© The Author(s), Published by EDP Sciences, 2024

1 Introduction

The duduk, a traditional Armenian instrument known for its deep and velvety timbre, is a double reed instrument with cylindrical bore, related to the Turkish mey and the Azerbaijani balaban. It gained popular recognition in the west through its use in film scores (cf. [1] Introduction).

In the general public, the duduk is often categorized as an oboe, which is contradicted by its bore shape, being cylindrical instead of conical. On the other hand, its bore may indicate a proximity to the clarinet, but this is contradicted by their differences in sound, in particular for high notes (cf. [1] p. 8).

As the instrument body is relatively simple, the part that singularizes the duduk is its reed which, among all reed instruments, is one of the largest. This clearly is a distinctive element as cylindrical instruments with smaller double reeds such as the Chinese guăn have a sound closer to that of the clarinet. The question is therefore to understand what makes the duduk reed unique.

The aim of the present article is the characterisation of the duduk reed using as few parameters as possible while keeping the main characteristic of the sound, which is his dark and velvety timbre. Such a model can then be used to understand which of these parameters are responsible for the timbre specificities. To achieve this, the first task is to describe its timbre and quantify its “darkness”. As a first step, this would make it possible to compare it to related and better studied instruments such as the clarinet or the oboe. Timbre is already the subject of a large literature that uses a wide range of tools. Owing to the high dimensionality of the notion of timbre, there is a plethora of descriptors related to its study (see e.g. [2]) among which it is often a difficult task to choose those most suitable, knowing that none of them capture the entirety of the notion of timbre. However, as the dark sound of the duduk has only few audible harmonics, it is possible to follow the evolution of each of them during musical gestures. This is therefore the selected representation in the present article.

Once the timbre is described, the next step is to get a proper model. The literature on the physics of the double reed is quite scarce as it is often considered to be close to the single reed. If this is partially true, the main difference comes from the geometry of the double reed and flow separation that can occur close to the tip (cf. [3–5]). Far from negligible, these effects can be measured (see [6]) and some refined models have been proposed to take into account the double reed peculiarities (cf. [7, 8]). Although these models are necessary to get a clear picture of the physics, we chose to keep a model as simple as possible, as long as it can reproduce the characteristics of the duduk sound. In particular, the reed parameters used in the present article are the four emerging from the three equations reed model detailed in Section 9.2.5.1 of [9] together with the input impedance of the resonator.

Our second goal is to calibrate these reed parameters. This is achieved using an impedance sensor and a new type of artificial mouth dedicated to the duduk. Various properties of the three equations model are then used to deduce the parameters using continuation techniques along the lines of [10]. The literature on reed parameters calibration is quite large, although mostly focused on single reed instruments (cf. e.g. [11–16]) and most of them are quite advanced by comparison to the one used in the present article.

The duduk geometry is described in Section 2 and its timbre is analysed in Section 3. Section 4 is dedicated to measurements of the duduk physical properties, such as its input impedances (for different fingerings), intonation characteristics, etc. Part of the data is obtained using a duduk specific artificial mouth, which is also described in this section. Section 5 focuses on a physical model and the calibration of the reed parameters it involves. Finally, Section 6 compares the experimental data obtained in Section 3 to those from the calibrated model in Section 5.

2 Description of the duduk

2.1 Physical description

The traditional duduk is a double reed wind instrument with a cylindrical bore having 10 holes, two on the back of the instrument and eight on the front, giving it a range of one octave and a fourth. The range can be extended in the upper register by a whole tone by squeezing the reed with the lips.

The duduk body is usually made of apricot wood and comes in many sizes ranging from piccolo in F (26 cm) to bass in A (64 cm). The most common duduk is the duduk in A (cf. [1] p. 115) and the dimensions of a typical duduk can be seen in Figure 1, together with its reed attached.

Figure 1 Description of an A duduk (maker: Simon). Measurements are given in millimetres.

The holes are all of the same size but their position along the cylinder is quite complex, as they have to accommodate for the reed (see Sect. 4.2 on impedance for more information). Moreover they have to be undercut to improve intonation.

2.2 The reed

The most prominent part of the duduk, and the one with the most variability, is its double reed known as ghamish, which is made of a single piece of bamboo and whose manufacture involves many complex stages (see [17]). Each reed is built only for a particular duduk model. A typical reed for an A duduk is displayed in Figure 2. It is 10.5 cm long, conical in shape at the one end that fits into the body, and flattened at the other. The sides of the flattened end are reinforced with tape, and a movable regulator is kept on the reed. Its purpose is to adjust the reed opening: when the regulator is pushed toward the body, it closes the reed and increases the pitch. Pushed the other way, the effect is opposite.

Figure 2 Typical reed for a duduk in A.

