PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

On the production mechanisms of the singer’s formant
Bernd J. KRÖGER
Department of Phoniatrics, Pedaudiology and Communication Disorders, RWTH Aachen University, Germany

ABSTRACT
A small extra tube directly located above the glottis (laryngeal tube), which is about six times smaller incross-sectional area than the pharyngeal-oral tube section and which has a length of about 1/6 of the overall
length of the vocal tract tube seems to be responsible for producing a broad (hyper-)formant at about 3 kHz,called the singer’s formant. From an acoustic viewpoint the singer’s formant has been described as aclustering of the third, fourth and fifth formant. Using a computer simulation model for sound production,
which includes a self-oscillating vocal fold model (source model), an acoustic wave propagation andradiation model (tube model or filter model) as well as source-filter interaction mechanisms it is possible toshow that a further mechanism is responsible for increasing the amplitude of partials in the frequency regionof 3 kHz. Our simulations indicate that in the case of normal to high subglottal pressure (normal speaking toloud singing voice) a specific source-filter interaction mechanism is an important contributor for increasing
partials in the region of 2 kHz to 5 kHz.
Keywords: Music, Singing, Voice, Singer’s Formant, Synthesis

1. INTRODUCTION
It is well known that female as well as male opera singers are able to form a singer’s formant (1, 2, 3).
This formant is located at about 3 kHz and is strong in intensity. The singer’s formant leads to a muchsmaller decay of the spectrum between 2 and 5 kHz as in the case of speaking voices or in the case of
singing by amateurs or in case of singing in other singing styles. The huge energy concentration in thefrequency region between 2 kHz and 5 kHz allows the opera singer to appear with a very sustainable voice.
The singer is thus well perceivable without microphone and in case of an accompanying classical orchestra.
From the production perspective the source of the singer’s formant seems to be a small extra tube rightabove the glottis (laryngeal tube, see Sundberg (1)) together with an abrupt change in cross sectional area atthe end of this tube, i.e., at the transition of this laryngeal tube to the pharyngeal cavity. It is assumed thatthe cross sectional area of the laryngeal tube is about 6 times less than the mean cross sectional area of therest of the vocal tract (pharyngeal and mouth cavity), thus forming an extra formant because of its acoustic
mismatch (high change in acoustic impedance) between this small laryngeal tube and the main tube
representing the vocal tract (pharyngx and oral region). Further studies were focused on the acoustic
characteristics of the vocal tract transfer function in the case of opera singers and postulate a clustering of
the third, fourth and fifth formant of the vocal tube as origin of the singer’s formant as well as interaction
effects between F1 and partials of F0 (2, 3).
The experiment described in this paper as well underpins the importance of a small laryngeal cavity
followed by a wide pharyngeal cavity but emphasizes the fact that the increase of formants in the region
between 2 kHz to 5 kHz is mainly related the so called “source-filter interaction” or “glottis-vocal tract
interaction” (4, 5, 6, 7).
Source filter interaction can be differentiated in three subcases, i.e., the case of acoustic-aerodynamic,
mechanic and no interaction. Acoustic interaction reflects the fact that the backward travelling acoustic
waves reflecting the standing waves (formants) influence the glottal flow at the level of the glottal tube.
This interaction mainly causes formant ripple within the opening phase of the glottal pulse (6). A major
aerodynamic interaction effect is that the peak of the glottal flow pulse is shifted to the right side incomparison to the peak of the glottal area pulse shape. This effect results from the inertia of thesubglottal-supraglottal air column and generates a specific flow shape mainly in the closing portion of theglottal cycle, i.e., more abrupt cutting off of glottal flow at the moment of glottal closure within a glottal
cycle. Mechanical interaction mainly occurs in case of an obstruction of airflow in the vocal tract and thus4554

occurs during consonantal full closures of the vocal tract. This decreases the glottal flow and leads to a
reduction up to a stop of vocal fold vibration during the time interval of vocal tract closure.
In this study the importance of the acoustic-aerodynamic source-filter interaction for the generation of asinger’s formant will be demonstrated using an articulatory-acoustic simulation model (8) capable ofsimulating glottal source and vocal tube aerodynamics, acoustics, and all of its interactions as describedabove.

