Acoust. Sci. & Tech. 38, 5 (2017) #2017 The Acoustical Society of Japan PAPER
Numerical simulation of the sawari tone on Chikuzen biwa
Tomoyasu Taguti Taguti Laboratory of Computation and Analysis, 2–16–1 Katsuragi, Kita-ku, Kobe, 651–1223 Japan (Received 17 August 2016, Accepted for publication 24 February 2017)
Abstract: The biwa, a family of Japanese lutes played with plectrum, produces a tone of peculiar timbre called the sawari tone caused by dynamic contact of the vibrating string with a small instrumental device called the sawari. This paper presents a physical model consisting of a linear string-resonator system to which the sawari is added so as to work as a unilateral constraint to the vibrating string. Included also is a stick-slip mechanism of plucking to simulate the so-called touch noise that exists as a precursor in the biwa tones (and in the tones of some other string instruments played with plectrum.) A numerical simulation was conducted in search for the condition that produces sawari-like tones as characterized by a unique temporal profile of the total amplitude and its constituting partials. The result showed that the system produces simulated tones that resemble a genuine sawari tone, called the reference tone, when the size of sawari is about the same as that of the instrument on which the reference tone was produced, under the condition that the model string is plucked at a point belonging to a specific portion off the bridge.
Keywords: Plucked string, Dynamic contact, Energy transfer among the partials, Touch noise by stick-slip plucking PACS number: 43.75.Gh 43.75.St 43.25.Ts [doi:10.1250/ast.38.237]
at both ends [2] showed that a small sawari surface 1. INTRODUCTION introduces rich overtones that do not exist initially. Here, The biwa is a family of Japanese lutes made of thick the model was formulated on the basis of two-dimensional and solid wood and played with plectrum. It has a long motion of string in the sense the string is confined to move history since the 8th century as known by the evidence of ‘vertically’ or in a plane perpendicular to the table of biwa. several ancient Gagaku biwas for Court music preserved in This formulation is considered reasonable for the study on the Shosoin Treasure House. the mechanical effect of sawari since the sawari works as a In the modern age, Satsuma biwa (since 16th c.) and unilateral constraint to the vibrating string in the direction Chikuzen biwa (since 19th c.), among others, have become perpendicular to the table. equipped with an instrumental device called the sawari. On the basis of the above two results, the present paper The sawari is a surface of very small size created on top of studies the effect of sawari on the production mechanism the nut and all frets, producing an instrumental sound of of the sawari tone numerically with a modified version of peculiar timbre, called the sawari tone, due to dynamic [2] in the following respects: (i) a linear damping in the contact with the vibrating string. string, (ii) a sawari surface with or without a partial An experimental study on the waveform of sawari depression, (iii) a resonator that terminates one end of the tones with a Chikuzen biwa [1] revealed that the partial’s string, and (iv) stick-slip dynamics in the plucking motion. amplitudes change undulatingly with time and unevenly The plucking generally induces a ‘horizontal’ component among the partials, where some of them are even growing of string motion (in the direction parallel to the table) in for a short timespan; as a result, the total amplitude addition to the vertical one. However, we shall formulate changes undulatingly with time and stays above a certain the model in a two-dimensional framework by neglecting magnitude before an eventual decay, differing from a the existence of the former for the same reason as in [2]. simple exponential-like decay significantly. With this model we shall conduct a parametric study in Then, a numerical simulation on a vibrating string fixed search for simulated tones whose waveforms resemble that of a genuine sawari tone. e-mail: [email protected] Other studies on the sawari and related subjects are
237 Acoust. Sci. & Tech. 38, 5 (2017)