The duduk is a notoriously difficult instrument to play in tune (cf. [1] p. 77). This can be explained in part by the complexity of the reed, its variability due to its construction and the fact it is made from an organic material. Indeed it is impossible to tune one duduk for all the reeds, and the musician has to compensate the intonation for each ghamish.

The reed is sensitive to humidity, and must be prepared before playing. As the hygrometry increases during playing, the regulator position must be adjusted to compensate the propensity of the reed to open. For the duduk reed, the effects are manifold, and the study of the variation of its physical parameters with hygrometry is therefore a difficult question (see e.g. [18–21]). However, we can say that:

• The reed is swelling (more in the transverse direction cf. [18] p. 246) and due to its shape and the bonding on the sides, it opens and bends to form an arc, which changes the stiffness. This also increases the reed volume.

• The Young modulus decreases (see [19] Fig. 3 for the longitudinal direction) and the density increases (cf. [19] Table I) with relative humidity.

• According to [21] Section 6.3, “The damping factor seems to remain constant regardless of relative humidity.”

2.3 Playing technique

The duduk is played with natural fingering (cf. Tab. 1), half notes and microtones are played either by closing partially some holes, or pinching the reed.

It is possible to play one tone higher by pressing the lips, and experienced musicians can play even one semi-tone above.

The lips are not curled inward and are positioned as close as possible to the tip, which is a relatively new trend (cf. [1] p. 116). This position allows for a precise control of the reed opening, which is necessary to correct intonation and timbre.

3 Analysis of the produced sound

The recordings were made in a studio (7 m × 7 m × 4.5 m) using a Neumann U87 microphone positioned 1.5 m in front of the musician’s mouth,with a professional duduk player trained in the traditional style.

3.1 Playing range

The musician was asked to play a bend as wide as possible for each fingering. An example is given for the fingering C+1 in Figure 3, where the top diagram represents the playing frequency, and the bottom one the RMS value of radiated sound. For this fingering, the range is around 230 cents up but is rarely used as such in a musical context as it is barely audible at its highest frequency. The bending ranges for all the fingerings are given in Figure 17.

Figure 3 Evolution of the playing frequency (top) and dynamics (bottom) of the duduk during a bend played mezzo for the fingering C+1.

This can be compared to the range of the modern clarinet that can be found for example in [22]: at most 70 cents for the Bflat clarinet, and 100 cents for the Oboe.

3.2 Timbre of a chromatic piano scale

The top part of Figure 4 represents the spectrogram of the complete chromatic scale of an A duduk, played piano, where the amplitude of each note has been normalized independently from the others. The fundamental is clearly the dominant part of the spectrum for each note. The second harmonic is quite small for low notes, as expected for a cylindrical resonator, and increases for the higher ones. The third harmonic has an inverse evolution, it is higher for low notes and almost inexistent for higher ones. Almost nothing is visible for higher harmonics.

Figure 4 Evolution of the duduk spectrum played piano for all the notes of the chromatic scale. Top: spectrogram, colour is proportional to the RMS value in decibel, bottom: relative RMS value contained in the harmonics, normalised by the total RMS value of the signal at that time. The non diatonic notes (played with half holes) are shown with pink background.

Although the spectrogram gives a lot of information, it is more qualitative than quantitative, and we need more precise data when comparing model and measurements. Owing to the signal periodicity and fast decay of the amplitude of the Fourier coefficients in the duduk spectrum (as can be seen in the spectrogram), it is possible to follow precisely the evolution of audible harmonics during musical phrases. For this, the Yin algorithm (cf. [23]) is used to extract the instantaneous frequency at each time, together with the corresponding waveform. The energy stored in each harmonic is then computed and normalised relative to the total waveform energy. The evolution of normalised audible harmonics are displayed in the bottom part of Figure 4, below the spectrogram, for the first five harmonics.

One striking result is that the energy is contained almost entirely in the fundamental, all the other harmonics are more than 10 dB below. This implies that the spectral centroid is close to the fundamental, which translates in a dark timbre. Moreover, removing the notes that are played with half holes (represented with pink background in Fig. 4) for which the musician’s breath is much more audible than for the other notes, we see that the harmonics are all below 20 dB of the fundamental up to the fingering F. From that note upward, the harmonic H₂ gains more prominence, and is audible, but poorly significant, for the highest notes.

3.3 Effects of the dynamics on timbre

The musician was asked to perform a crescendo for each fingering to measure the effect of the dynamic on timbre. The results are displayed in Figures 15 and 16 for the fingerings C and C+1, where they are compared to their theoretical versions. For the lowest note (fingering C, top of Fig. 15), up to an RMS value of 8 dB above the starting level, higher harmonics except H₂ follow a power law, which is reminiscent of Worman’s law (cf. [24] Section 21.3, although those are obtained for an internal pressure). As in the case of a clarinet, the timbre then gains many higher harmonics. Even when the sound is highly timbred, the fundamental is still 5 dB above the next harmonic.

