How the shape of the musical triangle influences its sound Open Access

The musical triangle—technically just a metal rod shaped into a triangle—produces enchanting and beautiful tones, raising deep and profound questions about the connection between sound and physics. A typical musical triangle is made of metal bent into a triangular shape with one open end. It is suspended at the top closed corner with a string or wire, and struck with a metal rod called a beater. The triangle is classified as an idiophone in the Hornbostell–Sachs classification system (von Hornbostel and Sachs, 1961), categorizing musical instruments based on sound-production principles. The triangle produces an indefinite pitch sound composed of multiple non-harmonic overtones.

The frequency characteristics are close to those of straight metal rods made with the same material and having the same length and diameter (Dunlop, 1984; Fletcher and Rossing, 1998). Although recent research has revealed that the frequency characteristics also depend on the angle of the triangles (Stanciu , 2021; Stanciu , 2022), the influence on the resonance due to the triangular shape has not been well investigated.

In this paper, we aim to reveal the influence of the triangular shape on their sounds. To investigate acoustic propagation around the triangle, we conducted an experiment measuring triangle sound by parallel phase-shifting interferometry (PPSI) (Ishikawa , 2016), which is an acousto-optic imaging method and can measure two-dimensional (2 D) sound pressure distribution. By analyzing the observed sound fields, we found that acoustic resonance occurs in the area of semi-open triangular air created by the musical triangle and confirmed that the sounds at these frequencies were louder and lasted longer.

To ensure the triangles fit within the measurement area, we selected the two triangles with respective lengths of 70 mm (T70, ST-7, Suzuki Musical Instrument MFG. CO., LTD., Hamamatsu, Shizuoka, Japan) and 100 mm (T100, ST-10, Suzuki Musical Instrument MFG. CO., LTD., Hamamatsu, Shizuoka, Japan) on the sides opposite the open end. Both triangles had a cross section diameter of 11 mm and were made of special steel and made with a special quenching process. The beater (SP-400, Suzuki Musical Instrument MFG. CO., LTD., Hamamatsu, Shizuoka, Japan), made of steel with rubber handles, was 133 mm long and had a cross section diameter of 7 mm. In a preliminary experiment, we also used a beater with a length of 163 mm and a cross sectional shape of 7 mm; however, since we found that the type of beater had no effect on the resonance, we used the shorter one in this experiment. The triangles and beater used in our experiment were manufactured by the Suzuki Corporation, Japan.

A schematic diagram of PPSI used in our experiment is shown in Fig. 1(a). Light emitted from a laser is divided into two paths: reference light and object light. The object light passes through the measurement area, and its phase is modulated by sound. The combination of the two lights is captured by a high-speed polarization camera (Onuma and Otani, 2014) as interference fringes. Since the high-speed camera has a maximum shutter speed of 1.5 M fps, it can capture transient sound fields and enables the measurement of various sound fields, such as those from loudspeakers and musical instruments, as well as flow-induced sounds, and Doppler effects (Ishikawa , 2018; Ishikawa , 2020; Tanigawa , 2020; Akutsu , 2023). The measurement area is limited by the diameter of the lens, optical flat, and mirror. Our current system has 200 mm in diameter.

The experimental setup is shown in Fig. 1(b). The triangle was suspended from a fixture with a string. An experimenter struck the triangle with the beater. The microphone was 300 mm from the triangle. Measurements with the microphone and camera in the PPSI system were performed synchronously. The sampling frequency of the microphone and camera were set to 50 kHz. The image size captured by the camera was $134×140$ pixels.

The spectrograms of the sound measured with the microphone are shown in Fig. 2(a). As in previous studies, we observed multiple non-harmonic overtones. Among the overtones, those at the frequencies indicated by the red arrows in Fig. 2(a) were louder and decayed more slowly than others at adjacent frequencies.

To investigate the cause of this feature, we analyzed the sound-field images captured by PPSI. Figures 2(b) and 2(c) show the visualized results measured by PPSI, which were extracted at each frequency by the bandpass filter centered on the peak frequency obtained by the microphone measurement. The color indicates the phase of light modulated by the sound. The left four images are the sound field from $t=10.00$ to $t=10.06$ ms with the interval of 0.02 ms. The right four images are those from $t=50.00$ to $t=50.06$ ms with the interval of 0.02 ms. The visualized sound fields of T70 and T100 are available as supplementary material (SuppPub1 and SuppPub2).

At the frequencies shown in Figs. 2(b) and 2(c), different characteristics of the sound field can be observed. At the frequency shown in Fig. 2(b), prominent waves were observed in the air formed by the triangle. On the other hand, at the frequency shown in Fig. 2(c), sound waves were spreading around the striking point. Additionally, while the sound waves inside the triangle observed in Fig. 2(b) were also observed around $t=50.00$ ms, the sound waves in Fig. 2(c) were hardly observable around $t=50.00$ ms. From these results, the sound fields at the frequencies shown in Fig. 2(b) were the standing waves due to the resonance inside the triangle formed by the triangles, because the frequencies at which the standing waves were observed decayed more slowly than the sound fields at the frequencies shown in Fig. 2(c).