where ð Þ is a continuous function satisfying that ð Þ¼0 if <0, and ð Þ 0 if 0. The present model sys- tem assumes the following where K0 is a positive constant [2]: 0if <0 ð Þ¼ ð2Þ Fig. 1 Model system. K0 if 0. The string under vibration is subject to dissipation of found in [3–5] (general descriptions and qualitative kinetic energy due to internal friction of material, friction discussions), [6] (pitch identification), [7–9] (the jawari with the air, etc. Precise analysis of these phenomena is in Indian instruments), and [10,11] (mathematical analysis quite difficult. To make the concerned equation simple and on the sawari-like objects.) practical, the law of linear damping proportional to the string velocity is assumed where the damping coefficient, 2. MODEL SYSTEM , will be determined so as to approximate the decay rate Figure 1 shows the configuration of our model system of the real ones. given in the x-z plane, where the x- and z-axes, respec- As a result of the above considerations, together with tively, are oriented to the horizontal and vertical directions. a small amplitude assumption on uðt; xÞ, the equation of The time variable is denoted by t. Plucking is made at motion in the free-vibration mode (t > tp) is: x ¼ x . The system assumes the simulated sawari tones p @2u @u @2u sounded on open string. þ ¼ T þ ðu Þð0 < x < LÞð3Þ @t2 @t @x2 2.1. Sawari The string is fixed at x ¼ 0, hence The sawari is modeled as the cross section of a small uðt; 0Þ¼r ð4Þ surface created on top of a cylindrical sphere of nut set next 0 to the origin x ¼ 0, being aligned in parallel to the string The condition at x ¼ L will be given in Eq. (14). with an optional partial depression. Let z ¼ sðxÞ denote this surface. For simplicity of notation, sðxÞ¼ 1is assumed 2.3. Resonator outside the nut. 2.3.1. Mechanical model The vibroacoustic response of the table of string 2.2. String instrument can be described practically with a set of The string is thin and flexible in length L having a transfer mobilities in the normal direction at various circular cross section, where and r0, respectively, denote transfer points on the table in reference to the driving the linear density and the radius of cross section. It is taut point at the bridge. Hereafter we shall be concerned with with tensile force T along the x-axis between the two ends the magnitude of complex-valued mobility unless other- x ¼ 0 and L with its bottom periphery contacting the wise stated, and call it the FRF (frequency response sawari surface at rest. The motion is allowed in the vertical function) for simplicity. direction only. Since the transfer FRF, t-FRF for short, varies at Let uðt; xÞ be the vertical position of the central axis of different points over the table, the totality of their spectral string, which is at rest for t 0. envelopes for a number of transfer points, as overlaid on a The string is given an initial motion with plectrum at single plate, manifests a very complex, multi-peak spectral x ¼ xp for a short period of time, 0 t tp (to be called the profile. This virtual spectral profile is called the aggregate plucking mode), then it is released (the free-vibration t-FRF. mode). Two quantities xp and tp, respectively, are called the The resonator is a mechanical network that approx- plucking point and the release time. The equation that imates the spectral characteristics of both the driving-point governs the motion of string necessarily changes before and FRF (d-FRF for short) and the aggregate t-FRF of the after tp. It must be noted that our model of plucking concerned biwa. involves a mechanism of stick-slip dynamics. See Sect. 2.4. The network has movable and fixed nodes, where each The string receives an upward repelling force, f , from movable node is assigned a mass that moves vertically in the sawari only when the condition uðt; xÞ sðxÞ < r0 the x-z plane of Fig. 1, while the fixed ones are allowed no holds. Hence the general formula of f is given as, with the motion. The arcs are assigned each a parallel combination def use of u ðt; xÞ¼r0 þ sðxÞ uðt; xÞ, of a spring and a dashpot both of which undergo the deformation equal to the difference of vertical displace- f ¼ ðu ðt; xÞÞ ð1Þ ments of the two nodes that define the concerned arc. The
