# What Might the Trombone Teach Us About the Singing Voice?-A Tutorial Review

Joe Wolfe, Henri Boutin, Noel Hanna, John Smith

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Joe Wolfe, Henri Boutin, Noel Hanna, John Smith. What Might the Trombone Teach Us About the Singing Voice?-A Tutorial Review. Journal of Voice, 2024, 10.1016/j.jvoice.2024.02.006. hal-05047411

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# What Might the Trombone Teach Us About the Singing Voice? — A Tutorial Review

*Joe Wolfe, †Henri Boutin, *Noel Hanna, and John Smith, *Sydney, Australia, and †1 Place Igor Stravinsky, Paris

**Summary:** The trombone and the male voice cover similar frequency ranges and, at a physical level, the basic anatomies of the voice and the trombone show some qualitative similarity: both have two vibrating flaps of muscular tissue (the vocal folds and the trombonist's lips, respectively), and in each case, these are loaded acoustically by resonant ducts both upstream and downstream. There are also large differences. For example, the downstream ducts differ in length. The trombone usually operates with an oscillation frequency close to that of one of the downstream resonances; this is only occasionally true of the voice. Because the lips of a trombone player are much more readily accessible for experiments, they have yielded more detailed measurements of longitudinal and transverse motion, AC and DC pressures, and flow under varying acoustic loads. In normal operation, the downstream motion of the lips or vocal folds leads the lateral opening motion, resulting in a sweeping flow that leads the flow through the aperture. The relative timing of these flow components is related to the phases of the pressure across the tissues and the downstream acoustic load. Further, the work done on trombonists' lips due to their sweeping motion makes an important contribution to maintaining oscillation with both inertive and compliant acoustic loads. This probably explains why trombonists can “lip” the pitch smoothly from above to below a downstream resonance. Similar calculations on measurements of vocal fold motion show a similar work contribution and suggest that this sweeping motion is significant in powering this component of laryngeal motion. Comparing and contrasting the operation of the two “instruments” gives new perspectives on the basic science of the voice, with practical applications including the use of resonances. This could be helpful to voice scientists but also useful background knowledge for singers and singing teachers.

**Key Words:** Trombone–Voice–Acoustic loads–Resonance tuning–Vocal tract–Sweeping flow.

## INTRODUCTION

At a physical level, the basic anatomies of the voice and the trombone-player system exhibit some similarity: each has two vibrating flaps of muscular tissue (the vocal folds and the trombonist's lips, respectively), whose oscillation is powered by, and in turn modulates, a flow of pressurized air from the lungs. In each case, these vibrating tissues are loaded acoustically by resonant ducts both upstream and as shown in Figure 1.

Where the two cases exhibit similar properties or behaviors, comparison may give a new viewpoint to voice research.¹,² Where they are different, asking why they are different may also give insight. A further motivation for comparing and contrasting these two systems in this paper is that the lips of a brass instrument player are much easier to observe experimentally,³⁻⁷ and so our quantitative knowledge of their behavior is richer in several ways (reviewed by Campbell et al⁸). Furthermore, it is possible to measure directly the acoustic impedance (the ratio of acoustic pressure to flow) of the up- and downstream ducts

for the trombonist's lips. Recently, the present team has made experimental measurements on trombonists to determine the steady and acoustic components of upstream and downstream pressure and flow. Simultaneously, high-speed video in two orthogonal planes showed the shape and motion of the lips, from which the lips' sweeping flow and the steady components of the airflow were calculated.⁷,⁹ In turn, this allows the calculation of the contribution of sweeping flow to the energy required for self-oscillation under both compliant and inertive loads.

Some of the analogous data are known for the voice, but measurements are usually less direct, and it is rare that several of these quantities are measured simultaneously. Consequently, comparing and contrasting them can give the voice scientist new insights into the operation of the voice. For the singer or singing teacher, this new perspective on the voice should give new understanding of their own remarkable musical instrument.

The unusual approach of the present paper is to examine the similarities and differences between the singing voice and the trombone-player system to obtain these new perspectives on the voice. The trombonist and the voice both exhibit subtleties that are important complications, but which tend to obscure the comparisons. For that reason, we take advantage of footnotes to deal with these important, but diversionary, complications.

## Schematic comparison

First, consider the similarities suggested in Figure 1.

(i) *The source.* In both voice and trombonist, two flaps of tissue (the vocal folds and the player's lips, respectively)

Accepted for publication February 6, 2024.

From the *School of Physics, UNSW Sydney, Sydney, Australia; and the †Sciences et Technologies de la Musique et du Son (UMR9912), Sorbonne Université, Ircam, CNRS, 1 Place Igor Stravinsky, 75004, Paris.

Address correspondence and reprint requests to Joe Wolfe, School of Physics, UNSW Sydney, Sydney 2052, Australia. E-mail: j.wolfe@unsw.edu.au

<footnote>a By convention, DC flow is from upstream to downstream. For the phono- nating voice, the lungs are upstream, even though they lie below the larynx. Journal of Voice, Vol XX, No XX, pp. xxx–xxx 0892-1997
© 2024 The Authors. Published by Elsevier Inc. on behalf of The Voice Foundation. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).https://doi.org/10.1016/j.jvoice.2024.02.006</footnote>
<figure><img src="image_3.png" /><figcaption>**FIGURE 1.** A physicist’s “idealized” schematic of a trombonist and a singer (right and above), with the anatomy vastly simplified. It is not to scale, and details of the oscillating components (lips and vocal folds) are not shown. The vocal tract acts as the upstream duct for the trombone’s source and as the downstream duct for the source of the voice, though with different boundary conditions. Regions involved in the acoustics are shown in black.</figcaption></figure>

undergo auto-oscillation, a nonlinear phenomenon. They act as a valve that modulates the airflow from the lungs, thereby converting some of the steady or DC power from the flow of pressurized air from the lungs into the acoustic or AC power of sound waves. In turn, the power to drive the oscillation is drawn from the pressure and flow. In voice science, the modulated flow from this auto-oscillating component is commonly known as the source, and this paper applies that term analogously to that produced by a trombonist’s auto-oscillating lips.

(ii) *The downstream duct.* Both singer and trombonist have a resonant acoustic duct downstream from the source. For the voice, this is the vocal tract (the airway from glottis to lips, not including the nasal passages for this paper); for the trombonist, it is the instrument. In each case, the duct has resonances that help radiate at least some frequencies effectively into the radiation field (ie, the surrounding air). In both cases, the downstream duct has variable geometry, and the resonant frequencies are thus controllable. The acoustic behavior of the ducts is usuallyᵇ nearly linear. However, the wall materials, the shape, the functions, and operation of these ducts in the two cases are very different.

(iii) *The upstream duct.* Both also have a resonant duct upstream from the source. For the voice, this is the subglottal tract or trachea leading towards the lungs. For the trombone, it is the player's vocal tract, and

sometimes the trachea, leading towards the lungs. In both cases, the effects of these upstream tracts are more subtle and also more difficult to study than those of the downstream ducts.