The evolution of timbre for fingering C+1 (top of Fig. 16) is quite different from that of a clarinet, even though it seems to follow the same linear evolution up to 8 dB where H₁ and H₂ are the only audible harmonics.

4 Measurements

4.1 Opening at rest

Because of its dependency to hygrometry (cf. Sect. 2.2) the reed opening is not well defined as it has to be adjusted regularly during playing using the reed regulator until an hygrometric equilibrium is attained and the player is satisfied with timbre and intonation.

The first step for measuring the reed opening is therefore to prepare into playing conditions by putting some water into the reed (never putting the reed in water) and then playing for some time. This process is difficult to do in a controlled way, even more so because the artificial mouth used in the experiment blows dry air, which dries, and therefore closes, the reed (cf. Sect. 4.4).

Once prepared, the opening area is estimated at 5.4 mm² using the image shown in Figure 5 and its width w is measured at 18 mm.

Figure 5 Typical reed opening during playing.

In the model, the opening section is considered to be rectangular, so that H is actually the opening mean along the width of the reed. Therefore H ≅ 0.3 mm is a typical value that ensures the duduk can produce a sound. Although the estimation on H using Figure 5 comes with an error of around 5% only, the uncertainty is actually greater as the reed opening changes with humidity. To be coherent with the other measurements, we therefore estimate the error margin for H to be around 30%, which gives a range of reed openings where the instrument produces a sound.

4.2 Hybrid impedance

The duduk full impedance is not readily accessible, as the reed cannot be easily included in the measurements with the impedance sensor, and cannot be modelled as a simple volume as its size is not small compared to that of the duduk body. It is therefore necessary to define a geometric model, which is used to build a hybrid model together with the measured impedances of the duduk body.

The reed model is composed of a conical shape near the duduk body, followed by a cylinder, cf. Figure 6. The complete impedance is then built in three steps:

• The impedance of the body is measured for each fingering using the CTTM impedance sensor (cf. [25]).

• The impedance of the part of the reed fitting in the duduk body is removed (green part of Fig. 6).

• The impedance of the interior of the reed is added.

Figure 6 Reed model used for the computation of impedance. The green part represents the reed part inserted in the body. Measurements are given in millimetres.

The removing and adding of impedances in the last two steps are done using the projected impedance formula (cf. [9] formula 4.29).

Typical impedance curves are displayed in Figure 7 for the fingerings G−1, G and C+1. For each of them, the value of the impedance at frequency twice the fundamental has a negligible magnitude compared to the peaks. This result is expected for the low notes (e.g. G−1) as all the holes are covered and the resonator is close to a cylinder, so that its impedance has peaks at odd multiples of the fundamental. However this is not so clear for the higher notes as the cylindrical part of the resonator is shorter.

Figure 7 Absolute value of the hybrid impedances for three different fingerings, one low (G−1, bottom row), one in the middle range (G, middle row) and one high (C+1, top row). The frequency of the fundamentals are given by dotted red lines, and their second and third harmonics in dotted green and blue lines.

For the low notes, the second peak frequency is relatively close to three times the fundamental, which is also expected for a cylindrical resonator. The resonator is thus closer to a simple tube like the clarinet. This changes for the high notes where the second peak shifts toward higher frequencies. The reed is therefore more important in the resonator. To get a clearer picture of what happens, Figure 8 displays the inharmonicity of the second peak frequency relatively to the fundamental, for the duduk and for the clarinet low register [26].

Figure 8 Inharmonicity of the impedance second peak relative to the first peak for each fingering (blue dots). For comparison, equivalent data for the clarinet are shown (orange crosses, from [26]).

Although the spectrogram from Figure 4 is drawn from a radiated sound pressure, and therefore cannot be linked easily to the impedance (cf. however Sect. 6.1), it is compatible with the fact that inharmonicity increases in Figure 8 with frequency together with a decrease of magnitude of H₃.

Same can be said for the magnitude of H₂ as it increases with frequency on the spectrogram, whereas hybrid impedances at twice the fundamental frequency diverge from the antiresonance (cf. Fig. 7).

4.3 Artificial mouth

Measurements are made using an artificial mouth built specifically for the duduk (see Fig. 9). It is composed of a latex membrane fixed on an aluminium barrel connected to a controlled air supply. The latex membrane is able to create an almost air tight joint at the reed tip with little force applied on it. Two pressure transducers Endevco 8507C-5 are used to measure the mouth and reed pressures. Adhesive paste is used to close holes of the duduk so that a complete diatonic scale can be played by removing them.

Figure 9 Duduk mounted on the artificial mouth, together with the air supply and the two pressure transducers.

4.4 Closing reed pressure

The closing reed pressure is the minimum static mouth pressure needed to close the reed. The procedure to obtain it is the following:

• first increase the mouth pressure until the reed closes,

• from that point, decrease slowly the mouth pressure until the reed opens, producing a sound.

