Seeing the sound of castanets: Acoustic resonances between shells captured by high-speed optical visualization with 1-mm resolution Open Access

Castanets are percussive instruments comprising two shells that produce light and high-pitched sounds on being clapped together.¹ The early evidence of the use of castanets in Spanish music can be found in the Cantigos de Santa Maria, which was written in the 13th century.² They are often used in Spanish music, particularly while performing regional folk dances, and occasionally in orchestral and popular music. However, despite their simplicity and familiarity, to the best of our knowledge, little is known about the physics of castanets. Understanding the physical mechanism will serve as the basis for developing their physical and numerical models, as well as the improvement of their design and fabrication processes. Herein, we report an experimental investigation of their sound generation.

The typical structure of a castanet is depicted in Fig. 1(a). The shells are often made from hardwood or glass fiber, and their shapes are circular or slightly oval. They are tied using a rope through holes in their bridge. There are various methods for playing castanets.^2,3 For the Spanish style, a pair of castanets is used. The higher-pitched one (female or hembra) is typically held in the right hand and the lower-pitched one (male or macho) in the left hand. A player holds the castanet on his/her palm by wrapping the rope around his/her thumb and tapping one shell by using other fingers, as depicted in Fig. 1(b). For orchestral music, percussionists often use a pair of shells that are mounted on a handle and tap them to make a sound. The shells collide with each other at their tips (the position where the two shells collide is called the point), and the sound is radiated. During the sound radiation, the bridge acts as a pivot; i.e., both the shells continue to contact each other at the bridge.

When their points are in contact at the moment of impact, the sides of the shells have small gaps of typically approximately a few millimeters [indicated by h₀ in Fig. 1(a)]. The inner side of the shell is hollow and is usually ellipsoidal, cylindrical, or of any other complicated shape. The small gap connects the inner hollow cavity to the surrounding air, and thus the system may be regarded as a type of acoustic resonator. This suggests that acoustic resonances can occur in the inner space and may be the cause of the pitch of the sound. Therefore, this study focuses on how the inner spaces and gaps of the castanets contribute to the sound thereof.

To investigate the sound radiation from musical instruments, the spatial profile of the sound field has been visualized using microphone arrays. For example, the sound-intensity fields around instruments were visualized via a four-point microphone or near-field acoustic holography, following which the magnitude and energy flow of the generated sound were characterized.^4–6 Surrounding spherical microphone arrays have been used to investigate the directivity, radiation pattern, and acoustics center of musical instruments.^7,8 Although such sound-field measurement methods provide useful information regarding the spatial characteristics of various types of musical instruments, the spatial resolution of microphone arrays is limited because the distance between microphones cannot be reduced to less than the diameter of the microphone. This essential limitation makes it difficult to spatially resolve the fine structures of sound fields; notably, spatial resolution is critical to investigating high-frequency components or small spaces, such as the gaps between the shells of castanets.

Optical technologies have gained considerable attention as an effective alternative to microphones for visualizing sound fields.^9–14 Light that propagates through a sound field experiences modulation because of the acousto–optic effect,^9,10 thereby enabling the contactless measurement of the sound field. Accordingly, because there is no need to place objects in the measurement area, we can achieve the measurement of a considerably near field and small space without any disturbance to the field. In addition, the spatial resolution in optical methods can be on the order of millimeters. Owing to these advantages, several studies have been performed on musical instruments by using optical methods. Gren et al.¹⁵ visualized the sound field around a violin that was excited by a mechanically controlled bow using a scanning laser vibrometer. Single-shot imaging of shock waves generated using a trumpet was achieved by Pandya et al.¹⁶ by using the Schlieren method. Recently, the gas flow and sound waves emitted from a whistle were simultaneously visualized via parallel phase-shifting interferometry (PPSI).^17–19 By applying these optical methods to measure the sound of a castanet, the sound field within the interior cavity can be resolved owing to the contactless nature and high spatial resolution of these methods.