238 T. TAGUTI: NUMERICAL SIMULATION OF THE SAWARI TONE network must have at least one fixed node in order to be free from a rigid-body motion, and only one node suffices. Let N be the number of movable nodes, and Xj be the (vertical) displacement of node j ( j ¼ 1; ...; N) from the static equilibrium. The fixed node is labeled node 0. It does not move, hence X0 ¼ 0. The arcs are identified each with a pair of nodes ð j; lÞ. They are unoriented, i.e., ð j; lÞ and ðl; jÞ N designate the same arc. Let j denote the set of all nodes l Fig. 2 Holding posture of the biwa with plectrum. of arcs ð j; lÞ. Let mj be the mass at node j ( j ¼ 1; ...; N), and let kjl and cjl, respectively, be the stiffness of spring and damping where 0 and c0, respectively, are the density and the sound coefficient of dashpot on arc ð j; lÞ. By definition, it satisfies velocity of air, and is the distance from the surface. that klj ¼ kjl and clj ¼ cjl. Let pN be the value of p at ¼ 0, i.e., In what follows, node 1 terminates the string, and node pN ¼ c X_N ðtÞð8Þ N takes the role of the table in the sense of aggregate 0 0 t-FRF; they are called the input and output nodes of the which represents the instantaneous amplitude of the virtual resonator. pressure wave on the virtual plane as a function in t.In When a force, g (a function in t), acts on node 1, Xj what follows it will be referred to as the sound pressure at ( j ¼ 1; ...; N) are subject to the following equations of the output node. motion, where X_j and X€j, respectively, are the first and second derivatives of Xj in t: 2.4. Plucking X 2.4.1. Modeling in stick-slip dynamics € _ _ mjXj þ ðcjlðXj XlÞþkjlðXj XlÞÞ ¼ gj ð5Þ Figure 2 illustrates a typical holding posture of the l2N j biwa with plectrum. Normally the plucking is made by with g1 ¼ g and gj ¼ 0 ( j ¼ 2; ...; N). pushing the string downward (toward the table), that is, Let g^ð!Þ and X^ jð!Þ ( j ¼ 1; ...; N), respectively, be the the plectrum first touches the string from above with its Fourier transforms of g and Xj. Then by definition, edge near the tip, then pushes the string while sliding the contact point toward the tip, and finally releases it at the tip Y^j ¼ i!X^ j=g^ ð j ¼ 1; ...; NÞð6Þ at some depth of string displacement. are the complex-valued mobilities in reference to node 1, The sliding motion of plectrum induces a stick-slip which in fact do not depend on g because Eq. (5) is a linear motion of string due to friction, which produces a peculiar system. sound due to noise-induced resonance on the instrument. Accordingly, the goal is to construct a network whose Ando [12] called such a sound the touch noise in his jY^1j and jY^N j, respectively, approximate the given d-FRF experimental study on the soh and 17-gen. and aggregate t-FRF. The phenomenon of friction has been studied from a 2.3.2. Alternative representation of kjl and cjl variety of perspectives in many mechanical systems [13]. For a movable node j of arc ð j; lÞ, the triplet fmj; kjl; cjlg However, since no sophisticated theory that may be can be associated with a free, damped oscillation of a mass- applicable to the present problem is found in the literature, spring-dashpot unit whose equation of motion is written for the Coulomb friction seems the only choice. We shall an unknown X as formulate it in the framework of our two-dimensional 2 model of string motion. mjðX€ þð2 fjlÞ=QjlX_ þð2 fjlÞ XÞ¼0 ð7Þ pffiffiffiffiffiffiffiffiffiffiffiffi Let fs and fk, respectively, denote the limiting static where fjl ¼ kjl=mj=ð2 Þ is the resonance frequency in friction force and the kinetic friction force, and put case of damping-free oscillation and Qjl ¼ 2 mj fjl=cjl fk ¼ fs, 0 < <1. Also, let ( L) be the thickness ðcjl > 0Þ is the quality factor of damping. We shall use of the vital edge of plectrum moving at speed ðtÞ on an fmj; fjl; Qjlg for synthesis of resonator. imaginary pathway of width passing over the interval 2.3.3. Sound pressure at the output node ½a; b with a ¼ xp =2 and b ¼ xp þ =2, and let S be The motion of node N is considered to emit a virtual the section of string in contact with the moving edge. We sound pressure wave in the following sense. shall deal with S as a lumped mass, denoting its vertical Suppose that a virtual plane attached to node N position by u ðtÞ (see Fig. 3). perpendicularly to the z-axis moves uniformly at velocity By integrating Eq. (3) over ½a; b and adding the X_N . Then the pressure, p, at a point close to the surface and friction term to the resulting right-hand side, it holds that, away from the periphery is given as p ¼ 0c0X_N ðt =c0Þ under the condition of ðtÞ < 0 for all the time,
239 Acoust. Sci. & Tech. 38, 5 (2017) z completes our model system. The output sound of the a b x system is pN of Eq. (8). 0 xp L The model system contains many pre-determined zu= variables related to geometry, material, acoustics and Sδ kinematics. Hereafter, they will be referred to as the problem parameters.