When compared quantitatively, a striking difference is obvious: although the trombone and a typical male voice share much of their pitch range, their downstream resonators have very different lengths: 2.7 m or more for the trombone, and around 0.17 m for the vocal tract. This could seem to raise a paradox: most men have a trombone-pitched voice located in a piccolo-sized instrument. Therefore, let us first consider the downstream ducts.

## THE ACOUSTIC BEHAVIOR OF THE DUCTS

### Downstream duct resonances

Qualitatively, the downstream impedance spectra for the trombone and voice (Figure 2A and 2B, respectively) both show the features expected of a pipe that is open at the remote end: the impedance magnitude is small at very low frequencies and then, as frequency increases, has a series of fairly sharp maxima and minima. Because of the very different lengths, the frequencies of the impedance peaks are very much lower on the trombone. However, and perhaps surprisingly, the magnitudes of some of the impedance peaks (the “strength” of the resonances) are very roughly comparable, between 10 and 100 MPa.s.m⁻³.

### Trombone

It is relatively easy to measure precisely the acoustic impedance of the trombone as seen by the source, that is, at the mouthpiece. Figure 2A shows that the peaks in the

<footnote>ᵇ Shockwaves result from nonlinear propagation. In the trombone, shockwaves are sometimes functionally important,¹⁰ and they appear to have analogs in elephant trumpeting,¹¹ but not in the human voice.</footnote>
magnitude of Z(f) are narrow and symmetric at lower frequencies. Because of the flare and the bell, the overall cross section of the trombone increases towards the radiating end, and this shape, plus the properties of the mouthpiece, has the (desirable) consequence that all but one of the significant peaks lie in a nearly harmonic series. (The exception is the first^c^ peak, which is not used in performance.) So, for n ≥ 2, resonance Rn or the nth impedance peak has frequency $f_n \approx n f_{\text{pedal}}$.

The lack of significant resonances above about 800 Hz (~G5) is a consequence of properties of the mouthpiece and the flare and bell. The mouthpiece has a cup, providing an acoustic compliance, and a constriction, contributing an inertia. Together, these are a Helmholtz resonator,^d^ one of whose effects is to attenuate severely the resonances above several hundred hertz.

All else equal, a narrow duct (like the input bore of the trombone or the vocal tract) has high acoustic impedance; the external field for sound radiation has low impedance. Transmitting energy effectively from high to low impedance (or vice versa) is called impedance matching. The flare and bell act as an impedance matching horn at the output end of the trombone. This effect works best for wavelengths comparable with or smaller than the dimensions of the bell. Consequently, at higher frequencies, waves traveling down the bore are well transmitted and poorly reflected at the bell; with only weak returning waves, the high resonances are further attenuated (see Figure 2A).

The bell's effective radiation of several of the higher harmonics contributes to the timbre of the instrument. Also, because hearing sensitivity increases with increasing frequency for the relevant range, the bell contributes to the impressive loudness of the trombone. For high frequencies, the bell also gives the instrument a high directivity and makes it loud for listeners in line with the bell (by removing the bell section and extending the slide to achieve the same pitch, one can readily demonstrate the directivity, loudness, and timbre effects of the bell^12^).

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**FIGURE 2.** Measured magnitudes of the downstream acoustic impedance spectra Z(f) of a trombone and a model vocal tract, and the sound pressure they can produce. **A.** The Z(f) of a trombone measured at the mouthpiece. **B.** That of a hardware model vocal tract measured at the larynx. **C.** The measured sound spectrum produced by the trombone when playing the note F3 (nominally 175 Hz) with the slide in the same position as (A). The trombone fundamental f₀ occurs close to a peak in Z(f), in this case R3. **D.** The measured sound spectrum for the model vocal tract when excited near the larynx at 172 Hz with a synthesized acoustic current that closely approximates a typical glottal waveform. Note that the harmonics falling near the resonances R1, R2 are larger than their neighbors and so produce the formants F1, F2.

## Vocal tract

In contrast, the vocal tract at the source is not directly accessible for comparable measurements,^c^ so, for purposes of comparison, the vocal tract input impedance shown in Figure 2B was measured on a 3D-printed model with an area function modeling the MRI data of one of the authors producing a sustained neutral /3/ vowel.^13^ The 3D-printed model means that its resonances are expected to have

<footnote>^c^ As Figure 2 shows, the frequency $f_1$ of the first impedance peak R1 lies below the frequency $f_{\text{pedal}}$ of the implied fundamental of the harmonic series. While it is difficult to play a note at $f_1$, a note at $f_{\text{pedal}}$, called the pedal note, is playable. For several n ≥ 2, the nth harmonics of the pedal note fall quite near the nth impedance peaks. This tight harmonic relationship (except for R1) is maintained when a player changes the playing frequency by moving the slide and thus changing the effective length of the instrument.</footnote>

<footnote>Why is $f_1$ so much lower than $f_2/2$ or $f_3/3$? If we consider a purely cylindrical lip-reed "instrument," we would expect the frequency ratio of the first two resonances to be 3:1, so this larger spacing should not be a surprise for lower frequency waves that "penetrate" less well into the bell and flare.</footnote>

<footnote>^d^ Taking the mouthpiece on its own, brass players often slap the rim of the mouthpiece against the palm of the hand to hear the "pop tone" at the Helmholtz frequency. The effective mouthpiece resonance is at lower frequency when the mouthpiece is fitted to the instrument.</footnote>

<footnote>^c^ It is possible to measure Z(f) for the tract with precision—through the lips.^14^ But this is the “wrong” end, acoustically speaking: to understand the effects of resonances on the source, the impedance at the source is more directly useful.</footnote>
smaller bandwidths than those of a human vocal tract, where the magnitude of the (complex) visco-thermal loss factor $\alpha$ is typically increased by a factor of five.¹⁴

The shorter length of the vocal tract means that its resonances occur at higher frequencies than those of the trombone. Furthermore, they are not usually harmonically related, and can be varied independently of each other depending on the vowel to be produced. The first impedance peak $R1$ is largely controlled by the lip aperture and $R2$ largely by the position, forward or back, of the tongue constriction.

Does the voice have something like a bell for impedance matching at high frequency? First, one could argue that the tract overall is an impedance matcher at or near its resonance frequencies, so it acts to radiate any harmonics falling near the frequencies $Rn$—a feature to which we shall return. Further, for styles of singing in which a large mouth aperture is maintained over most vowels, the whole tract acts as a megaphone, increasing sound radiation on axis and increasing loudness for listeners facing the singer.¹⁵ Finally, an impedance matching role has also been attributed to the epilaryngeal tube or supraglottal constriction, a short, narrow section between the glottis and the main waveguide of the tract. For classically trained male singers, a lowered larynx produces a supraglottal constriction whose quarter-wave resonance boosts harmonics with frequencies at roughly 3 kHz (~G7), producing the singer's formant or vocal ring, increasing loudness as well as changing the timbre.¹⁶,¹⁷

## Downstream ducts and the output sound

Figure 2C and 2D shows the sound pressure spectrum that can result when these two different downstream ducts are excited by the relevant source to produce the note around F3 in the low mid-range of both trombone and male voice (175 and 172 Hz, respectively). Despite the great difference in $Z(f)$, the sounds produced have considerable similarities.