In this study, via PPSI, we investigated the physics of castanets by visualizing the sound fields generated by them. We employed a measurement system with the spatial resolution of 1.1 mm and frame rate of 100 000 fps. The sound fields around the castanets were measured while playing the castanets. By carefully aligning the tilt of the castanets, the sound fields within the small gap were successfully visualized. The frequency analysis of the recorded sound-field videos identified two acoustic resonances of the castanets. By comparing the experimental results with theoretical models, it was found that the first mode could be explained on the basis of the Helmholtz resonance with a time-varying resonator shape due to the movement of the shells after impact, while the second mode could be considered as the standing wave mode caused by an interior volume of the shells.

Three hembra castanets, designated A, B, and C, were used in this study. Their photographs are shown in Fig. 1(c), and their properties are listed in Table I. The instruments are made by different manufacturers and have different combinations of the material and hollow shape as follows: castanet A is made from ebony and has a round hollow; castanet B is made from fiberglass and has a round hollow; castanet C is made from fiberglass and has a cylindrical hollow. The sizes of castanets A and B were almost the same, while castanet C was bigger than the others.

The temporal waveforms measured using a microphone and their scalograms are plotted in the middle and bottom of Fig. 1(c), respectively. The microphone recordings described here and the optical experiments described in Sec. III were performed in the same room, where the walls and ceiling were covered using sound-absorption materials, and the floor was carpeted. The castanets were played by the first author. A quarter-inch microphone was placed approximately 40 cm from the castanets, and the sounds were recorded at the sampling frequency of 100 kHz. For time–frequency analysis, we used the wavelet transform to clearly resolve a frequency chirp within a few milliseconds from the impact. The scalograms were calculated using Morlet wavelets.

The three scalograms depicted in Fig. 1(c) have two common components. The first frequency components appear between 1000 and 2000 Hz. Their instantaneous frequencies increase with time for several milliseconds. The second frequency components are appear at the frequency of approximately 5000 Hz for castanets A and B and at 4000 Hz for castanet C. Higher-frequency components exist around the moment of the impact, while their structures differ among the instruments.

In Secs. III and IV, we refer to the two common frequency components as the first mode and second mode, and their (instantaneous) frequencies are indicated by f₁ and f₂, respectively.

For visualizing the sound fields around the castanets, we employed an optical method that comprised PPSI^20,21 and a high-speed polarization camera.²² The method has recently been used for visualizing sound fields.^14,17–19 In Fig. 2(a), we depict the measurement system, which forms the Fizeau-type optical interferometer. The expanded laser beam was projected onto the measurement area. Sound modulates the phase of the incident light, as described by the acousto–optic theory.¹⁰ The returned light is combined with the reference light and is incident onto the high-speed polarization camera. Notably, the camera is equipped with a linear polarizer array mounted on an image sensor, and it enables single-shot and quantitative imaging of a two-dimensional (2D) optical phase map upon using it along with a polarization interferometer. A detailed description of the optical system can be found in the literature.^14,23 By post-processing the observed images,^19,24,25 the high-speed video of a sound field can be obtained.

The measurement conditions are as follows: The laser beam had a wavelength of 532 nm and power of 200 mW. The beam diameter in the measurement area was 100 mm. The pixel dimension of the camera was 92 × 92, and each pixel of the camera imaged 1.1 × 1.1 mm² of the measured area. The frame rate of the camera was set to 100 000 fps. The castanets were positioned at the center of the measurement area. The player carefully aligned the tilt of the shells so that the light could propagate through the gaps between the shells. The recording timing of the camera was synchronized with the microphone recording, whose data are depicted in Fig. 1(c).

An example of the obtained image is depicted in Fig. 2(b). The outer circle represents the laser beam with diameter is 100 mm in the measurement area. The color of each pixel represents the value of the optical phase, which is proportional to the line integral of the sound pressure along the laser beam.¹⁴ Notably, the pixels where objects block the measuring light have no information, and the objects can be recognized as shadows in the recorded images. The color ranges of the following results were separately adjusted for ensuring visibility.

In this section, the imaging results of castanet A are presented. The imaging results of the three castanets are given as supplementary materials.²⁶ The sound-field videos of castanets A, B, and C, are given as Mm. 1, Mm. 2, and Mm. 3, respectively. As can be seen from the supplementary materials and videos, the following results can be applied to the three castanets.