Fig. 3 Configuration of S , ¼ b a L. 2.6. Numerical Scheme The governing equations are solved numerically by use of Newmark’s scheme [15] with ¼ 0 on uniform d2u du þ ¼ g f ð9Þ spatial meshes. Because of ¼ 0, it is explicit in the dt2 dt ab F displacement variables, and implicit in the velocity and 2 2 where gab and fF, respectively, are the forces due to the acceleration variables where @ u=@x is approximated by displacement of string and due to the friction such that the second central difference quotient. It is considered that both the numerical stability and @u @u gab ¼ T þ ð10Þ accuracy of the entire scheme depend on the part of scheme @x @x 8 a 0 bþ0 for the string. Hence, a large M for spatial mesh spacing > minðgab du =dt d =dt; fsÞ x ¼ L=M under which a temporal mesh t ¼ < pffiffiffiffiffiffiffiffiffiffiffiffi x= ðT= Þ (the Courant number is unity) is assumed; fF ¼ > for du =dt ¼ ðtÞ ð11Þ :> M ¼ 4;000 is taken in the following simulation, which f du=dt > ðtÞ k for makes x ¼ 0:2 mm and t ¼ 0:90188 ms in the case of In the state of static friction, fF is gab du =dt C3#. The computed values of pN ðtÞ and others are down- d =dt, which may take any value in the range between 0 sampled to 48 kHz to store into files in PCM-format. and f , hence we have the top two rows in the right-hand s 3. SIMULATION side of Eq. (11). In the state of kinetic friction, it is fk, hence we have the bottom row. Writing the equation of 3.1. Aim and Scope motion in this form owes to [14]. The central theme is the role of sawari sðxÞ for Hereafter, we shall assume that fs may vary with time, production of sawari-like tones. The target tones are those hence write it as fsðtÞ. Then the five quantities xp, tp, ðtÞ, produced on open first and second strings tuned to C3# fsðtÞ and describe the plectrum motion. and G2#, respectively. Here, the case of C3# will be Assuming that is a constant independent of the other studied in detail in Sects. 3.2 and 3.3, followed by a brief variables, the set P: summary on the case of G2# in Sect. 3.4. As the base of evaluating the quality of pN , a genuine P ¼fxp; tp; ðtÞg where ðtÞ¼ð ðtÞ; fsðtÞÞ ð12Þ sawari tone on open first string tuned to pitch C3#, called describes the maneuver of plectrum. We shall call P the the reference tone, and the FRFs of the instrument on plectrum practice, and ðtÞ the plectrum trajectory. which it was produced, called the associated instrument, 2.4.2. Equation for uðt; xÞ in the plucking mode are provided. See Appendix A for details. In the plucking mode (0 t tp), the following set of Since P drives the string, the combined effect of sðxÞ three equations replaces Eq. (3). and P has to be dealt with. The entire space of permissible 8 2 2 instances of all problem parameters is too large, so the > @ u @u @ u > þ ¼ T þ ðu Þð0 < x < aÞ search domain will be restricted to the set (Cartesian < @t2 @t @x2 product) fsðxÞg fxpg with all other problem parameters > u ¼ u ; u is subject to Eq.(9) ða x bÞ ð13Þ > 2 2 fixed each to a single instance. :> @ u @u @ u þ ¼ T ðb < x < LÞ @t2 @t @x2 3.2. Instances of Problem Parameters Pitch and sound pressure level of tone. The tone pitch is 2.5. String Motion Coupled with the Resonator C3# (F0 ¼ 138:6 Hz). Terminating the string at x ¼ L with the resonator such In order to fix a single instance of ðtÞ, we shall assume that two levels ‘0 and ‘1 (re. 0 dB = 20 mPa) of the sound 8 pressure of pN , i.e., ‘0 in the period of touch noise and ‘1 < disp.: X1ðtÞ¼uðt; LÞ at the ‘main’ onset coming right after the touch noise. : @u ð14Þ force: gðtÞ¼ T ðt; LÞ Setting ‘1 ¼ 85 dB seems appropriate as a normal con- @x dition of playing, while ‘0 is determined from an assumed
240 T. TAGUTI: NUMERICAL SIMULATION OF THE SAWARI TONE
Fig. 4 Network representing the resonator. Fig. 6 Sawari: z ¼ sðxÞ.