This method is applied using the artificial mouth. The recording is displayed in Figure 10, the closing reed pressure is estimated at P_cr = 3500 Pa

Figure 10 Description of the closing reed pressure measurement : the mouth pressure is slowly increased (in green) until the reed pressure (in orange) drops to 0. Then the mouth pressure is decreased (in red) until the systems produces sound at time t0 (dashed gray line). The closing reed pressure Pcr is the mouth pressure at time t0 (dashed blue line).

Due to the reed drying, the value of P_cr is not constant. As its measurement is quite fast compared to the evolution of the reed hygrometry, the procedure can be done multiple times before the reed closes completely. This is used to monitor the evolution of the closing reed pressure as a function of time. Result is displayed in Figure 11, together with its linear regression that has an R² value of 0.98, indicating a linear dependence with respect to time.

Figure 11 Evolution of the closing reed pressure with time (blue dots), together with linear regression (dashed orange line).

As the measurements are mainly done after some delay since the reed preparation and the opening of air supply, we took a typical value of P_cr = 4500 Pa.

4.5 Intonation

For each fingering, the duduk is played with the artificial mouth with constant pressure, adhesives are removed one by one and playing frequencies are recorded. The complete process lasts for about 50 s which, according to Figure 11, implies a change in closing reed pressure around 30%. Playing frequencies are drawn in Figure 12 with respect to the impedance first peak frequency for the corresponding fingering (see Sect. 4.2).

Figure 12 Playing frequency for each fingering, drawn with respect to the first peak frequency of the corresponding input impedance. Surrounded in green are the notes following the expected linear relation, and in red the three highest notes.

For the low octave, region from G−1 to G surrounded by the dotted green curve in Figure 12, the playing frequency follows the impedance first peak frequency. This is the expected relation for reed instruments, cf. for example [26] for the clarinet.

However, for the three highest notes (surrounded by the dotted red curve) this relation is clearly false, and the input impedance is no longer sufficient to determine the playing frequency. This means that, contrary to the clarinet, the reed dynamic plays an important role for the duduk intonation [10].

5 Physical model

The model chosen for the duduk is given by the three equations presented in (1) (cf. Sect. 9.2.5.1 of [9]) linking together the reed opening h, the mouth pressure p_m, the reed pressure p and the flow through the reed u.

$$\{\begin{array}{l}\ddot{h}+\frac{{\omega }_{\mathrm{r}}}{Q_{\mathrm{r}}}\dot{h}+{\omega }_{\mathrm{r}}^2\left(h-H\right)=-\left(p_m-p\right)\frac{{\omega }_{\mathrm{r}}^2}{K}\\ \widehat{p}\left(\omega \right)=Z\left(\omega \right)\widehat{u}\left(\omega \right)\\ u=w{h}^{+}\mathrm{sgn}\left(p_m-p\right)\sqrt{\frac{2\left|p_m-p\right|}{\rho }}\end{array}.$$(1)

The first equation represents the reed seen as a one degree of freedom oscillator, with H the reed opening at rest, Q_r its quality factor, K its stiffness and ω_r its frequency. The second equation is the definition of the impedance written in the Fourier domain, and Z has the form

$$Z\left(\omega \right)=Z_{\mathrm{c}}\sum _n\mathrm{}\frac{C_n}{\mathrm{j\omega }-s_n}$$(2)

where Z_c represents the characteristic impedance $Z_{\mathrm{c}}=\frac{\rho c_0}{\pi R^2}$ with ρ = 1.2 kg · m ⁻³, c₀ = 340 m · s⁻¹ and R is the resonator entry radius. Values of s_n and C_n are either obtained by fitting the impedance in the case of experimental impedance (cf. Sect. 4.2) or from theory in the case of a tube (cf. Sect. 5.1).

The last equation is the description of the valve effect obtained through Bernoulli’s equation, where w is the reed width, h⁺(t) = max (h(t), 0) (so that wh⁺(t) represents the opening section when h(t) ≥ 0) and ρ is the air density. According to Sections 4.1 and 4.4 we have H ≈ 0.3 mm and P_cr ≈ 4500 Pa.

From the relation P_cr = KH the reed stiffness is estimated at K ≈ 15 MPa · m⁻¹ depending on humidity.

Because of the potential change in the flow direction which happens when H grows, the last equation must be regularised along the lines of [27], by replacing the square root:

$$\mathrm{sgn}\left(p_{\mathrm{m}}-p\right)\sqrt{\left|p_{\mathrm{m}}-p\right|}⟶\frac{p_{\mathrm{m}}-p}{\sqrt[4]{(p_{\mathrm{m}}-p{)}^2+\epsilon P_{\mathrm{cr}}^2}}$$(3)

where ε is a small dimensionless parameter, taken for the simulations to be 10⁻⁶.