The visualization results for castanet A are depicted in Fig. 3. To eliminate static and low-frequency noises,¹⁷ the differences between two successive images were shown. The sound field at the moment of impact is depicted in Fig. 3(a). The attack of the sound of the castanet was clearly visualized. The initial pressure wave (bright and dark parts) horizontally propagates along the surface of the shells. The propagated distance of the wavefront, obtained from the images, was approximately 6 mm within 0.01 ms. The propagation speed significantly exceeded the speed of sound in air, suggesting that the initial wavefront is attributed to the shell-surface deformation. The visualization results obtained 2 ms after the impact are depicted in Fig. 3(b). The sound field between the shells was captured. The amplitude of the sound field between the shells is higher than that outside the shell. The sound fields around the shells are complicated owing to the superposition of multiple frequency components, and, therefore, it is difficult to recognize individual wavefronts.

To extract the sound fields that correspond to either the first or second mode from the visualizations, a time-directional bandpass filter, which passes either f₁ or f₂, was applied to each pixel of the images. The filters were fourth-order infinite impulse response filters with a bandwidth of 500 Hz. The results of the first mode are depicted in Fig. 4(b). The intervals of the images are 140 μs, which approximately correspond to the quarter of the period of f₁. It is clearly seen that the air between the shells strongly vibrates with the period of f₁. In addition, compared with the air between the shells, the air around the shells vibrates with a smaller amplitude. The temporal waveforms extracted from a single pixel between the shells and a single pixel outside the shells are plotted in Fig. 4(d). The amplitude between the shells is significantly larger than that outside the shells, and the phases of both the waveforms differ by 90 degrees. Because both the shells form a relatively large volume inside them and the small gap that connects the inner volume and the surrounding air, this phenomenon can be regarded as the Helmholtz resonance. A theoretical analysis based on these results will be provided in Sec. IV A.

The results for the second mode are depicted in Fig. 4(c). The intervals of the images are 50 μs, which approximately correspond to the quarter of the period of f₂. The phase of the vibration between the shells is opposite to that of the node at the center of the major axis. The temporal waveforms from the pixels at both the sides of the node and near the point are plotted in Fig. 4(e). The phases of the sound on both the sides are opposite to each other, and that near the point differs by 90 degrees from them. In addition, it can be seen from the images that a weak spherical wave is radiated from the point.

For the second mode, we should visualize the sound field from other directions to investigate the spatial profile along the optical path. Unfortunately, sound fields between the shells cannot be captured from other directions, as there is no gap through which the measurement light propagates. We performed repeated measurements with the three instruments used in this study and with the same instrument, and we confirmed that the appearance of this mode was consistent.

The shape of the castanet—an internal cavity is connected to the surrounding air by a narrow and long slit—is similar to that of the Japanese wooden drum called the mokugyo. Sunohara et al.²⁷ modeled the side port of the mokugyo as the cylindrical hole, and their model showed good agreement with the measured values. Therefore, we model the castanets by following the idea of the previous study.

The scalograms in Fig. 1(c) show that the frequencies of the first modes increase with time. We consider that the frequency sweep is attributed to the movement of the shells during the sound radiation. Because both the shells move in the opposite direction because of the rebound after the impact, the height of the gap between the shells, h, increases with time. The resonant frequencies of each instrument, calculated using Eq. (1) as a function of h, are plotted in Fig. 6(a). The filled area indicates the combined standard uncertainties associated with the measurement of the geometry of the instruments.²⁹ Expectedly, the resonant frequency increased as h increased.

To further investigate the frequency sweep, h should be obtained from the experimental data. Because PPSI is capable of visualizing not only the phase of the light (sound field) but also the intensity thereof (shapes of objects),³⁰ we calculated h using the intensity videos by employing image-processing techniques (see  Appendix A for details). The instantaneous frequencies were also estimated using the microphone data. We employed the two methods using short-time Fourier transform (STFT) and analytic signal, respectively, as explained in  Appendix B. In Fig. 6(b), we depict the instantaneous frequencies calculated from Eq. (1) using the estimated time-varying gap height h (⁠ $f HR$⁠), using the STFT (⁠ $f STFT$⁠), and using the analytic signal (⁠ $f Analytic$⁠). The timings of the data were aligned such that the attacks were at the origin of the time scale. The results indicate that $f HR$ generally matches with both $f STFT$ and $f Analytic$ within the experimental uncertainties. This suggests that the model of the Helmholtz resonator with time-varying gap height can explain the first mode.