2. METHOD
2.1 Model
The articulatory-acoustic simulation model used here (8, 9) comprises a subglottal, glottal andsupraglottal component (Fig. 1, left). The subglottal system models the lungs and the tracheal tube. Lung or
pulmonary pressure is inserted at the bottom of the tracheal tube and the subglottal pressure results
underneath the glottis. The vocal fold system includes a self-oscillating two-mass model modeling thevibrating part of the glottis (Fig.1, right) and an extra shunt arranged in parallel modeling the cartilagenous
part of the glottal slit (not shown in Fig. 1). The supraglottal part comprises an acoustic model of thepharyngeal, oral and nasal tube including all occurring mechanical, aerodynamic and acoustic loss
mechanisms, including a noise frication generator (e.g. for producing fricatives) and including a radiation
model for sound radiation at the mouth and at the nostrils. Vocal tract shapes can be formed using anarticulatory model as described in Kröger et al. (8).
Three different versions of the model exist: i) direct insertion of a prescribed time-varying glottal flowin the vocal tract tube (see Veldhuis (10), no source-filter interaction, this variant of the model is called
model 1). ii) Direct insertion of a prescribed time-varying glottal area A1 (see Fig. 1; this variant of themodel is called model 2). Here, glottal flow results from the aerodynamic-acoustic laws. Source-filter
interaction (11, 7) occurs as a byproduct in this model because glottal flow is directly part of thesubglottal-glottal-supraglottal vocal tube (see Fig. 1). iii) Self-oscillating glottis model (model 3,physiological model, see Pelorson (12)): the aerodynamic-acoustic part is same as model 2 but in addition
the vibration of the vocal folds and thus glottal area as function of time is calculated from aerodynamic andmechanical forces acting on the two masses (13, 12, 11, 7).

Figure 1 –The articulatory-acoustic simulation model (left side) and the mechanical part of the self-oscillating
glottis model (right side; part of model 3). Left side: u+ and u- represent forward and backward traveling partial
flow waves, ug represents glottal flow, um and un represent air flow at mouth and nostrils, psub represents subglottalpressure and psp represents radiated sound pressure from the mouth and nostrils. GA, CT and PP represent glottal
control parameters glottal aperture, cord tension and pulmonary (or lung) pressure. A 0, A1 , A2, … Amax represent
cross-sectional area of the subglottal tube section, of the glottal tube section, as well as of the supraglottal tube
sections. Noise is inserted if a narrow constriction together with a high airflow occurs in the oral tube (noise
frication generator). Right side: A two-mass model represents the mechanical part of the self-oscillating glottis
model and comprises two damped spring-mass pairs (mi, ri, si, i=1,2) as well as a coupling spring with spring
constant kc. Model 2: A1 is directly controlled by an input time function. Model 1: Glottal area is set to a low value
in comparison to A0 and A2 in order to allow a small backward travelling wave u-1 as it occurs in the open phase ofglottal cycle. Glottal flow is inserted in tube section 1 (glottal tube section) as forward traveling partial wave u +1.

