Geometrical effects on the tuning of Chinese and Korean stone chimes

Stone chimes have been cherished musical instruments for centuries in several Asian countries. They have taken a number of different forms, but generally they have two legs which meet at a vertex. The longer leg is often called the drum, because it is where the chime is struck, and the shorter leg is called the femur or thigh.

Several sets of stone chimes have been found in ancient Chinese tombs dating from 400 BC and before. A bianqing of 32 stones was found in the tomb of the Marquis Yi of Zheng (the same tomb that contained the renowned set of 65 bells), dating from about 433 BC. Although many stones were found broken, Chinese scholars were able to determine their dimensions, and from these, Lehr determined scaling laws for their various dimensions.¹ Stones are scaled in lengths as well as thicknesses.

Another ancient bianqing from the Warring States period (450–221 BC) of the Zhou dynasty was found more or less intact, so that the fundamental frequencies of most of the stones, as well as the dimensions, could be determined.² Vertex angles of these stones ranged from $142to155deg$⁠.

The Korean pyeongyeong consists of a set of 16 hard stones cut in the shape of an inverted letter L. The stones are hung from a wooden stand and struck, near the lower end of the longer side, by a mallet tipped with horn. The upper ridge of the longer side of a stone is about 1.5 times longer than the shorter side. Unlike most Chinese stone chimes, the pyeongyeong stones in a set have nearly the same shape except for the highest stone in which the long edge may be slightly shortened, but are scaled in thickness.³ 

Old pyeongyeongs are preserved in museums as national treasures, but replicas are built and played in concerts. Earlier we analyzed a pyeongyeong built by Hyungon Kim and played by the National Traditional Orchestra of Korea. The fundamental frequency was shown to be directly proportional to thickness over most of the range.⁴ The second mode was found to be tuned to 1.5 times the nominal frequency, the third mode to about 2.3 times the nominal frequency, and the fourth mode about 3 times the nominal frequency up to the 12th stone.⁴ 

In this paper, we report on studies designed to show how the vibrational frequencies of L-shaped chime stones, such as are found in the Korean pyeongyeong and the Chinese bianqing, depend upon the details of their geometry.

The shape of a pyeongyeong stone is said to be the “shape of the heaven that curves to cover the earth.” The 115° angle between the drum part and the femur part and the concave curved base form a smooth L shape. In Korea, the pyeongyeong has been built in this smooth L shape for some five centuries, whereas Chinese bianqing stone chimes have varied in shape from time to time. Old stones from before the Christian era mostly have a straight base, or an arc-shaped base, for example, whereas stones from the Qing Dynasty (1644–1911) are L shaped.

According to Kuttner, stones with a straight base [Figs. 1(a) and 1(b)] are older than those with an arc-shaped base [Fig. 1(c)], and he concluded that the curved base resulted from experiments to improve the acoustical qualities of the stones. Also, the angle between the drum part and the femur part became narrower in stones from the Qing Dynasty and later.⁵ 

In this study, we examine the effects of geometry on the tuning and the sound quality of stone chimes, such as the Korean pyeongyeong and the Chinese bianqing. Modal shapes and modal frequencies are estimated by means of finite element methods (FEM) analysis, and these results are compared to modal shapes and frequencies of existing stones, determined by holographic interferometry and by experimental modal testing. Modal shapes and frequencies of stones of different shapes are calculated by finite element methods. Of particular interests are the effects of varying the vertex angle and the curvature of the base.

Since the sound spectra and mode shapes of several pyeongyeong replica stones have previously been measured,⁴ we used these stones as models for finite element analyses. The stone of lowest frequency is tuned to C5 $(523Hz)$⁠, and we use that stone for (FEM) modeling and designate it as PGC 115, since the vertex angle $α$ is 115°. The modal frequencies calculated for this stone are compared to the modal frequencies determined from the sound spectrum, from holographic interferometry, and from experimental modal testing using the (STAR) system. The agreement is very good, as shown in Table I.

The upper seventh stone from the tomb of Marquis Yi is supposed to be tuned to C, but it was broken when it was excavated.⁶ The size of each side, angle, and thickness are reported by the Museum of Hubei Province.⁶ Since the vertex angle $α$ is 163°, the model for this stone is designated as BQC 163. Keeping the principle lengths A, B, E, and F of stone the same, vertex angle $α$ was then changed to 140° and 115°, and these models are designated BQC 140 and BQC 115, respectively. FEM was used to determine the mode frequencies and mode shapes of the three bianqing models and the pyeongyeong model. The four models are shown in the upper row in Fig. 2.