Then, sðxÞ is BðxÞ with an optional depression of width d da on the left side of the level surface of width d, deep enough to be free from contact with the string. The resulting surface of interval [d da; d] is called the active surface with its active width da. It is the surface the string touches at rest, and is the main area that works as a unilateral constraint to the string in motion. The entire surface of ½0; d is called the nominal surface with its Fig. 5 d-FRF (broken line) and t-FRF (solid line) of the nominal width d. resonator (re. 0 dB = 1 m s 1N 1). The four resonance The following values are assumed: R ¼ 25 mm which frequencies are labeled F1 , F2 , F3 and F4 from low to is the estimated value of the nut on the associated high. instrument, while d and da take on integers in the units of mm such that 1 d 10 and 1 da d amount of level difference ‘ ¼ ‘1 ‘0. We shall adopt ð15Þ together with d ¼ da ¼ 0 ðno sawariÞ ‘ ¼ 15 dB as discussed below, hence ‘0 ¼ 70 dB. String and plectrum. r0 ¼ 0:68 mm, ¼ 1:4 g/m, K0 ¼ Plectrum practice. 3:2 106 N/m2, L ¼ 0:8 m, ¼ 0:8 mm, ¼ 4:0 s 1, ¼ (i) Plucking point. Normally the plucking is made at a point 2 2 0:3 and Tð¼ 4 L F0Þ¼68:8 N. The first three are taken about 1=8 1=5 of string length off the bridge. Hence xp is from [16] and the next two are considered to be typical set to take on the following 12 points: ones. The is determined by a preliminary computation so xp ¼ð1 ÞL that the decay rate of p approximates that of real sawari ð16Þ N where ¼ 0:10 ð0:01Þ 0:21: tones, while is a small number taken for granted among those reported under various combinations of two hard (The right-hand side of formula in Eq. (16) designates materials. the set of values in terms of the minimum, the step-size and Resonator. The resonator is constructed as an approxima- the maximum from left to right.) tion to the associated instrument. Figures 4 and 5, (ii) Release time and plectrum trajectory. respectively, present the network and FRFs of the A single instance of ftp; ðtÞg is established based on resonator, and Fig. 12 presents the FRFs of the instrument. the observation on several real sawari tones and several The approximation is made in terms of the frequencies preliminary runs of simulation as follows. and levels of four major resonance peaks but the Q-factors The touch noise is observed to last tp0 ¼ 20 30 ms, which are not available in the source data. followed by a sharp rise of 15 20 dB reaching the Sawari. Figure 6 shows the sawari z ¼ sðxÞ, which is a maximum in the next short period of tp1 ¼ 20 ms or two-dimensional model of a real sawari created on top of thereabouts. Although the exact release time of plectrum nut made of tarred bamboo. can not be identified by such a waveform analysis, tp0 þ tp1 More specifically, let z ¼ BðxÞ be such that must be its upper bound. The present simulation assumes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ 40 ms together with ‘ ¼ 15 dB. BðxÞ¼min 0; R2 ðx d=2Þ2 R2 d2=4 ; p Since pN is considered to be stationary if ðtÞ is a which is a curvedp lineffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi originating from an arc of radius R constant vector-valued function ðtÞ¼ c (t 0) provid- centered at ðd=2; R2 d2=4Þ. It holds that BðxÞ¼0 for ed that the stick-slip mechanism works effectively, ðtÞ x 2½0; d , and BðxÞ < 0 otherwise. for the production run is taken to be a piecewise linear
241 Acoust. Sci. & Tech. 38, 5 (2017)
Fig. 7 16 cases of AðtÞ with dðdaÞ= of Eq. (18). Fig. 8 16 cases of AðtÞ with dðdaÞ= of Eq. (19).
0 function in t with two time points tp ¼ 25 ms and tp ¼ 40 ðd; Þ2f6; 7; 8; 9g f0:15; 0:16; 0:17; 0:18g; da ¼ d ð19Þ ms such that( 0 where d ¼ 6 and 9 are on the periphery of merit. Figure 8 c0; ð0 t tpÞ ðtÞ¼ 0 ð17Þ shows AðtÞ with dðdaÞ= of Eq. (19). c0 þ EðtÞð c1 c0Þ; ðtp < t tpÞ Regarding the effect of depression (da < d)onAðtÞ,it where c0 and c1 are two constant vectors that realize turned out, by examining all the 672 cases, that the extent the low and high sound pressure levels respectively, and of undulation (or shaking for large d) is reduced as the 0 0 EðtÞ¼ðt tpÞ=ðtp tpÞ. depression is enlarged (i.e., as da is decreased), the effect A parametric study was conducted on the set f g appearing significant for da 3 if d 7. f fsg f g in search for c0 and c1. As a result, c0 ¼ At this point we shall study the temporal behavior of ð 0:03; 0:07Þ and c1 ¼ð 0:28; 0:70Þ, respectively, were the partials. Let qm be the sound pressure level of the m-th chosen so as to produce the desired sound pressure levels in partial of pN . Figure 9 presents the temporal development the mean over . See Appendix B for details. Here we note of qm ðm ¼ 1; ...; 16Þ in the case 8(8)/0.17 (graph I, no that the actual sound pressure level with a particular depression), together with those of 8(2)/0.17 (II, partial could be different from ‘0 in the touch noise or ‘1 at the depression) and 0(0)/0.17 (III, no sawari). The curves of ‘main’ onset. qm are labeled with numerals to designate the partial numbers except some qm with less dominating sound 3.3. Production Run pressure levels which are