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2.2 Simulations
In this study we simulated combinations of (i) two different lung pressure levels, (ii) two different typesof source-filter interactions (no interaction vs. acoustic-aerodynamic interaction), (iii) different vocal tract
shapes (schwa-sound an /a/-sound) and (iv) occurrence or absence of the narrow laryngeal tube, i.e. wide(or no) and narrow laryngeal tube. Two different values of lung pressure were set, representing a normal
loud voice, here called speaking voice (pulmonary (or lung) pressure PP = 600 Pa) and a loud voice, here
called singing voice (PP = 1200 Pa). Two different types of interactions were modeled in order todifferentiate the cases no interaction vs. full acoustic-aerodynamic source-filter interaction. These different
cases or models are labeled here as “model 1” (directly imposed glottal flow, no source-filter interaction)
and as “model 2” (directly imposed glottal area, source-filter interaction). Two types of vocal tract shapes
were used, i.e. that of a schwa-sound (constant cross sectional shape of vocal tract tube, Fig. 2) and that of a“technical /a/-sound” (see Fig. 2). The technical /a/ comprises two sections or tubes with constant crosssection, i.e. a relatively narrow and a wide tube. The claim for producing an acoustically correct “cardinal/a/” is reached, if both tubes are of comparable length (14, 15). In addition each cross sectional tube wasrealized with a wide (i.e., no) and with a narrow laryngeal tube. These 8 cases used in the simulation
experiment are subsumed in table 1. The self-oscillating glottis model is not used in this simulation study.
Table 1 – Simulation cases for each model (model 1 without source-filter interaction and model 2 including
source-filter interaction)

caselung pressure

vocal tract shapelaryngeal tube

1low (speech)

schwa

wide

2

low (speech)

schwa

narrow3

low (speech)

/a/

wide

4

low (speech)/a/narrow

5high (singing)

schwa

wide

6

high (singing)

schwa

narrow

7high (singing)

/a/

wide

8

high (singing)/a/narrow

Figure 2 – The area function for schwa vowel (vocal tract tube with constant cross-sectional shape of 3 cm2, red)and for the technical /a/-vowel (blue) with wide (top) and with narrow laryngeal extra-tube (bottom). The nasal
tract is not coupled. All vowel productions are non-nasalized.

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Figure 3 shows typical results for simulations using no source-filter interaction (model 1) and fullacoustic-aerodynamic source-filter interaction (model 2). It can be seen that source-filter interaction leads
to a modification of glottal flow, which can be found in a pronounced way in the first time derivative ofglottal flow. This acoustic-aerodynamic interaction causes a formant ripple, i.e., an additional ripple in thetime signal caused by the pressure and flow variations of the vocal tract which result from the vocal tract
resonances (4, 5, 6). We can clearly see that the ripple occurring during the opening phase of the glottal
cycle is in a frequency region of about 1000 Hz to 1500 Hz which covers the region of the second formant
in case of the /a/.

Figure 3 – Time signals for glottal flow ug and its time derivative ug’ in a model without (model 1, top: a) andwith source-filter interaction (model 2, bottom: b) for vowel /a/ and pulmonary pressure PP = 1200 Pa. In case ofmodel 2 the glottal area function is plotted in addition to flow and time derivative of flow. The time axis is 50
msec and thus 5 glottal cycles are given (F0 = 100 Hz).

3. RESULTS
The analysis of the 8 spectra for the 8 simulations displayed in Fig. 4 reveals that in the case of nosource-filter interaction (model 1) a fourth formant occurs in the case of narrow larynx-pharynx tube which
replaces the fourth and fifth formant in the case of wide larynx-pharynx tube and the mean amplitude
between 2 kHz and 5 kHz is a little increased by about 5 dB. This holds for speech and singing in the caseof the simulation of the schwa vowel while no such increase in is observed in the case of simulation of thevowel /a/.
Analyzing the 8 spectra for the 8 simulations displayed in Fig. 5 we can see that in this case of
acoustic-aerodynamic source-filter interaction (model 2) a strong increase in amplitude occurs within thefrequency region of 2 kHz to 5 kHz in both cases, i.e. in case of schwa-vowel as well as in the case ofvowel /a/. Thus, the occurring narrow laryngeal tube together with an occurring source-filter interaction
seems to increase the mean amplitude in this frequency range by about 15 dB to 20 dB for speech andsinging. In the case of the schwa vowel two formants replace the fourth formant in case of existence of thenarrow laryngeal cavity while in case of /a/ vowel not such a clear situation can be observed. Here no extra
formant occurs but a strong increase in amplitude in the frequency region between 2 kHz and 5 k Hz occurs
as well.