There are some issues requiring further investigation of the present model. First, according to the scalograms in Fig. 1(c), the frequency chirps ended around 5 ms after the impacts. An increase of h cannot explain this. It suggests that the Helmholtz resonator model may not be applicable when h is large. Second, the scalograms in Fig. 1(c) show that the first mode of the castanet C was dumped heavily than that of castanets A and B. To explain this, not only the geometry of the shells as modeled in this paper but also the collision and vibration of the shells should be considered. Third, the frequencies calculated using the model tended to be higher than those obtained using the microphone data; this occurs if the gap height h is overestimated. For example, because the gap height decreases toward the bridge and becomes zero at the bridge, the estimated values may be greater than the average gap height along the circumference. Furthermore, image imperfections such as aberrations and halos can result in estimation biases. This may have caused the overestimation; notably, evaluating and improving the image-processing scheme, as well as investigating model errors, will help address the overestimation.

The spatial profiles of the models and measured data are compared. The magnitudes of the second modes extracted from the PPSI data are depicted in the top of Fig. 7. In addition, the spatial profiles of the magnitudes along the white lines in the images are plotted at the bottom of Fig. 7. In the bottom plots, the line integrals of the modal shapes of both the spherical and cylindrical cavities are also shown. The direction of the line integral is parallel to the nodal diameter. The origins of the horizontal axes are at the positions of the nodes, and the negative directions are toward the point. The profile of castanet B and that near the bridge of castanet C could not be clearly extracted because of the small gap height and light diffraction. According to the plots, the line integrals of the two models have similar profiles, which matches well with the experimental data around the nodes. However, the magnitude of the experimental data does not become zero at –a. This is attributed to the sound fields outside the shells.

The discussions so far suggest that the sound radiation from the gaps of the shells has to be taken into account. The deviations of the model from measurement should be caused by a finite radiation impedance due to the gaps between the shells. Considering the radiation impedance of the gap as well as ellipsoidal cavity modes for castanets A and B will provide a further understanding of the second mode.

Because of the contactless and high-spatial-resolution imaging provided by PPSI and the high-speed camera, the sound fields between the shells of the castanets were successively visualized. By extracting the two dominant frequency components from the visualizations, it was observed that two acoustic resonances occurred between the shells. The first mode exhibited a frequency chirp of several hundred hertz for several milliseconds after impact. This behavior can be modeled by considering Helmholtz resonance with time-varying resonator shape. The frequencies of the second modes were almost time invariant. The optical visualizations and discussions revealed that the second modes are the standing wave modes formed by an interior volume of the shells having a single nodal diameter perpendicular to the major axis. These physical phenomena involved in the sound radiation from the castanets could be observed and identified owing to the unique features of the optical-imaging methods, such as millimeter resolution, capturing an instantaneous pressure field, and imaging a sound field that is considerably close to objects. We believe that the optical method will be a powerful tool for studying other musical instruments and various acoustic phenomena.

This work was supported in part by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Research Fellow (16J06772) and in part by the Waseda Research Institute for Science and Engineering, Grant-in-Aid for Young Scientists (Early Bird).

The heights of the gaps between the shells were calculated using the intensity videos recorded via PPSI. The procedure is depicted in Fig. 8(a).

An example of an intensity image is depicted in Fig. 8(b). The pixels that correspond to the shells and the finger of the player have small pixel values, as these objects block the measurement light. After trimming and thresholding the intensity images, the edges of the shells were detected using the Canny edge detection technique.³² The detected edge is depicted in Fig. 8(c). Then, the shells and gap were filled with the value one, and the number of the filled pixels along the lines that were perpendicular to the major axis were detected using the Radon transform algorithm.³³ The filled image and lines used for counting the number of pixels are depicted in Fig. 8(d). The angle of the perpendicular lines was manually determined, and eight parallel lines were used for the gap-height calculation. These operations are performed for each frame.

The instantaneous frequencies of the first mode were calculated using the microphone data by employing the following two methods.