An online real time simulation tool using this model and the physical parameters for the duduk is accessible at https://perso.univ-lemans.fr/~smauge/Acoustics/yapsmi.js/instruments/duduk/.

5.1 Estimation of the reed frequency

The reed frequency can be estimated by studying its effect on the evolution of the playing frequency F_play with respect to the frequency F₁ of the input impedance first peak. Following [10], the duduk is replaced by a cylinder of varying length L so that F₁ becomes a continuous parameter whose influence on the playing frequency can be studied using the continuation software auto-07p (cf. [28]).

Consider a cylinder of length L (varying) of radius R = 6. If c₀ denotes the speed of sound in air, a simplified model for the resonance modes in the cylinder is given by

$$\begin{array}{cc}{C}_n=c_0/L& s_n=j2\pi f_n+c_0\alpha \left(f_n\right).\end{array}$$(4)

with $f_n=\frac{(2n-1)c_0}{4L}$ (cf. [19] Eq. (5.147)) where α(f_n) is the absorption coefficient in the tube for which a good approximation is given by

$$\alpha \left(f\right)=3·10^{-5}\frac{\sqrt{f}}{R}.$$(5)

According to [9] equation (12.133), radiation losses are of the order of (2πf_nR/c₀)², which is much smaller than visco-thermal losses given by α(f_n)L for low frequencies. Radiation losses are therefore neglected in the model. The software auto-07p is then used to perform a Linear Stability Analysis (LSA) of the the regularised system (1).

Here, the point of interest in the LSA is the first Hopf bifurcation: it is the destabilisation threshold. Taking this threshold as the basis for the playing frequency F_play ensures that only the first mode of the impedance is prevalent. In particular, the differences between the duduk and the cylinder are minimized. The sum in (2) can therefore be taken over only two modes.

As already two reed parameters (H and K) have been fixed in the previous sections, only two remains. To study the reed frequency F_r, we fix a temporary value of the quality factor Q_r = 3 within the range of what is known in the literature (cf. [29]).

The playing frequency F_play is drawn for different reed frequencies F_r in Figure 13 with respect to F₁ for a cylinder of varying length. It is obtained with auto-07p by first finding the threshold pressure for one value of F₁, and then continuing this Hopf bifurcation with a varying F₁.

Figure 13 Playing frequency Fplay of the cylindrical model as a function of the input impedance first peak F1 for a cylinder of varying length, for Qr = 3 and three different reed frequencies, from bottom to top: Fr = 460 Hz in red, F1 = 500 Hz in green and F1 = 480 Hz in blue. The black line represents the equality Fpray = F1, and the dash-dotted lines represent the maximum playing frequency given by the reed frequencies.

The curves in colour are close to the black line F₁ = F_play for frequencies up to 300 Hz and begin to diverge after that point. The playing frequencies then tend toward a maximum specified by reed frequency (dash-dotted line).

From Figure 13, we see that playing frequency cannot be greater than the reed frequency, so that F_play < F_r for every playable frequency, and therefore F_r > 471 Hz. Moreover, the curvature of the blue curve, obtained for F_r = 480 Hz, gives the closest match to the experimental data (blue dots).

To obtain Figure 13, the quality factor had to be fixed, but it can be shown that it has negligible impact on the intonation diagram compared to that of the reed frequency.

The estimation of F_r depends on the value of K which comes therefore with some uncertainty. It can be shown that F_r is actually a function of the dimensionless parameter

$$\zeta =w{Z}_{\mathrm{c}}\sqrt{\frac{2}{\rho P_{\mathrm{cr}}}}$$(6)

which is used in auto-07p to solve the model (cf. [27] Appendix B). This equation implies that if P_cr comes with an uncertainty of 30%, that of F_r should be around 15% at most, and probably less as the playing frequency cannot be higher than the reed frequency.

This low reed frequency explains why the duduk has no higher registers, as it represents the maximum playable frequency.

5.2 Estimation of the quality factor

The quality factor is estimated by fitting the theoretical bifurcation diagram obtained with auto-07p with the experimental one.

In Figure 14, both types of bifurcation diagrams are shown for the fingering C+1. This fingering was chosen because it has the shortest body length, and the influence of the reed parameters should be maximal. The plain curve represents the measurements done with the artificial mouth by doing a decrescendo from high pressure. As the decrease in mouth pressure is performed relatively fast (around 8 s) compared to the drying of the reed (cf. Fig. 11), all the physical parameters are supposed to be constant: according to Figure 11, we can expect a decrease of the closing reed pressure around only 300 Pa.

Figure 14 Experimental and theoretical bifurcation diagrams for fingering C+1 (top) and C (bottom). The dashed lines represent the bifurcation diagrams for different values of the reed quality factor obtained with physical parameters H = 0.30 mm, K = 15 MPa · m−1, Fr = 480 Hz and, from right to left, Qr = 2.3, 2.8, 3.3, 3.8. The plain lines represent the measurements.