With the same angle $α=115°$⁠, the base lines were changed from resembling the smooth arc of the pyeongyeong (BQ115 in Fig. 2) to a near right angle shape (BQ 115ra), three oblique lines (BQ 115to) and a straight line (BQ 115sb) as shown in the second row in Fig. 2. FEM was again used to determine the mode frequencies and mode shapes of the three new bianqing models.

The commercial program ANSYS Mechanical 8.1 was used for the FEM analysis. Since a gyeong is a flat plate of constant thickness, we adapted the three-dimensional eight-node Shell 93 element for modeling it. The shell element 93 was found to give the most accurate results. For the PGC 115 model, 13 key points were created to make the keypoints for modeling by ANSYS. We chose quadrilateral element shapes to construct the mesh.

Systematic ways of varying the vertex angle and base line were introduced. The first set of 19 alternative gyeong models was constructed by adding 5° increments to the vertex angle from 90°, the shape of an L, to 180°, the shape of a bar. In this series of shapes the baseline curvature was kept the same. Next we estimated the effect of baseline curvature on mode frequencies and shapes as the baseline was systematically varied.³ 

Table II shows the dependence of the modal frequencies on vertex angle as calculated by FEM. The models correspond to those in Fig. 2. Also shown are the frequencies relative to the lowest mode of vibration. Note that decreasing the vertex angle lowers the ratios of the higher modes and also brings them nearer to a harmonic relationship. This may have made them more sonorous. The ratios for BQC 115 are reasonably close to those of the Korean pyeongyeong model PGC 115.

Table III shows the dependence of the modal frequencies on baseline curvature. Also shown are the frequencies relative to the lowest mode of vibration. The differences between the smooth arc (BQC 115), the near right angle (BQC 115ra), and the three oblique lines (BQC 115to) are small, but in the stone with the straight base (BQC 115sb) the second, third, and fourth mode frequencies are raised substantially compared to the fundamental. The sound of this stone model would be expected to be quite different from the others and also different from the pyeongyeong stone PGC 115.

More details of the dependence of mode shapes and frequencies on vertex angle were provided by varying the vertex angle in five-degree increments from 90° to 180°. Figure 3 shows how the mode shape and frequency of the lowest mode changes as the vertex angle changes from 90° to 180°. Similar plots for the higher modes are shown online in file . Figure 4(a) shows how the mode frequencies depend on the vertex angle. A polynomial trend line calculated for best fit, also shown, is of the form $y=0.002x3−0.0405x2−3.0059x+1122.2$⁠. Figure 4(a) shows the fundamental frequency, and Fig. 4(b) shows the frequency ratios of the higher modes to the fundamental.

The frequency ratios of the various partials might be expected to determine the timbre of the sound. A vertex angle of about 115° was adopted by both the makers of Korean pyeongyoung and the Chinese bianqing of the Qing dynasty and later.

From examining existing Chinese bianqing, it appears that the L shape with a gently curved base gradually evolved over several centuries. The 18th century stone in Fig. 1(d) has a vertex angle of approximately 113°, as measured from the photograph. This is very close to the angle of 115° measured in Korean pyeongyeong stones of about the same age, which is not surprising since the earliest pyeongyeong were imported from China. This vertex angle was probably arrived at by attempting to maximize sonority.

The dependence of modal frequency, and especially the ratios of the higher mode frequencies to the fundamental, on chime shape suggests that the ancient Chinese and Korean chime makers tuned their chimes by adjusting the shape of the chimes to obtain the most pleasing sound. This was probably done through centuries in China, and when the Koreans began making stone chimes, they were guided by the Chinese chimes having the best sound. The most important parameter is the vertex angle, with the curvature of the base having a much smaller influence.

By means of finite element methods (FEM) analysis, we have shown how changing the vertex angle $α$ and the curvature of the base changes the relative frequencies of the modes of vibration of stone chimes (lithophones) and hence the character of the sound. Pyeongyeong stone chimes in Korea have maintained pretty much the same shape during the past five centuries, and the same appears to be true of Chinese bianqing. Our calculations show that relative modal frequencies are changed considerably by changing the vertex angle, but are less sensitive to changes in curvature of the base. The L shape with vertex angle around 115° appears to produce stones with maximum sonority.

The fabricators of ancient lithophones no doubt discovered the same thing, albeit by years and years of trial and error.

The agreement between mode frequencies and mode shapes calculated by FEM and measured by holographic interferometry and by experimental modal testing under playing conditions was found to be quite satisfactory, and this validated the use of FEM to study the effects of varying the model shapes.