To confirm that the standing waves are derived from resonance modes, we conducted a numerical simulation using the finite element method to calculate the eigenfrequencies of the resonance modes. The simulations were conducted with COMSOL Multiphysics^® software v.6.1 (COMSOL AB, Stockholm, Sweden). The geometry was extracted from three-dimensional (3 D) models described in the left panel of Fig. 3(a). The geometric parameter values were set to $θ=62 °$⁠, d₂ = 23.5 mm, d₂ = 23 mm, φ_t = 11 mm for both T70 and T100. The other parameters were set as l₁ = 33 mm and l₂ = 45 mm for T70, and l₁ = 64 mm and l_s = 78 mm for T100. For the sake of simplicity, the simulation was performed in a 2 D plane run through the central axis of the triangles, and the geometry was defined in the 2 D plane as shown in the right panel in Fig. 3(a). As for the boundary conditions, the edges in contact with the triangle were rigid walls, and the triangle opening was the inlet, to which a constant pressure of $1 Pa$ was applied. To calculate the eigenfrequencies of the resonance modes, an ARPACK (Lehoucq , 1998) eigenfrequency solver implemented in COMSOL Multiphysics^® software was used to calculate the eigenfrequencies of resonance modes. The range of the eigenfrequencies in real parts was set to from 0 to 37.5 kHz.

The results are shown in Fig. 3(b). The top row shows the experimental data, which are real parts of complex amplitudes obtained using the Fourier transform along the time direction. The bottom row shows selected simulated eigenmodes. As can be seen in Fig. 3(b), the simulation results show spatially similar patterns at frequencies close to those observed in the experiment. However, the simulation frequencies did not perfectly match the measured ones. This discrepancy can be attributed to the fact that the frequencies responsible for generating standing waves in the measured data were largely influenced by the eigenfrequencies of the triangular rods. Therefore, we can infer that standing waves were observed in the measured data at frequencies close to the eigenmodes of the triangle air among the eigenfrequencies of the triangle. From these results, the observed waves are the resonances in the air within the triangle.

To summarize the results, we refer again to the spectrogram in Fig. 2(a). The frequencies indicated by the red arrows are the frequencies at which the standing waves occurred. Thus, the acoustic resonances at those frequencies lead to louder and longer sounds.

To confirm the process of resonance generation, the transient sound field immediately after the impact is shown in Fig. 4. These are the data obtained by applying a high-pass filter with a cutoff frequency of 1 kHz to cut low-frequency noise. The transient sound fields of T70 and T100 are available as supplementary material (SuppPub3 and SuppPub4). Immediately after the impact, the sound waves propagated through the rod of the triangle (⁠ $t=7.90$ ms of T70 and $t=7.92$ ms of T100). When the wave reached the bend, the direction of travel changed according to the triangular shape, and the first sound wave interfered with the subsequent sound wave (⁠ $t=7.94$ ms of T70 and $t=7.98$ ms of T100). These results suggest that the standing waves were formed by the combinations of radiations from the three sides of the triangle.

We interpret these experimental observations as the resonance occurring in air inside the triangles. However, concerns remain regarding the small sound-pressure differences between inside and outside the triangle and the inability to identify the conditions for the resonance.

As can be seen from Fig. 2(b), radiations were emitted outside the triangle at the comparable amplitudes as inside the triangle [e.g.,11570 and 14400 Hz in Fig. 2(b)]. Ideally, if resonance occurs, the sound pressure inside should be greater than that outside. The triangular air being a semi-open space may be influencing this; however, further investigation should be performed for this feature.

Further experiments and analysis of the conditions under which resonance occurs are also an open question. As shown in Figs. 2(b) and 2(c), there were frequencies at which resonance occurred and others at which it did not. We hypothesize that the resonance occurs when the eigenfrequency of the triangle coincides with the resonance mode of air inside the triangle. For a more detailed discussion, further investigation of the relationship between the eigenfrequency of a triangle and resonance mode of air inside the triangle should be clarified, for example, by synchronous measurement with the vibration of the triangle.

We experimentally discovered that resonances occur in the air formed by a musical triangle. These results suggest that the reason why the shape is triangular is that the resonances do not occur with a straight rod. Although the interpretation of the experimental results is open to discussion as noted in the Discussion section, we have observed that resonance also occurs in the semi-open spaces formed by a triangle. In addition to triangles, the semi-open air resonance phenomenon may also occur with percussion instruments with semi-open spaces, such as tambourines, castanets, and hi-hat cymbals.

See the supplementary material for the visualized sound fields of T70 and T100, respectively, and for the transient sound fields of T70 and T100, respectively.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.