### Trombone

The source for the trombone is the vibrating lips of a player. In the absence of the instrument, and with just a mouthpiece over the lips, a trombonist can vary the oscillation frequency $f_0$ of the lips smoothly over a few octaves by modifying the embouchure, which includes the tension and geometry of the lip muscles, the shape of the tongue, and the blowing pressure, and including what players describe as “breath support.” With the instrument present, however, the player is very aware of its strong resonances and usually the lips oscillate at a playing frequency $f_0$ a few hertz above that of one of the impedance peaks of the instrument's bore ($n \ge 2$). At this frequency, the impedance magnitude is nearly maximal, and it is compliant: the acoustic current leads the AC mouthpiece pressure in phase.⁹ Smooth, continuous, large changes of the embouchure, and other parameters that control the pitch do not produce smooth changes in pitch but rather jumps from the

vicinity of one resonance to the next. This allows the player to “bugle” notes in the harmonic series without moving the slide. Near a resonance, however, players can easily shift $f_0$ from slightly above to slightly below the resonance, called “lipping up and down.”⁹,¹⁰

The weakness of the higher resonances (Figure 2A) has implications for pitch stability in the high range of the instrument: players can play in this range but say they “don’t get much help” from the instrument about intonation.

### Voice

The model vocal tract discussed above cannot be connected to actual vocal folds, so the source was a synthesised acoustic current¹⁹ that closely approximates a typical glottal waveform as calculated from inverse filtering.²⁰ It had a glottal contact quotient (CQ) of 0.5—see²¹ for details.

A male singer usually sings at frequencies below—and often well below—the lowest acoustic resonance of the tract. However, singers of both sexes have little difficulty singing above the lowest resonance: a small mouth opening gives $R1$ around 300 Hz or lower (~D4 or lower) and even basses can usually sing above this. Often, though not always, they can glide smoothly through a downstream resonance—see below.

So, with respect to the nearest resonance $Ri$, both “instruments” can often operate easily at either a frequency $f_0$ close to but below $Ri$ with inertive (mass-like) acoustic loads or at $f_0$ close to but above $Ri$, with compliant (spring-like) loads. For the trombone, the load is usually compliant, for the voice, usually inertive. For the trombone, which plays almost always near a resonance, the impedance is always large. For the voice, especially the male voice, $f_0$ is often well below the nearest resonance, so the downstream impedance can be small. This difference is related to the different functions: at least one of the resonances of the trombone largely determines the pitch, while the function of the downstream resonances of the voice is usually phonetic rather than melodic. This function is described, especially in the simplified source-filter model, as filtering: those harmonics that fall close to a resonance $Ri$ are boosted and produce a formant $Fi$.²,²²

The duct walls of a brass instrument are almost rigid. However, wall vibrations make a very small contribution to the sound and to a player's perceptions.⁸ The finite mass and stiffness of the cheeks and the walls of the vocal tract allow wall vibrations that contribute to acoustic losses and lead to a mechano-acoustic resonance and anti-resonance at low frequencies (~20 and ~200 Hz (~E0 and G3), respectively¹⁴; these may influence plosives.²³

<footnote>f Beyond “lipping up and down,” many experienced brass players are also able to vary $f_0$ smoothly over a large range without using the slide, called a “lip *glissando*.” This is rarely requested of trombonists because the instrument’s slide allows smooth changes of pitch. Trumpeters sometimes perform a “half-valve lip *gliss*.”; they partly depress one or more valves, which weakens the resonances, and makes the *glissando* easier.¹⁸</footnote>
At higher frequencies, however, the contribution of the
mechanical properties of the walls is usually small. A
plastic trombone is detectably different from a brass one,
yet perhaps surprisingly similar, at least for the listener.
Modelers of the vocal tract often make the approximation
that, for frequencies above a few hundred Hz (~D4), the
duct walls are essentially immobile. Apart from the very
different lengths, the biggest difference between the
downstream resonators of trombone and voice is the many
and subtle possible variations in the geometry of the latter.

Resonance and pitch: resonance tuning and impedance matching

The glottis and the trombonist's lip aperture are both small and so have a high characteristic impedance; the external radiation fields (at mouth and bell, respectively) have low impedance. How to transmit sound efficiently from source to output? In the trombone, the flare and bell do this for a broad range of frequencies. Vocal tract resonances are also good impedance matchers—provided that the source frequency or one of its harmonics falls near that of a resonance.

In normal speech and much singing, the resonances and
pitch are controlled independently; one can sing different
vowels at the same pitch (vary *Ri* but not *f*<sub>o</sub>) or vocalize on
the same vowel (vary *f*<sub>o</sub> with approximately the same *Ri*). In
contrast, resonance tuning means adjusting the frequency of a
downstream resonance to lie close to that of the fundamental
or a higher harmonic of the source. This is done with some
loss of phonetic information: in resonance tuning, the note
rather than the vowel determines at least one resonance. One
likely advantage of resonance tuning is the voice efficiency
resulting from the impedance matching mentioned above.
Another is that high impedance at the source, and perhaps
especially a large inertive impedance, may assist vibration of
the vocal folds.²⁴ Either effect on its own is expected to give
louder output for the same vocal effort.

Voice

The most common case of resonance tuning is adjusting the
first resonance to lie close to the fundamental (R1: f<sub>o</sub>
tuning). When f<sub>o</sub> rises above R1 for the vowel suggested by
the lyrics, R1 is increased so that it falls close to f<sub>o</sub> (see
Figure 3). R1: f<sub>o</sub> tuning is practised by sopranos, whether
trained or not, over the range from about C5 to C6.<sup>25,26</sup>
Altos and sometimes tenors and baritones use it, especially
for close vowels, when the pitch range allows it.<sup>27</sup> In the
minority of sopranos who sing substantially above C6,
some tune the second resonance close to the fundamental
(R2:f<sub>o</sub> tuning).<sup>28,29</sup>

R1:2f₀ tuning means tuning the first resonance close to the
second harmonic. Altos often use this strategy in part of
their range, and it is characteristic in Bulgarian folk
singing³⁰ and belting.³¹ Tenors and baritones sometimes use
R1:2f₀, R1:3f₀, and other R1:nf₀.²⁷ All these give a useful
boost to the harmonic in question, and therefore have an
important effect on the spectrum and so the timbre.

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

**FIGURE 3.** The tuning of resonances when playing the trombone and when singing. Black dots indicate the lowest frequency resonance used for that value of *f*<sub>o</sub>, and red dots indicate the higher frequency resonances. The open black circle indicates the “missing” lowest resonance for the pedal note on the trombone. The singing example shows a case where R1: *f*<sub>o</sub> and R2:2*f*<sub>o</sub> tuning occurred simultaneously.