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Figure 4: Spectra for schwa- and /a/-vowel and both laryngeal tube sizes using the model without
acoustic-aerodynamic interaction (model1) for speech (a: PP = 600 Pa) and singing (b: Pa = 1200 Pa) (cases 1 to
8 in table 1). Frequency scale: 0 to 5 kHz; relative amplitude scale covers a range of 80 dB.

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Figure 5: Spectra for schwa- and /a/-vowel and both laryngeal tube sizes using the model withacoustic-aerodynamic interaction (model 2) for speech (a: PP = 600 Pa) and singing (b: p sub = 1200 Pa) (cases 1
to 8 in table 1). Frequency scale: 0 to 5 kHz; relative amplitude scale covers a range of 80 dB.

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4. DISCUSSION
No clear effect can be found for the modification of formants by changing from a wide to a narrow
larynx-pharynx tube in case of low subglottal pressure and in the case of a simple articulatory-acoustic
model which does not include acoustic-aerodynamic source-filter interaction. Thus, the change of formant
intensity in the region of 2 kHz to 5 kHz mainly depends on the model used, i.e. whether the model with orwithout source-filter interaction is used. The intensity in this region increases by 15 dB to 20 dB in case ofsource-filter interaction (model 2) while it increases in maximum by 5 dB in case of no source-filter
interaction (model 1). Furthermore the change of pulmonary pressure from 600 Pa (normal pulmonarypressure as occurs during speaking) to 1200 Pa (high pulmonary pressure as may occur e.g. in opera
singing) has no strong effect on increase in intensity within the singer’s formant frequency region in case of
no source-filter interaction. But in the case of source-filter interaction this leads to a small effect, i.e., to anincrease of 5 dB in that frequency range.
Thus this simulation study illustrates that source-filter interaction has a considerable effect onestablishing a singer’s formant in combination with the occurrence of a narrow laryngeal tube. But it is notclear whether this effect is caused by the clearly visible effect of formant ripple in the opening phase of theglottal cycle (Fig. 3) or by other effects resulting from acoustic-aerodynamic source-filter interaction. Thus
it should be checked whether the source-filter interaction effect influences the closing phase parameters ofglottal flow, i.e. the parameters Ee, Td and Ta in terms of the LF-model (16, 17). More simulations andanalyses of simulated as well as of natural data are needed in order to confirm or reject this hypothesis.

5. CONCLUSIONS AND FURTHER WORK
It can be concluded that the occurrence of source-filter interaction beside insertion of a narrow laryngeal
extra tube initiates the effect on increase in intensity of about 15 dB in the frequency region of the singer’s
formant. But no clear or rule-based effect can be observed for a shift of formants or an insertion of an extra
formant in the case of an insertion of the laryngeal narrow tube on the basis of our simulation experiments.
Like in other studies here a clear separation of cross sectional area of the laryngeal tube (narrow tube) andthe further pharyngeal tube (wide tube, wide sinus Morgagni, see Sundberg (1, 2) and Sundberg et al. (3))
needs to occur within the area function of the vocal tract and thus the articulatory maneuvers which initiate
such an occurrence of a narrow laryngeal extra tube need to be discovered and learned by singers. This may
be the reason why only trained opera singers are capable of producing the singer’s formant.
The effects speculated on in this paper need a further and more detailed investigation. It is necessary tocheck the increase in intensity in the frequency range of the singer’s formant by doing simulations withmore vocal tract shapes (more vocalic configurations beside schwa- and /a/-sound) as well as to check theeffects ad different F0. The simulations introduced here are done only on schwa vowel and technical /a/vowel at fundamental frequency F0 = 100 Hz. Thus a lot of combinations of F0 values with different vocal
tract shapes should be checked in further studies in order to evaluate the effect of source-filter interaction
and of insertion of a narrow laryngeal tube section in more detail.

ACKNOWLEDGEMENTS
This paper is dedicated to all classic style singers and especially to Rauf Berman (Maastricht, Netherlands).

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