The dashed curves on the figure represent the theoretical diagrams for different quality factors. They are obtained by continuation along the parameter p_m using auto-07p with the hybrid input impedance of Section 4.2.

Figure 14 shows that the threshold pressure is a decreasing function of the quality factor. For all values of Q_r, there is a notable discrepancy between theoretical and experimental bifurcation diagrams for fingering C + 1. This may be due to the fact that the reed is the predominant part of the resonator for fingering C + 1, and to the hybrid model of input impedance which is too coarse.

For the low note C, the dynamic is well respected whatever the theoretical quality factor, and it seems the best choice would be around Q_r = 2.3. However, as the reed quality factor does not change with fingering, we fix Q_r = 3.3 as a compromise that is not too bad for low notes and keeps a good fit of the threshold pressure for fingering C+1.

Drawing equivalent diagrams to Figure 14 for extreme value of K and F_r gives a variation range for Q_r around 25%. A precise value of Q_r is actually difficult to draw from this experiment, but we found that taking it between 2.3 and 3.8 do not change qualitatively the results obtained hereafter.

6 Comparison of the model with experiments

The reed parameters values used in the following section are those obtained previously using an average closing reed pressure, and are summarised in Table 2. They are used together with the hybrid impedances computed in Section 4.2 to compare the model (Eq. (1)) to the recordings detailed in Section 2.

All the figures obtained in this section are computed using the continuation program auto-07p to follow the evolution of a periodic solution with respect to one particular parameter. From these simulations, the normalised audible harmonics (cf. Sect. 3.2) are computed and drawn together with the evolution of frequency, in a similar fashion as diagrams of Section 2.

6.1 Radiated sound

The goal of this section is to compare the radiated sounds produced by the musician with those coming from the model, it is therefore necessary to have a model for the transfer function for each fingering. This is achieved by considering the duduk as a cylinder of radius R and length L comprising the reed and the duduk body up to the first opened hole.

The transfer matrix is then of the form

$$\left(\begin{array}{l}\widehat{p}\\ \widehat{u}\end{array}\right)=\left(\begin{array}{ll}\cos \left(k_{\mathrm{c}}L\right)& j{Z}_{\mathrm{c}}\sin \left(k_{\mathrm{c}}L\right)\\ \frac{j\sin \left(k_{\mathrm{c}}L\right)}{Z_{\mathrm{c}}}& \cos \left(k_{\mathrm{c}}L\right)\end{array}\right)\left(\begin{array}{l}{\widehat{p}}_{\mathrm{out}}={\widehat{u}}_{\mathrm{out}}{Z}_{\mathrm{r}}\\ {\widehat{u}}_{\mathrm{out}}\end{array}\right)$$(7)

with $k_{\mathrm{c}}=\frac{\omega }{c_0}+(1-j)\alpha \left(f\right)$ the complex wave number (α being given by equation (5), and Z_r the radiation impedance. According to [9], equation (12.133), a good approximation of Z_r is

$$Z_{\mathrm{r}}=Z_{\mathrm{c}}(0.6133\mathrm{jkR}+(\mathrm{kR}{)}^2/4)$$(8)

with $Z_{\mathrm{c}}=\frac{\rho c}{\pi R^2}$ the characteristic impedance and $k=\frac{\omega }{c_0}$ the real wave number. Therefore

$${\widehat{u}}_{\mathrm{out}}=\frac{\widehat{p}}{Z_{\mathrm{r}}\cos \left(k_{\mathrm{c}}L\right)+j{Z}_{\mathrm{c}}\sin \left(k_{\mathrm{c}}L\right)}$$(9)

Under the assumption that the tube radiates as a monopole and using equation (12.12) of [9] describing the far field pressure at a distance r from the tube end, we get

$$\mathrm{j\rho }\omega \frac{\widehat{p}}{Z_{\mathrm{r}}\cos \left(k_{\mathrm{c}}L\right)+j{Z}_{\mathrm{c}}\sin \left(k_{\mathrm{c}}L\right)}\frac{e^{-\mathrm{jkr}}}{4\pi r}.$$(10)

6.2 Mouth pressure and crescendo

The musician’s crescendos are studied here as a simple increase in mouth pressure. In a real situation, it is probable that the musician changes many reed parameters during a crescendo, for example to correct the intonation. Together with the simplicity of the model and the hybrid impedance, this implies that we cannot expect a perfect fit between experiment and theory, especially for higher harmonics.