Trombone

Trombonists learn early to adjust the embouchure and other
playing parameters so that f<sub>o</sub> lies close to, but slightly above,
one of the resonances. There is then no need to adjust a
higher resonance to one of the higher harmonics: an intrinsic
result of the trombone's bore profile is that it produces a
harmonic relationship among most of the resonances in the
low and middle ranges of the instrument. For example, when
playing the note F3, the trombonist has R3 ≅ f<sub>o</sub>, R6 ≅ 2f<sub>o</sub>,
R9 ≅ 3f<sub>o</sub>, and R12 ≅ 4f<sub>o</sub> (there is no R15—see Figure 2A). In
the language of resonance tuning, this might be called si-
multaneous R3:f<sub>o</sub>, R6:2f<sub>o</sub>, R9:3f<sub>o</sub>, and R12:4f<sub>o</sub> tuning.

In the high range of the trombone, the absence of high
resonances means that there are no resonances at 2f<sub>o</sub> or
higher. In the low and middle ranges, however, at least one
or more of the higher resonances help radiate power at the
higher harmonics. They also influence the motion of the
lips; indeed, for the pedal note Bb1, there is no resonance
corresponding to f<sub>o</sub> (see Figure 3), and it is the impedance
peaks at second and higher harmonics that allow the
trombone to produce a stable note at this pitch.

The closest reported analogy to this situation for singers
would be a strategy used by some sopranos: simultaneous
$R1:f_0$ and $R2:2f_0$ tuning;²⁸ see Figure 3B in the range 600–800 Hz. Titze¹⁵ points out that, while wind instruments have multiple resonances located close to harmonics, singers can adjust the mouth shape so that several harmonics have loads with advantageous, inertive phase. Single resonance tuning could give favorable acoustic load to one of the harmonics; its impedance matching effect (on any harmonic) would also give greater output for less effort.

A practical question: Can the voice compete for loudness with the trombone? For notes of low and medium pitch, the trombone has the advantage of “automatic” tuning of bore resonances to the lowest several harmonics. The vocal tract is shorter, so the singer’s resonances are more widely spaced, so multiple resonance tuning is impractical in this range. For high notes in both trombone and voice, successively more power is in the fundamental, so tuning of just one (or two) resonances is possible—and adequate—in both instruments. Consequently, the trombone might expect competition from, for example, a soprano singing in the high range and practicing one of the soprano tuning strategies described above, or a tenor in the high range practicing $R1:2f_0$. And, for those singers not aiming for trombone loudness, resonance tuning can give more power for less effort.

## The upstream ducts

### Trombone

For the trombone and other lip-valve instruments, the player's vocal tract (viewed from the lips) is the upstream duct, and its impedance spectrum at the lips ($Z_u$) can be varied by changing the shape of the tongue and the degree of opening of the glottis. Endoscopy of experienced wind players³³,³⁴ and players of the didjeridu³⁵ suggests that, during performance, experienced players may keep the glottal aperture small compared with respiration. Measurements and modeling of vocal tract influence, consistent with a nearly-closed glottis, are reported for advanced techniques on trombone, didjeridu, clarinet, and saxophone.³⁶⁻⁴⁰ A large reduction in the glottis size has the effect of strengthening resonances measured at the lips and reducing the involvement of the subglottal tract; it also increases the frequency spacing of resonances.¹³

For normal playing over much of the range, the first three peaks in the impedance of the vocal tracts of trombonists⁶ (measured at the lips) fall at roughly 200, 700, and 900 Hz (approximately G3, D5, and A5) and are largely independent of the playing frequency (eg, Figure 4). Their magnitudes are varied and fall in the range roughly 0.2–10 MPa.s.m⁻³.⁷ However, especially in the high range, where bore resonances are weaker, trombonists can increase the magnitude of $Z_u$ at the playing frequency (or one of its harmonics) to be comparable with or greater than that of the bore.³⁶

<figure><img src="image_7.png" /><figcaption>**FIGURE 4.** The upper graph shows the measured magnitude of the upstream acoustic impedance spectrum $Z(f)$ of a trombone player measured during performance at 114 Hz (Bb2).⁷ The lower graph shows measured⁴¹,⁴² or fitted¹³ impedance spectra of the subglottal tract of living subjects.</figcaption></figure>

### Voice

The trachea and respiratory system form the upstream duct for the voice. It is usually impossible to measure its impedance spectrum directly on an undisturbed living subject, but measurements have been made on patients through a tracheostoma⁴¹ or by using an endotracheal tube⁴²—see Figure 4. (Both measurements required some correction and were effectively made from a position upstream of the glottis so their resonance frequencies would be expected to be higher than that experienced by the glottis.) The impedance varies over a similar range with typically two broad resonances below 2 kHz (~B6). The fitted curve¹³ was obtained by fitting a simple model to the difference in the impedance spectra measured through the lips when a participant opened and closed the glottis.

The subglottal resonance frequencies have also been estimated from formants in the subglottal pressure.⁴³ They can also be estimated from measurements made using mechanical excitation of the outside of the neck⁴⁴,⁴⁵ and tracheal puncture.⁴⁶ The first few peaks lie at about 640, 1400, and 2100 Hz (~E5, F6, and C7),⁴⁷ and the values are expected to vary relatively little for a given healthy adult, because the geometry of the trachea is not very varied.

Why are the magnitudes of peaks of the upstream impedance for the trombonist (ie, that of the vocal tract) rather weaker than those for the voice (the trachea)? One answer may be that the (at least partially) open glottis necessary to play a wind instrument. Partial opening of the glottis decreases the magnitude of the impedance peaks and greater opening also increases their number.¹³

## THE SOURCE

In both trombone and voice, when the source aperture is open, air passes through a constriction (the lip aperture for

<footnote>g In this study, four experienced players and three beginners played a Yamaha YBL 321 instrument with a mouthpiece fabricated for the experiments.⁷,⁹ The playing range covered Bb2 to F4 (114–355 Hz).</footnote>
the trombone, the glottis for the voice). Passing through this confinement, it travels quickly, at speed v, to which it is accelerated by a fall in pressure from the lungs (for the voice) or mouth (for the trombone) to the constriction. From conservation of mechanical energy, the pressure in this aperture is therefore lower than that upstream by the “Bernoulli term,” $P_B \approx \rho v^2/2$, where $\rho$ is the density of air. In both cases, there is likely to be turbulence immediately downstream, so a large fraction of that kinetic energy is subsequently lost, with the result that pressure is lower in the aperture and downstream than it is upstream. During the open phase, other modest contributions to the pressure difference across the lips or folds come from the viscosity of air and inertial effects of the varying flow rate. The distribution of pressure over the surfaces of the lips or folds exerts time- and spatially-varying forces on the tissues. These forces plus the internal forces associated with mechanical deformations of the tissue give rise to auto-oscillation.