difficult to label because of The production run processed 672 cases arising from tangled curves, and some others with dominating sound all combinations of (d; da; ). In what follows AðtÞ denotes pressure levels are given also the respective numerals in the temporal envelope of the amplitude of pN, and dðdaÞ= parentheses. The same rule will apply to Fig. 10 also. stands for (d; da; ). Figure 7 overlooks the overall para- Graph I of Fig. 9 in comparison with III readily shows metric dependence of AðtÞ with the following 16 cases out that the sawari causes to boost the amplitude undulatingly of the 672 cases: and to elongate the lifespan of the 6th partial considerably, ð Þ2f g f g ¼ ð Þ as well as those of the 11th and 12th partials to a lesser d; 0; 4; 7; 10 0:12; 0:15; 0:18; 0:21 ; da d: 18 degree. Note that the frequency of the 6th partial is near F1 , The figure reveals that the profile of AðtÞ does not and those of the 11th and 12th partials are near F2 . deviate very much from a negative exponential-like (See Fig. 5.) Also, graph II in comparison with I shows a envelope for d ¼ 4, but does deviate from it noticeably reduction in the effect of sawari with a smaller da. in an undulating way for d ¼ 7 or in a shaking way rather than undulating for d ¼ 10. 3.4. Open Second String Among these cases, AðtÞ with 7(7)/0.15 and 7(7)/0.18 The following problem parameters substitute for those deserve a special attention because of their slowly given in Sect. 3.2: the tone pitch is G2# (F0 ¼ 103:8 Hz); 6 2 undulating nature with staying above a certain magnitude r0 ¼ 0:73 mm, ¼ 1:67 g/m, K0 ¼ 2:0 10 N/m and 2 2 for a short timespan. There are other cases of dðdaÞ= that Tð¼ 4 L F0Þ¼46:2 N. Although these data call for a produce AðtÞ having such a characteristic profile. They are new ðtÞ to meet the required ‘0 and ‘1 for pN , we shall confined in the set: use the same ðtÞ as for C3# for convenience.
242 T. TAGUTI: NUMERICAL SIMULATION OF THE SAWARI TONE
Fig. 11 The reference tone. Graphs I: Envelope of total amplitude, II: Power spectrum of touch noise, and III: Temporal development of the first 16 partials. (re. 0 dB = arbitrary.)
of the ðtÞ established for the first string which is thinner Fig. 9 Temporal development of the first 16 partials of than the second string. p . Cases 8(8)/0.17 (graph I, no depression), 8(2)/ N Figure 10 presents the temporal developments of qm 0.17 (II, partial depression) and 0(0)/0.17 (III, no ðm ¼ 1; ...; 20Þ in the cases 8(8)/0.13 (graph I, no depres- sawari). sion) and 0(0)/0.13 (II, no sawari). The sawari causes to boost the amplitude undulatingly and to elongate the lifespan of the 8th partial (near F1 ) considerably, as well as those of the 15th and 16th partials (near F2 ). 4. DISCUSSION AND CONCLUSIONS It has been shown that the sawari has a unique effect on the waveform of pN in the time-frequency domain. There seems to occur an unsteady, partial exchange of energy among the partials. Unless the active width da is very small, the extent of sawari’s effect depends primarily on its nominal width d combined with the plucking point, giving a stronger influence on the partials whose fre- quencies are near the resonance frequencies of the resonator. Fig. 10 Temporal development of the first 20 partials of By looking at Fig. 8 in comparison with graph I of p on the open second string (G2#). Cases 8(8)/0.13 N Fig. 11, the simulated tones pN for d ¼ 7 and 8 share with (graph I, no depression) and 0(0)/0.13 (II, no sawari). the reference tone more or less a unique temporal profile of the total amplitude characterized by a slow undulation with The result showed that the effect of sawari parallels staying above a certain magnitude for a short timespan. almost the case of C3#, i.e., the profile of AðtÞ has an The partials’ temporal development in graphs I and II of undulating nature similar to those of Fig. 8 when Fig. 9 as well as graph I of Fig. 10 have some resemblance to graph III of Fig. 11. ðd; Þ2f6; 7; 8; 9g f0:10; 0:11; 0:12; 0:13; 0:14g; d ¼ d a However, there is an apparent difference between ð20Þ graph I (or II) of Fig. 9 and graph III of Fig. 11. The major where d ¼ 6 and 9 are on the periphery of merit. reason for this must be attributed to the model resonator The sound pressure levels of pN are lower than those in (Fig. 4) and the lack of resonant open strings in the system, the case of C3# by about 5 dB. The reason must be the use as well as the assumed plectrum trajectory.
