more
brilliant the voice type, the higher the singer's formantaround 2.5 kHz for bass and baritones, 2.8 kHz for tenors, and
3 kHz for altos and sopranos [46].
In this study, we found F1 and F3 to be most significantly different between Stradivari and other violins. F3 may be
related to the BH mode of the bridge-body vibration, and also to singer's formant. If we apply the concept of singer's
formant, then Strad violins (F3=279243 Hz, meanSEM) would be comparable to a tenor, and Group 4 [modern]
(F3=253243 Hz) would be comparable to a baritone. It appears to correlate with the general opinion of concert
violinists that Strad violins sound more brilliant and project better, and it may also imply that Stradivari violins produce
higher BH mode resonances during actual playing.
On the other hand, F1 appears to be related to the corpus bending modes B1- (~475 Hz) and B1+ (~541 Hz) [47]. The
higher F1 of Strads (54717 Hz, meanSEM) compared to Group 3+4 (49412 Hz) may either reflect a frequency shift
in these modes, or a change in the relative amplitude of these two modes. Schleske has observed that B1+ can differ
significantly between different violins, and considers it the most important mode in controlling tonal color. Based on his
observations, if B1+ is below 510 Hz, it may lead to a soft, dark sound; above 550 Hz, it may lead to a bright sound,
+
possibly harshness [48]. If F1 represents B1 mode during actual playing, then our data appears to be consistent with
Schleske's general observation, which implies that higher F1 may also contribute to brighter tone color.
Generally speaking, the vibration modes of violins can only be measured in laboratory settings using mechanical
excitation, such as hitting the bridge with a force hammer. When the violin is held by a violinist in actual playing, it is
difficult to make exact physical measurements. We do not yet understand if vibration modes driven by hammer
excitation can accurately reflect actual violin sound during natural playing. The string, the bow, the hold, and bowing
technique will influence violin vibration and add a lot of variability. There is currently little understanding of what physical
differences may contribute to the higher formants produced by Stradivari violins under natural playing conditions.

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IV. Conclusion
In this study, we found that the LPC algorithm can be successfully applied to analyze the spectral envelope of violin
notes to identify broad resonance peaks, or formants. Analyzing the scale recordings of 14 professional quality violins,
we found that violin formants display several notable similarities to human voice formants: 1) violins and voices typically
display four or five steady-state formants below 5.5 kHz; 2) the first four violin formants have similar frequencies as the
first four voice formants; 3) violins and voices exhibit similarly large formant variations, especially in F1 and F2. There is
much formant variability between different notes on the same violin, as well as between different violins. In human
voices, there is also much formant variability due to vocal tract differences (generally shorter in women than men) and
the vocalization of different vowels.
These similarities may help explain why people can often hear vowel-like qualities in violin notes and are able to
match different violin notes to different vowels [14, 27]. This also implies that formant variations of violin notes may
influence our perception of tone quality. Comparing the formants of different violins, we observed that Stradivari violins
generally produce higher formants (F1-F4), especially in F1 and F3. These differences are statistically significant,
implying that there may be underlying physical differences that distinguish Stradivari violins.
Violin formants F1, F3, and F4 appear to correspond to the dominant vibration modes of the violin, which are the B1
corpus bending mode (including B1- and B1+), the BH mode, and the bridge-filter mode, respectively [33]. Having
higher formant frequencies implies that Stradivari violins can produce higher dominant modes during actual playing, but
the underlying relationship between formants and normal modes remains mostly unclear. There is also little
understanding of how differences in violin construction or material properties may lead to higher formants, and how
higher formants may affect tone perception.
In human voice perception, there is an apparent association between higher formants and brightness, which makes
the female voice brighter than the male voice, and front vowels brighter than back vowels [38-39]. It is very interesting
to note that Strad violins display greater tendencies to emulate the formants of female front vowels, which is the
brightest vowel group. Many concert violinists have subjectively reported that top quality Strad violins are characterized
by brilliant tonal qualities. Hence, there may be a correlation between the formant properties of Stradivari violins and
their perceived tonal qualities.
We also examined four fine examples of antique Italian violins (Group 3), which are older than the Strads tested but
lack Strad-type tonal characteristics (according to concert violinists). Their formants are lower than those of Stradivari
violins, and more comparable to modern professional quality violins. This suggests that aging by itself in well-crafted
Italian instruments is not sufficient to generate the higher formants of Strad violins. On the other hand, concert violinists
also suggest that there are some rare violins not made by Stradivari that possess Strad-like tonal characteristics. When
we analyzed such instruments (Group 2), we found that they produce somewhat higher formants than other
professional quality violins, and, just like the Strads, they have greater tendencies to emulate the formants of female
front vowels. Taken together, the tendency to produce higher formants and to emulate female front vowels appears to
correlate better with Strad-like tonal brilliance described by concert violinists than with instrument age.
To conclude, we have demonstrated that there are measurable, objective differences in the resonance properties of
Stradivari violins compared to other professional quality violins, both old and new, within the group of instruments
selected for this study. It is therefore plausible that the perceived tonal distinction of Stradivari violins frequently
described by concert violinists may originate from actual physical differences, not just psychological bias. Having
higher formants appears to correlate with the tonal qualities attributed to Stradivari violins, generally described as
brilliance. The greater tendency of Strad and DG violins to emulate female front vowels also seems to correlate with
their favorable perception by concert violinists. Our work illustrates a novel analytical approach to violin tone by
focusing on the spectral envelope of individual notes, rather than the average response curve of the instrument, which
can be carried out without special equipment. Further work will be required to extend our new analytical approach to a
larger collection of instruments to see if our preliminary observations can be generalized.

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V. Acknowledgments
We thank the Chi Mei Museum for lending the valuable violins for the recording session, and Chu-Hsuan Feng for the
violin performance. We thank Joseph Nagyvary for recording advice and in-depth discussions, Anders Buen for useful
discussions about his published data, Chien-Hung Tu for recording assistance, and David Chiang for manuscript
comments.

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