## Lateral and longitudinal motion

In comparison with the voice,⁹ it is relatively easy to observe and quantify the aperture area and aspects of the motion of a trombonist's lips in the transverse and longitudinal directions using a mouthpiece fitted with windows and making simultaneous video recordings in two or three orthogonal planes.⁵⁻⁷

Researchers studying brass instruments have further advantages over their colleagues in voice science: it is easy to fit microphones and/or pressure transducers in the mouthpiece, and only slightly perturbing to place them inside the player's mouth, giving measurements of both acoustic and DC components of pressure immediately up- and downstream from the source. Further, because the downstream impedance is already known, or can be measured a few seconds later,⁴⁹ the downstream pressure allows the calculation of the acoustic flow into the mouthpiece. Analogous measurements of the immediately up- and downstream pressure for the voice are considerably more perturbing for an experimental participant, and the up- and downstream impedances are less well known.

We now look at the operation of the trombonist's lips, keeping in mind the considerable differences between voice and trombone, but looking out for effects where the trombone might be relevant for understanding the voice.

## The trombone

Figure 5A-D shows two complete cycles of the note Bb2 played on the trombone by an experienced player at normal

pitch, that is, with a compliant downstream load. The greater access that the trombone allows means that the relevant pressures, flows, and lip geometry can be measured simultaneously.⁷,⁹ Samples of the images taken from the side and front view are also shown (Figure 5E). The pressure measurements include both the DC and acoustic components. The DC flow is not measured directly, rather it is known because an instant when the volume of lips inside the mouthpiece is neither increasing nor decreasing occurs during the period during which the lips are closed. The calculation of the lip volume in the mouthpiece uses images from Figure 5E. These measurements were made for normal playing, meaning at a frequency slightly above the bore resonance and thus with a somewhat compliant downstream load (we discuss inertive loads below).

Figure 5A shows aspects of the motion of the upper and lower lips; their maximum lip displacement in the x (longitudinal, downstream) direction and the aperture between them. Freeze-frame images of the lips are shown in Figure 5E. These show that the lips, especially in this case the lower lip, move downstream (into the mouthpiece) while still closed. This explains how the total flow begins before the lips open (see Figure 5B); it is a consequence of the motion of the closed lips in the longitudinal direction producing a sweeping flow. The lips then open, allowing an aperture flow between the lips before they eventually close after retracting back towards the teeth. This phase difference between x and z motion is important for the energetics of auto-oscillation and we return to it below. Figure 5B shows the total flow and its two components: the aperture flow passing between the open lips and the sweeping component of flow due to the motion of the lips in the mouthpiece.

In this example, the playing was at a comfortable loudness level for which the player used a DC blowing pressure of a little more than 1 kPa. During the time when the lip aperture is closed, the pressure difference across the source is around 3 kPa, a consequence of the resonant standing waves downstream. Near resonance, downstream standing waves produce large peaks in the downstream impedance: a small AC flow gives a large AC pressure. In Figure 5C, a pulse of strong suction in the mouthpiece arrives at the lips when they are closed (image 1) and closest to the teeth. This applies a large force acting to pull the lips forward (downstream). One lesson here for voice is an indication that resonance tuning can provide a strong longitudinal force component on the vocal folds and might thus contribute to determining or stabilizing the pitch, a little like the way it does in brass instruments.

## The vocal folds

The transverse motion of the vocal folds may be measured using nasendoscopy with high-speed video; such measurements have yielded much of our knowledge about the transverse motion. Measuring the longitudinal vibration of the folds is more difficult than for the trombone. Measurements using laser triangulation during endoscopy

<footnote>h An interesting example of an experimental technique from voice science being used to study lip motion in musical instruments is the use of the electroglottograph (EGG). Usually used to measure the time-varying electrical admittance across the neck at the larynx, electrodes connected to an EGG have been used across the player's lips for the didjeridu⁴⁸ and trombone.¹ The electrical admittance is related to the contact area between the vocal folds (for voice) or lips (for an instrument). However, the relation is not simple.¹</footnote>
<figure><img src="image_8.png" /></figure>

**FIGURE 5.** Lip motion, pressure, flow, and work for a trombonist playing Bb2 (at 114 Hz). **A.** (Top left) The forward displacement of the upper and lower lips, the aperture between them and the AC component of the volume of lip tissue in the mouthpiece. **B.** (Lower left) The total flow into the bore and its two components; the sweeping flow due solely to the motion of the lips and the flow through the aperture between them. **C.** The up- and downstream pressures; that is, those in the mouth and mouthpiece, respectively. Using the simple model described later, **D.** the instantaneous sweeping power applied to the lips by the pressure difference across them and the integral of this quantity over two complete cycles. This integral is the PV work supplied to the lips by the pressure difference and the longitudinal component of the sweeping action. **E.** Images taken at the indicated points along the time axis of **B**. Here and in Figure 6, the minimum in pressure in the mouthpiece is chosen as the zero of time. Adapted from⁹.

have reported longitudinal movement with an amplitude of order 1 mm.⁵⁰,⁵¹ George et al⁵² measured both the longitudinal and transverse motion of the folds using laser scanning endoscopy. Their measurements also show that (as with the trombonist's lips) the opening motion of the folds lags in phase the downstream (upward) oscillation. Further, Granqvist et al⁵³ has found non-zero flow during periods of vocal fold closure, consistent with downstream (upward) motion of the folds while closed.

**The phase of the acoustic loads**

The downstream impedance of the trombone as seen by the lips is a strong function of frequency (see Figure 2A). At frequencies near the peaks in the impedance of a duct (eg, the trombone, the vocal, and subglottal tracts: see Figures 2 and 4), the magnitude and phase of the acoustic load change rapidly with small changes in frequency. At frequencies a little below an impedance maximum, the pressure leads the flow (the input pressure has its maximum
<figure><img src="image_9.png" /><figcaption>**FIGURE 6.** The relationship between mouthpiece pressure and flow when a trombonist plays at three different stable pitches without changing the configuration of the trombone. The upper figures show the measured pressure in the mouthpiece. The lower figures show the total flow into the bore and its two components; the flow due to lip motion alone and the flow through the aperture between them. Arrows indicate the maximum values of pressure and the total flow; pressure leads flow for inertive load (left) and lags flow for compliant load (center and right).</figcaption></figure>

Using data from⁹.

a little earlier in time). Thus, the load is inertive or mass-like (a pressure must be applied first to move a compact mass of air and to create a flow). At frequencies a little above the impedance maximum, the flow leads the pressure in phase. Here, the load is compliant or spring-like (flow into a closed volume of air compresses it like a spring and thus increases the pressure). When the frequency is exactly at resonance, the magnitude of the impedance is greatest and the load is purely resistive (flow and pressure in phase). Because the oscillatory pressures at the source act on the vocal folds or lips, the phase and the magnitude of the load impedances are potentially important in producing stable vibration.

In order to oscillate at different frequencies around an impedance maximum, the phase between the pressure in the mouthpiece and the total flow from the source must match the phase of the load impedance (further, to maintain oscillation, work must be done on the oscillating components of the source to replace the energy lost in deformation and collision—see later).