243 Acoust. Sci. & Tech. 38, 5 (2017)
Modern Biwas (Kasama-shoin, Tokyo, 1998) (in Japanese). [6] K. Kishi, Y. Tohnai and M. Yamada, ‘‘The ‘Satsuma biwa’,’’ Proc. 3rd. Jt. Meet. ASA and ASJ, Honolulu, December, pp. 325–330 (1996). [7] C. V. Raman, ‘‘On some Indian stringed instruments,’’ Proc. Indian Assoc. Cult. Sci., 7, 29–33 (1921). [8] R. Burridge, J. Kappraff and C. Morshedi, ‘‘The sitar string: A vibrating string with a one-sided inelastic constraint,’’ SIAM J. Appl. Math., 42, 1231–1251 (1982). [9] C. Valette, ‘‘The mechanics of vibrating strings,’’ in Mechanics of Musical Instruments, A. Hirschberg, J. Kergomard and G. Weinreich, Eds. (Springer-Verlag, Vienna and New York, Fig. 12 d-FRF (broken line) and seven t-FRFs (solid 1995), pp. 115–183. lines) of the associated instrument (re. 0 dB = [10] L. Amelio and G. Prouse, ‘‘Study of the motion of a string 1 1 1m s N ). vibrating against an obstacle,’’ Rend. Math., 8, 563–585 (1975). [11] M. Schatzman, ‘‘A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave In fact, the aggregate t-FRF in Fig. 5 has only 4 obstacle,’’ J. Math Anal. Appl., 73, 138–191 (1980). [12] M. Ando, ‘‘Temporal analysis of the plucking action on the Soh spectral peaks for approximation to Fig. 12 that has a dense and 17-gen,’’ Annu. Rep., Dep. Music, Tokyo Univ. Arts, 8, 45– set of spectral peaks in the frequency domain extending 87 (1982) (in Japanese). above 3 kHz. The piecewise linear ðtÞ in Eq. (17) could [13] R. A. Ibrahim, ‘‘Friction-induced vibration, chatter, squeal, and have been too simple, e.g., an initial, impulsive behavior chaos. Part II: Dynamics and modeling,’’ Appl. Mech. Rev., 47, 227–253 (1994). (hitting the string with plectrum) has not been taken into [14] Ch. Wensrich, ‘‘Slip-stick motion in harmonic oscillator consideration in ðtÞ. Nevertheless, the range of d in chains subject to Coulomb friction,’’ Trib. Int., 39, 490–495 Eq. (19), d ¼ 7 and 8 in particular, gives a good corre- (2006). spondence with d ¼ 8 (approx.) of the reference tone. [15] N. M. Newmark, ‘‘A method of computation for structural Regarding the touch noise, graph II of Fig. 11 has dynamics,’’ Proc. ASCE, 85, No. EM3, 67–94 (1959). [16] T. Taguti, H. Fujita and K. Tohnai, ‘‘Elastic deformation of silk many spectral peaks around the dominant resonance string for biwa by lateral compressive force,’’ Mem. Konan frequencies of Fig. 12. A similar property holds in the Univ. Sci. Eng. Ser., 53(1), pp. 57–66 (2006) (in Japanese). simulated tones as discussed in Appendix B. [17] T. Taguti and Y. (K.) Tohnai, ‘‘Vibro-acoustic analysis of In conclusion, we claim that the proposed model several biwas,’’ Proc. Int. Symp. Musical Acoustics ISMA 2007, Barcelona, September, Paper 3-P1-10 (2007). system describes the essential physical elements for the production of the sawari tone, and that a more realistic APPENDIX sawari-like tone can be made with a revised model enhanced with those physical components discussed above. A. Reference tone and its associated instrument Also a three-demensional modeling of string motion is a The reference tone is a sawari tone taken from [1]. The challenging subject. tone, which was labeled S3, was rated as best among four sawari tones at pitch C3# on open first string of an ACKNOWLEDGMENT excellent Chikuzen 5-string biwa in a parametric study on The author is deeply indebted to Mr. Yoshimasa the width of sawari. (Kakuryo) Tohnai for his stimulating discussion about the By following the notation in Fig. 6, it was produced physical and musical understanding of the subject. with d ¼ 8 and da ¼ 3 approximately. (The nominal width d ¼ 8 was not reported in [1].) REFERENCES Here, four unplucked strings (tuned below perfect 4th, [1] T. Taguti and Y. (K.) Tohnai, ‘‘Acoustical analysis of the unison, above major 2nd, and above perfect 5th relative to sawari tone of Chikuzen biwa,’’ Acoust. Sci. & Tech., 22, 199– 207 (2001). C3#) were left open, so it must contain the resonating tones [2] T. Taguti, ‘‘Dynamics of simple string subject to unilateral on these strings. Also, there was no measurement on the constraint: A model analysis of sawari mechanism,’’ Acoust. plucking action of the player. Hence we have had no Sci. & Tech., 29, 203–214 (2008). physical data about it. [3] E. Kikkawa, ‘‘On the sawari of shamisen,’’ in Aesthetics of Shamisen and An Episode on the Birth of the Department of Graphs I, II and III of Fig. 11, respectively, are the 0 Japanese Music at Tokyo University of Fine Arts and Music envelope of total amplitude pN , the power spectrum of (Shuppan Geijutsu-sha, Tokyo, 1997), pp. 113–136 (in touch noise (duration 30 ms), and the temporal develop- Japanese). ment of the rms-values q0 in dB (re. 0 dB = arbitrary) of [4] Y. Ando, ‘‘Structure of Chikuzen biwas,’’ Ongakugaku (J. Jpn. m the first 16 partials, of the reference tone. Here, the sound Musicol. Soc.), 25, 137–152 (1979) (in Japanese). [5] K. Tohnai, Study on Japanese Modern Biwas (Kasama-shoin, pressure level of pN was about 80 dB at maximum. Tokyo, 1994) (in Japanese); Sequel of Study on Japanese Figure 12 is the FRFs of the associated instrument [17].