Normalised audible harmonics of the radiated pressure are shown in Figure 15 for the fingering C (low note). The top figure represents the measurements detailed in Section 3.3, and the middle one the theoretical diagram obtained from the model given by equation (1) with the parameters of Table 2, and filtered through the transfer function given in equation (10). The x-axis origin is fixed to best follow that of the measurements. For comparison, the bottom figure represent a duduk with a clarinet reed model obtained from the duduk model by changing the reed impedance (cf. Sect. 4.2), increasing the reed frequency to 2000 Hz, the quality factor to 10 and keeping all the other parameters unchanged. Note that neither the reed nor the tube geometry are optimised for a clarinet, so that a perfect fit with a clarinet model is not expected. For this duduk with a clarinet reed model, the bifurcation diagram shows that the branch originating from the first Hopf bifurcation begins with unstable solutions. As a consequence, there is no pianissimo dynamics in the bottom row of the figure, and the minimum playing pressure cannot be used as a reference for the x-axis. Therefore, for the duduk with a clarinet reed model, the RMS value reference is chosen so that the beating reed threshold aligns with the others, around 8 dB. For this low fingering, the reed part in the hybrid impedance has less influence than the cylindrical body. The timbre evolution is therefore similar to what is expected for a clarinet, that is a growth of the harmonics amplitude as the pressure to the power of the harmonic order (known as Worman theorem, see [9] p. 511). Both top and middle diagrams of Figure 15 give compatible evolutions for the first three harmonics H₁, H₂ and H₃: for low pressures, H₁ dominates all the others and H₃ increases regularly up to a point where it is about 5 dB lower than H₁. Harmonic H₂ is constantly 15 dB smaller than H₁. Note that the radiated even harmonics (dashed lines in Fig. 15) are actually difficult to model precisely as they depend on the transfer function, which is highly simplified in our case. The duduk with a clarinet reed model shown in the bottom row of the figure, has much more prominent higher odd harmonics, which is compatible with the behaviour of the clarinet for low notes (see [30], Sect. 3.2.3), even though the duduk with a clarinet reed model is not a faithful clarinet model. Moreover, even for small dynamic, H₂ is only 5 dB lower than H₁.

Figure 15 Normalised audible harmonics of the radiated sound pressure for the fingering C. The top row represents the measurements, and the reference of the horizontal scale is the RMS value of the recorded pressure at the beginning of the recording. The middle row represents the model and the reference of the horizontal scale is an arbitrary RMS value chosen so that both diagrams have the same H1 at 14 dB. The bottom row represents the duduk with a clarinet reed model, and its x-axis is normalised to have the same beating reed threshold as the other rows.

An equivalent diagram is shown in Figure 16 for the fingering C+1 (high note). For this high fingering, the reed part of the hybrid impedance is more dominant than for low notes. Because of the approximate nature of our the reed impedance model, more differences are to be expected. The first two harmonics of the first two rows, representing the experiment and the duduk model, have similar evolutions and amplitudes, while higher harmonics are less well described. Note that for low dynamics, before the beating reed threshold that occurs between 8 dB and 10 dB, the spectrum of the radiated sound contains mainly H₁ and H₂ in both top and middle rows, except maybe for H₄ and H₆ but, as stated earlier, the even harmonics of the radiated sound are difficult to predict. In comparison, the bottom row has a much more pronounced change of timbre after the beating reed threshold, with H₅ becoming the prominent contribution.

Figure 16 Normalised audible harmonics of the radiated sound pressure obtained from the model for the fingering C+1. See Figure 15 for a description of the scales.

6.3 Reed opening and bend

To bend a note, the musician presses his lips to close the reed. This probably changes many reed parameters such as the reed opening, and the reed stiffness (because of the added mass). As the change in stiffness is difficult to measure we chose to focus only on the reed opening.

For each fingering, the playing frequency range is computed using continuation. First, auto is initialised to a solution with a specified mouth pressure (50% of the closing reed pressure) with the reed parameters given in Table 2. Then the reed opening H is decreased until the reed pressure drops to 0.

The theoretical bending range for each fingering is shown in Figure 17, together with the measurements extracted from the recordings done with the musician.

Figure 17 Evolution of the duduk playing frequency range during a bend played mezzo for all the fingerings. Physical duduk in blue crosses, and model in orange dots.

The evolutions of bending range are similar for the model and for the duduk up to the fingering A+1. The gap of up to 50 cents could be explained by the differences of duduk and reed, changes in the musician’s mouth pressure, etc.

The bending range in the simulation becomes smaller for the last two notes which is not the case in practice. This reveals the approximate nature of our model.

7 Conclusion

The present study proposes a minimal model for the duduk, together with a full set of reed parameters, that is able to reproduce qualitatively some characteristics of the duduk, namely its intonation and the strong predominance of the first harmonic in its spectrum across each fingering and dynamic. In particular, it highlights its unusually low reed frequency compared to other instruments, which seems to be the main cause for the dark timbre of the duduk. This reed resonance frequency induces a non linear relation between playing frequency and inverse of the resonator length far from the clarinet model and is a consequence of the reed manufacture, notably its size.