## Trombone

Trombonists play a little above the frequency of a resonance in normal playing but can also easily play a little below. Mechanical lips in a trombone playing system⁵⁴ reproduce aspects of this behavior. Human players, when asked to “lip up” or “lip down” (ie, to raise or to lower the playing frequency without changing the slide position and therefore the downstream resonance), can lip down from their normal pitch rather further than up. The range that they usually cover, however, has limits that have approximately the same value of impedance magnitude (roughly

15% of the peak value) and also of phase (about 70° either side of zero: from pressure lagging flow by roughly 70° when lipping up through zero to leading flow by 70° when lipping down).⁹

The limits to the range invite the questions: does the lip vibration require a moderately high impedance load? Or does it require a load with phase angle in a certain range? Or some combination of both? And what is the relation between lip motion and the phase of the downstream load?

Figure 6 shows the relationship between mouthpiece pressure and flow for normal playing, and also when playing significantly above or below the impedance maximum. For normal playing, with a somewhat compliant load, the total flow into the mouthpiece leads the mouthpiece pressure. Here, the total flow begins while the lips are closed, as indicated by the sweeping flow produced by the motion of the lips into the mouthpiece. The lips then open soon after they start moving forward (about 1 ms after).

To play at a higher frequency, where the load is more compliant, the lips enter the mouthpiece more rapidly, and thus the initial sweeping flow is greater. The lips also open earlier with respect to their forward motion (only about 0.4 ms after), and the resultant earlier aperture flow produces a maximum in the total flow that occurs sooner after the minimum in mouthpiece pressure (chosen as $t=0$ in Figure 6) and thus further ahead in phase of the mouthpiece pressure.

For playing at a lower frequency, the load is inertive and flow into the mouthpiece must lag the mouthpiece pressure. In this case, the lips only open towards the end of their forward motion (after about 1.3 ms). The lips also enter the mouthpiece more slowly, reducing the initial sweeping flow due to lip motion. The aperture flow thus lags the
downstream displacement of the lips by a larger interval
than for normal playing.

These cases show how changes in the timing of the lip
opening and the speed at which the lips enter the mouth-
piece can significantly vary the phase of the total flow with
respect to the downstream pressure.

Voice

In speech and in much singing, unlike the case of a musical
instrument, the resonances of the load and f₀ are to some
extent controlled independently: for a sustained note in
singing, the pitch (and originally the composer) specifies f₀
while the vowel (and thus the librettist/lyricist) suggests
values for the resonances R1 and R2. Further, and in
contrast with brass players, singers can usually vary f₀
smoothly over a wide range without changing the geometry
of the downstream resonator and largely independently of
its resonances.

It is more difficult to determine the phase of the load on
the singing source because direct measurements are usually
impossible. Some generalizations are possible, however:
For most of the male voice range and for much of the fe-
male range, f₀ lies well below R1 so the load at the source is
somewhat inertive for the fundamental frequency. Titze²⁴
has argued that singers may prefer to sing with the vocal
source loaded downstream by an inertive impedance, which
occurs on the low frequency side of a vocal tract resonance.

Singing or playing across a downstream resonance

The input impedance of the trombone as seen by the lips is
a strong function of frequency (see Figure 2A). As noted
above, brass players, whose vibrating lips might be de-
scribed as “driven” near resonances of the bore, can
smoothly and easily glide across a resonance (“lipping up
and down”)⁹: there is no discontinuity suggestive of an
abrupt change in mode of operation.

What about singers? Titze²⁴ and Titze et al³² asked
participants to sing a *portamento* so that f₀ crossed the R1
of the intended vowel, or a slow diphthong at constant
pitch, with the initial and final vowels chosen so that R1
smoothly crossed f₀. They found that bifurcations in pho-
nation were more likely (but not always present) in the
region where R1 ~ f₀, particularly for men, who would
have less experience with R1 ~ f₀ from speech.

The present team⁵⁵ repeated these experiments with some
extensions. We also observed some statistically increased
occurrence of vocal instabilities when f₀ was a little above
R1 but also observed smooth variations in the range where
pitch-resonance crossings were expected. Referring to
Figure 2: perhaps what disturbed some singers was the
sudden reduction in Z above resonance, which implies ei-
ther a sudden reduction in the transglottal pressure or an
increase in the flow, and an overall lower singing efficiency.

Singers also encountered difficulties near the lower limit of
laryngeal mechanism M2 (the head voice).

We also asked the participants to sing their *portamentos*
into an acoustically “infinite” duct, with the result that
there was effectively no downstream resonance corre-
sponding to R1, and the downstream impedance was lar-
gely resistive at these low frequencies, with pressure and
flow quite close in phase. Here, instabilities were reduced,
and when they did occur, they were concentrated near the
lower limit of M2.⁵⁵ So the conclusion from both studies is
that the crossover of f₀ and R1 can sometimes pose diffi-
culties for some singers, and that, for them, removing the
resonance removes these difficulties.

However, the results of these, and several other studies,
have generally showed that, while instabilities are statisti-
cally more frequent near R1, they are not inevitable: there
are no consistent and clear-cut instabilities when the fun-
damental frequency, or a harmonic, crosses the resonance
frequency of the tract.

So, the trombone-player system and the voice are not
totally dissimilar, and we might look to the former to help
understand those many voices that do glide smoothly
across resonances; we do this later.

Possible effects of an upstream resonance

Most of the singing range of men and altos lies below the
lowest acoustic impedance peak of the subglottal tract
(~600 Hz or D5). In the glide studies discussed in the pre-
ceding section, no enhancement of instabilities was ob-
served when f₀ was in the vicinity of the subglottal
impedance peaks. Perhaps it is not important. Or perhaps
sopranos—the singers in whose standard range this
~600 Hz resonance occurs—are so used to this unvarying
challenge that they have subconsciously developed ways of
dealing with it. The work done on vocal folds due to their
sweeping motion (discussed later) may also contribute. Can
musical instruments give any insight? In their case, we have
the advantage that the upstream acoustical impedance
spectrum and the upstream pressure can be measured
during performance.

Because of continuity, the flow out of the upstream duct
equals that flowing into the downstream duct, so the ratios
of acoustic pressures and impedances have the same mag-
nitude: Z_up/Z_down = -p_up/p_down. For a trombone, Figure 5
shows much lower acoustic pressure upstream (mouth)
than downstream (mouthpiece). This is usual: (i) trombo-
nists usually play near a downstream resonance and not an
upstream one and (ii) the upstream resonances are usually
weaker than those downstream. However, in the high range
of the trombone, when the trombone's resonances are
weak, the acoustic pressure upstream can equal or exceed
that in the mouthpiece.³⁶,³⁷ These authors conclude that
players often arrange the vocal tract so that |Z_up/Z_down| > 1
when playing in the high range (eg, D5 ~ 595 Hz). In the
range, we studied⁷ (up to F4 ~ 355 Hz), we found no
evidence of players tuning a tract resonance to match the frequency of the note played.