244 T. TAGUTI: NUMERICAL SIMULATION OF THE SAWARI TONE
Fig. 13 Result with cðtÞ¼ð 0:03 m/s; 0:07 NÞ and ¼ 0:16 under d ¼ ds ¼ 0 (no sawari). Graph I: Waveforms of pN ðtÞ, uðt; xpÞ and vðt; xpÞ, and II: Power spectrum of pN ðtÞ in the plucking mode.
Fig. 14 S ½ c and P ½ c for 0:04 fs,c 1:0 plotted as a function of j cj (0:01 j cj 0:30).
B. Determining c that produces a touch noise pN at ‘ dB The nine response curves of S ½ c of graph I tell that We are concerned with the sound pressure level of pN the stick-slip mechanism works effectively only when j cj under constant trajectories, together with the associated lies in a limited range that depends on fs,c, where the peaks number of stick-off events in the mean over . attain about 26 at maximum except case i, in which the Let cðtÞ be a constant trajectory such that maximum would take place at a speed larger than 0.3 m/s. Theoretically the stick-off events never take place in a cðtÞ¼ð c; fs,cÞðt 0Þ finite timespan if j cj is small enough, or only once at the where c and fs,c are positive constants. very beginning if it is large enough. Graph I depicts this With the use of this cðtÞ, let P½ ; c be the sound fact in the range 0:01 j cj 0:3 m/s. 00 00 pressure level over 0 t tp (tp ¼ 50 ms) when the string In the cases a f, S½ c attains a constant 7 for a is plucked at xp ¼ð1 ÞL under d ¼ da ¼ 0, and sim- certain interval of j cj. It implies that the stick-off events ilarly, let S½ ; c be the number of stick-off events. take place about at the rate of the pitch frequency of C3# Figure 13 presents the result where the first 50 ms is in (138.6 Hz), although the oscillation did not appear to be the plucking mode with cðtÞ¼ð 0:03 m/s; 0:07 NÞ and periodic in every case. ¼ 0:16, followed by another 50 ms in the free-vibration The above experimental study leads to the following mode. choice for c0 and c1 of ðtÞ in Eq. (17): It is readily seen that, in the plucking mode, the c0 ¼ A ð21Þ waveforms of fpN ðtÞ; uðt; xpÞ; vðt; xpÞg are non-periodic c1 ¼ B0 ð22Þ (graph I). The P½ ; c and S½ ; c for ¼ 0:16 are found to be 68.6 dB and 13. Hence, pN is regarded as a where A ¼ð 0:03; 0:07Þ and B0 ¼ð 0:28; 0:70Þ are the touch noise at 68.6 dB in the plucking mode. Its power two points as marked in graph II of Fig. 14. spectrum shown in graph II has 4 spectral peaks around The reason for Eq. (21) is that A is one of the points F1 F4 (see Fig. 5.) It was verified that such a phenom- associated with P ¼ ‘0 (‘0 ¼ 70) at a high level of S½ c . enon is observed in any other cases when the stick-slip However, the reason for Eq. (21.2) cannot parallel that for mechanism works effectively. Eq. (22). Since ðtÞ changes rapidly during the period 0 Let S½ c be the mean value of stick-off events over 12 tp t tp, ðtÞ terminated by a point like B associated instances of , ¼ 0:10 ð0:01Þ 0:21; similarly, let P ½ c with P ¼ ‘1 may not realize the target level ‘1 (‘1 ¼ 85)at be the mean value of P½ ; c . Graphs I and II of Fig. 14, the ‘main’ onset of the waveform of pN . After several trial 0 0 0 respectively, present S½ c and P½ c as functions of j cj runs, we chose B associated with P ¼ ‘1, ‘1 ¼ 90, larger for 9 instances of fs,c up to 1.00 N. than ‘1 by 5, instead.
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