Acoust. Sci. & Tech. 38, 5 (2017) #2017 The Acoustical Society of Japan PAPER Numerical simulation of the sawari tone on Chikuzen biwa Tomoyasu Tagutià 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. Ãe-mail: [email protected] Other studies on the sawari and related subjects are 237 Acoust. Sci. & Tech. 38, 5 (2017) where ðÞ is a continuous function satisfying that ðÞ¼0 if <0, and ðÞ0 if 0. The present model sys- tem assumes the following where K0 is a positive constant [2]: 0if<0 ðÞ¼ 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 ðuÀÞð0 < 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, sðxÞ¼1is 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; xÞsðxÞ < 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 À def use of u ðt; xÞ¼r0 þ sðxÞuðt; xÞ, of a spring and a dashpot both of which undergo the deformation equal to the difference of vertical displace- f ¼ ðuÀðt; 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.