Upstream resonances can affect the timbre of the output sound. This is spectacularly the case in the didjeridu,<fnref n="1" /> an Australian indigenous instrument whose source (the airflow modulated by the player's oscillating lips) operates much like that of a trombone. Usually, cyclic breathing ("circular breathing") is used to play a continuous low drone at a constant pitch close to the instrument's lowest resonance. The musical interest comes from rhythmic variations in timbre, including those provided by changes in the player's vocal tract, and associated with the cyclic breathing. The instrument's impedance is low, even at resonances, because of its large bore diameter and irregular internal shape. Consequently, the impedance of the vocal tract (upstream) can easily exceed that of the instrument (downstream), especially on instruments judged to be of high quality.⁵⁶ Measurements show that large impedance peaks in the vocal tract reduce the amplitude of the instrument's high harmonics that lie at frequencies nearby. When the tract impedance at the lips is high, little acoustic power is transmitted through the lip aperture, so an impedance peak in the vocal tract creates an antiformant (a minimum in the spectral amplitude) in the sound.³⁸,⁵⁷

Determining the influence of the subglottal tract on the voice is experimentally difficult. The major problems are that the resonances are not very sharp, are difficult to determine precisely and, unlike the vocal tract when it acts as the upstream tract for lip-valves, they cannot be significantly varied by the subject. They also occur in a frequency range that is also covered by the downstream resonances of the supraglottal tract.

How should we expect peaks in impedance in the subglottal tract to affect the sound of the voice? For a low note sung, an argument like that cited above would suggest that subglottal impedance peaks might inhibit the magnitude of harmonics that fall nearby, sometimes creating antiformants in the sound. Such antiformants might escape notice, however, for high $f_0$ and for vowels such as /i/ that have little power in the range of the expected antiformant.

Compared with the didjeridu, the relationship between subglottal behavior and the voice spectrum is possibly complicated by the relatively small glottis in normal speech and singing; this might tend to isolate the subglottal duct acoustically from the vocal tract.

Conversely, when trachea and vocal tract are strongly coupled, as in cases of aspiration or whispering, resonances of the combined longer duct may be excited by turbulence at the vocal folds. For measurements made at the lips, opening the glottis adds the trachea to the vocal tract and roughly doubles the effective resonant length, which

approximately doubles the number of resonances measured over the same frequency range.¹³

Several studies describe how the resonances of the subglottal ducts may influence the voice spectrum, but the effects are subtle and variable² and there does not appear to be a clear consensus. Fant et al⁵⁸ report that the second resonance of the subglottal tract contributes to the output sound an extra formant above $F2$ (ie, above the formant produced by $R2$ of the supraglottal tract) and point out that the subglottal resonance is hard to identify and that it may invalidate estimation of $F2$. They stated that "the extent to which subglottal resonances influence the properties of the voice source is not clear yet." Sundberg et al⁴⁶ found that "radiated vowel spectra did not show clear influence by the subglottal resonances." Using physical models, Zhang et al⁵⁹ observed strong interactions between fold vibrations and subglottal resonances for relatively long upstream ducts. Stevens,⁶⁰ analyzing a woman speaker pronouncing /I/ in "bit," reports a small extra formant near 1600 Hz (~G6) due to the second subglottal resonance, which falls near that frequency. On the other hand, Chi and Sonderegger⁶¹ looked for attenuation of the speech spectral envelope (an antiformant) in the vicinity of the second subglottal resonance at 1300-1600 Hz (~E6-G6). Four of the six subjects studied showed the effect, but the effect was small and subtle signal processing was required to detect it. Combining sound recordings, an accelerometer on the neck and impedance spectrometry,¹³ attempts in our lab to identify antiformants and formants clearly associated with the subglottal resonances have been inconclusive.

Why is the upstream impedance effect so strong in the radiated sound of the didjeridu but subtle and hard to measure in that of the voice? Part of the reason may be the relatively large aperture between the lips of the didjeridu player,³⁸ and the low $f_0$ (about 50-80 Hz, ~G1-E2).

## Longitudinal motion and its contribution to auto-oscillation

### A simple model concerning the energetics of oscillation

Strong and Dudley⁶² report a simplified model for the lateral and longitudinal motion of a trumpet player's lips. Two flaps can swing outwards like a door (and thus comprise an "outward striking valve," < +, - > in the notation of Fletcher⁶³). These flaps also oscillate along their length, with a phase difference between the two motions. Adachi and Sato⁶⁴ have used a qualitatively similar model for both a brass player's lips and a singer's vocal folds⁶⁵ (see also⁶⁶). We have used this model to quantify aspects of the oscillation energetics, first for trombone then for the voice.

Figure 7, after,⁹ shows a highly simplified model for vocal folds or a trombonist's lips. A pressure $P_1$ acts on the upstream side and $P_2$ on the downstream side. As noted above, turbulence downstream with the approximation of negligible pressure recovery means that the pressure in the channel is also approximately $P_2$. With this approximation, the transverse ($z$) displacements do no work, that is, the

<footnote n="1">¹ The didjeridu may have more to teach us about the voice because, using the technique that involves simultaneous phonation and playing, one can observe how easily the acoustic standing waves controlled by the lips can influence the vibration of the vocal folds.⁴⁸</footnote>
<figure><img src="image_10.png" /><figcaption>**FIGURE 7.** One proposed model for vocal folds or (rotated 90° clockwise) a trombonist's lips. The rectangles at left and right are stationary tissue, the dashed line is an axis of approximate symmetry. The oscillation giving *x* (vertical) displacement leads to the compressive oscillation (mainly *z*) in phase. The tissue sweeps out, in the *x* direction, a positive volume in each cycle (i→ii→iii→iv→i). We call this flow, whose integral is non-zero and positive around a whole cycle, the longitudinal sweeping flow.</figcaption></figure>

$PΔV$ work associated with all stippled areas in Figure 7 is zero.

The phase by which the longitudinal (downstream) motion leads that of the lateral (opening) motion is exaggerated to 90° in the sketch, for clarity. The key point is that the flaps (lips or folds) are closed when they begin to move downstream, open while downstream and close near their most upstream position. This difference in the phase of longitudinal and lateral motion means that the flow through the aperture, the flow swept by the moving tissues, and the total flow all have different phases, as discussed above—see Figure 6. Further, the closed tissues moving downstream sweep more air in the longitudinal direction (i–ii) than the open (and therefore shorter) tissues sweep upstream on their return (iii–iv). Thus, around a complete cycle, the folds or lips sweep out a positive longitudinal¹ swept volume $ΔV_x$ in the *x* direction. The pressure difference does positive work around a whole cycle. In contrast, because the aperture and downstream pressures are approximately equal, volume swept in the transverse direction involves negligible $PΔV_x$ work. The total effect is that the $PΔV$ work associated with the sweeping motion of the vocal folds is positive for inertial as well as for compliant loads.⁹

When the impedances either side are non-negligible, this sweeping work due to the longitudinal sweeping flow can be divided into two terms: that due to $ΔP$, the DC component of $P_1 - P_2$, and that due to its oscillating component, $Δp$; see ⁹ for details. The work component associated with oscillating pressure is positive when the load is compliant and negative when it is inertial. However, the work done by the average or DC component of the pressure is always positive for the cases we measure. This means that the total sweeping work can be positive for an inertive load.