So far, the model is able to simulate quite well the duduk dynamic and intonation, and approximate parts of the timbre and playing range. To go further, the geometrical reed model should be refined, and the physical model should include the flow induced by the reed motion, vortices occurring at the reed tip and the pressure recovery (cf. [8]). To improve the model accuracy, measurements should also be improved, most notably by using wet air that keep the reed characteristics stable. Finally, it would be interesting to have a more precise way to follow the musician’s technique and the implied equivalent reed parameters, for example using an instrumented mouthpiece (cf. [31]). This could provide a quantitative explanation for the bend, which may depend on more than one parameter to produce such a large range of playing frequencies.

Acknowledgments

The authors wish to thank Christophe Vergez and Tom Collinot for their help, the “Plateforme MAS” in Marseille for providing the studio and the recording gear, and the duduk player Michaël Vemian. We would also like to thank the referees for their careful reading and helpful remarks.

Conflicts of interest

The authors declare that there is no conflict of interest.

Data availability statement

The data are available from the corresponding author on request.

References

All Tables

All Figures

	Figure 1 Description of an A duduk (maker: Simon). Measurements are given in millimetres.
In the text

	Figure 2 Typical reed for a duduk in A.
In the text

	Figure 3 Evolution of the playing frequency (top) and dynamics (bottom) of the duduk during a bend played mezzo for the fingering C+1.
In the text

	Figure 4 Evolution of the duduk spectrum played piano for all the notes of the chromatic scale. Top: spectrogram, colour is proportional to the RMS value in decibel, bottom: relative RMS value contained in the harmonics, normalised by the total RMS value of the signal at that time. The non diatonic notes (played with half holes) are shown with pink background.
In the text

	Figure 5 Typical reed opening during playing.
In the text

	Figure 6 Reed model used for the computation of impedance. The green part represents the reed part inserted in the body. Measurements are given in millimetres.
In the text

	Figure 7 Absolute value of the hybrid impedances for three different fingerings, one low (G−1, bottom row), one in the middle range (G, middle row) and one high (C+1, top row). The frequency of the fundamentals are given by dotted red lines, and their second and third harmonics in dotted green and blue lines.
In the text

	Figure 8 Inharmonicity of the impedance second peak relative to the first peak for each fingering (blue dots). For comparison, equivalent data for the clarinet are shown (orange crosses, from [26]).
In the text

	Figure 9 Duduk mounted on the artificial mouth, together with the air supply and the two pressure transducers.
In the text

	Figure 10 Description of the closing reed pressure measurement : the mouth pressure is slowly increased (in green) until the reed pressure (in orange) drops to 0. Then the mouth pressure is decreased (in red) until the systems produces sound at time t0 (dashed gray line). The closing reed pressure Pcr is the mouth pressure at time t0 (dashed blue line).
In the text

	Figure 11 Evolution of the closing reed pressure with time (blue dots), together with linear regression (dashed orange line).
In the text

	Figure 12 Playing frequency for each fingering, drawn with respect to the first peak frequency of the corresponding input impedance. Surrounded in green are the notes following the expected linear relation, and in red the three highest notes.
In the text

Figure 13 Playing frequency Fplay of the cylindrical model as a function of the input impedance first peak F1 for a cylinder of varying length, for Qr = 3 and three different reed frequencies, from bottom to top: Fr = 460 Hz in red, F1 = 500 Hz in green and F1 = 480 Hz in blue. The black line represents the equality Fpray = F1, and the dash-dotted lines represent the maximum playing frequency given by the reed frequencies.

In the text

	Figure 14 Experimental and theoretical bifurcation diagrams for fingering C+1 (top) and C (bottom). The dashed lines represent the bifurcation diagrams for different values of the reed quality factor obtained with physical parameters H = 0.30 mm, K = 15 MPa · m−1, Fr = 480 Hz and, from right to left, Qr = 2.3, 2.8, 3.3, 3.8. The plain lines represent the measurements.
In the text

Figure 15 Normalised audible harmonics of the radiated sound pressure for the fingering C. The top row represents the measurements, and the reference of the horizontal scale is the RMS value of the recorded pressure at the beginning of the recording. The middle row represents the model and the reference of the horizontal scale is an arbitrary RMS value chosen so that both diagrams have the same H1 at 14 dB. The bottom row represents the duduk with a clarinet reed model, and its x-axis is normalised to have the same beating reed threshold as the other rows.

In the text

	Figure 16 Normalised audible harmonics of the radiated sound pressure obtained from the model for the fingering C+1. See Figure 15 for a description of the scales.
In the text

	Figure 17 Evolution of the duduk playing frequency range during a bend played mezzo for all the fingerings. Physical duduk in blue crosses, and model in orange dots.
In the text