## Lips and the trombone

How much power is needed to sustain auto-oscillation? Once per cycle, the trombonist's lips collide with each other. The images suggest that these collisions must be inelastic and convert a non-negligible fraction of the lips' kinetic energy into heat. Further losses are incurred by the hysteresis in bending incompletely elastic tissues. Velut et al⁶⁷ review the literature and give values of Q or quality factor of order one for human lips. This suggests that an energy comparable with that of the total kinetic energy of the tissues must be supplied in each cycle. In this context, it is interesting to note that the longitudinal sweeping work for one cycle of the trombonist's lips is of the same order as, and usually greater than, the maximum kinetic energy of both lips,⁹ indicating that it is an important contribution to auto-oscillation. The lip kinetic energy is about 20 times smaller than the energy per cycle supplied to the input of the instrument by the flow from the mouth: the power required to drive the vibrating lips can be much less than the power modulated by the vibration, the latter being the acoustic input power to the trombone or, analogously, the singer's vocal tract.

## Vocal folds and the voice

George et al⁵² kindly supplied the raw data for their laser-ranging nasendoscopy study of human vocal fold vibration; this allowed us⁶⁸ to calculate the average longitudinal sweeping flow of 34 mL·s⁻¹. The laser scanning procedure made it impossible for George et al⁵² to measure simultaneously the total flow or the transglottal pressure. However, if the transglottal flow had a typical value⁶⁹ of 300 mL·s⁻¹, the calculated longitudinal sweeping flow would be ~10% of the total laryngeal flow. It is difficult to estimate the kinetic energy and power terms for the vocal folds, but as with the trombone, a sweeping power component that was an order of magnitude smaller than the total acoustic power would be significant in maintaining auto-oscillation.⁶⁸ For a transglottal pressure of about 1 kPa, the sweeping power would be about half a milliwatt,

<footnote>j Note the difference between the longitudinal sweeping flow and the total sweeping flow discussed earlier. The integral of the latter around a complete cycle is zero: the amount of tissue in the mouthpiece does not increase indefinitely.</footnote>
which again might be of the order required to replace the kinetic energy lost on fold-fold collisions in each cycle.

Other models of the lips and the voice concern primarily the lateral motion (eg, van den Berg⁷⁰; Flanagan and Landgraf⁷¹; Titze²⁴). Voice source models often have two or more masses at different longitudinal positions, vibrating in different phase, sometimes in conjunction with longitudinal vibration. Transverse motion of these sorts was not clearly observed in our trombone experiments, possibly because the side views like those in Figure 5E do not show enough of the lip interior, or possibly because such motion was not very large.

Titze²⁴ argues that two-mass models of the vocal fold with lateral motion in different phases produce more stable oscillation when the acoustic load is inertive rather than compliant. It is possible that several different types of lateral and transverse motion occur in the vocal folds, either simultaneously or under different conditions. If the George et al⁵² results are typical, then similarly to the trombone, the sweeping work would be larger for compliant loads than for inertive, and might explain why some singers have little difficulty producing glides across a resonance.

## Conclusions: functional similarities and differences

* **Downstream loads.** While downstream loads control the pitch of lip-reed instruments at lower frequencies, this seems to be rarely or never the case for the normal voice. For both trombone and voice, the downstream resonances achieve impedance matching for select frequencies. (For the trombone, this is enhanced by the bell and flare.) When acting on higher harmonics, resonances produce formants in the voice, and generally louder high harmonics for the trombonist. For singers, resonance tuning provides impedance matching for the fundamental, and sometimes other harmonics. For brass instruments, normal playing involves a fundamental frequency very close to a resonance and, except for high notes, some harmonics lie close to resonances as well. Vibration of the walls usually has small effects in both cases, but can add a zeroth formant to the voice.

Singers have a right to be jealous: for brass players, the analogy of resonance tuning “comes with the instrument” but, while R1:f₀ tuning is easy for sopranos and altos in the high range, other resonance tunings are more difficult but may be worthwhile for vocal efficiency and stability.

* **Upstream loads.** In much of the trombone range, the effects of upstream impedance loads at the fundamental appear to be small for normal playing. Their effects on higher harmonics are very important in the didjeridu but much smaller in other lip-reed instruments. Measurements of analogous effects on the voice (antiformants or formants due to subglottal impedance peaks) are difficult to observe and not always consistent.

* **Source-filter interactions.** Brass players can glide the f₀ smoothly over the high impedance portion of a downstream resonance and thus play easily with either inertive or compliant loads. This requires changing the phase angle between the pressure in the mouthpiece and the acoustic current flow into the instrument bore. In turn, this involves altering the relative timing and magnitude of the sweeping flow, which depends on the timing of lip opening and the speed at which the lips enter the mouthpiece. However, players find it difficult to play where the impedance is low between resonances and the angle of the impedance is greater than about 70° in either direction.

Many singers who can sing high enough can glide the f₀ smoothly over an upstream resonance. Gliding f₀ up over a downstream resonance sometimes causes vocal instabilities, perhaps because the acoustic reactance suddenly changes sign, or because the impedance magnitude changes. However, many singers can glide f₀ smoothly over a downstream resonance; perhaps, it is the presence of a sweeping flow component that allows the phase angle between the downstream pressure and flow to be matched to the impedance of the vocal tract in a manner similar to that employed by a trombonist's lips.

Source-filter interaction by impedance matching is usually present in brass playing, and present in singing when resonance tuning is used. Singers who observe the pitch stability and output efficiency that resonance tuning gives brass instruments can be inspired to learn or to improve resonance tuning themselves, where it is available.

* **Source energetics.** In both the voice and brass instruments, the out-of-phase motions of the tissues in the longitudinal and transverse directions are important to the energetics of auto-oscillation. The power generated by the combination of the flow swept longitudinally by the tissues and the pressure across the source contributes power to maintain auto-oscillation, especially, but not only, for compliant acoustic loads. This contributes to the ability of brass players to slide easily from compliant to inertive downstream loads; it might also explain why some singers can glide smoothly across vocal tract resonances.

This aspect of voice source energetics is perhaps of particular interest to voice scientists, but could also be of interest to singers—and especially to singing teachers, who may occasionally be asked what makes the vocal folds vibrate.

## Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Joe Wolfe reports that financial support was provided by the Australian Research Council. If there are other authors, they declare that they have no
known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

We gratefully acknowledge the support of the Australian Research Council and thank all our volunteer musicians and singers. This work was conducted on Bidjigal country and we acknowledge their elders past and present.

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