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Effect of vibration properties of a resonance board on piano timbre.

Abstract

The resonance board is one of the most important components of a piano and greatly affects the piano's timbre. The vibration properties of eight resonance boards made from Picea glehnii, Picea jezoensis, Picea spinulosa, and Picea sitchensis were analyzed using the spectrum analysis method of the fast Fourier transform following vibration theory. The modulus of elasticity (E) and velocity of vibration transmission (C) of each resonance board were calculated. After the resonance boards were incorporated into pianos, the mean amplitude, ratio of maximum to minimum amplitudes, and mean ratio of successive amplitudes of resonant frequencies for 36 typical keys of each piano were obtained from the frequency spectrum. The correlation between a piano's timbre and the resonance board's vibration properties was analyzed. The results show that the piano timbre improves with increasing E and C, especially in the y direction of the resonance board.

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A soundboard amplifies the soft sound produced by the strings of a piano. The pleasant sound of a piano depends on the construction, raw materials, and processing of the soundboard (Jin 2002). The soundboard is made up of a resonance board, ribs, and frame. The resonance board is a thin wooden plate about 8 mm thick, made of spruce strips with widths of 80 to 100 mm, which are usually glued together edge to edge.

In order to obtain good performance of a resonance board, the vibration properties of wood need to be known. There are presently two theories that can be used to evaluate the vibration properties of wood: beam vibration theory and plate vibration theory. Vibration parameters, such as the resonance frequency, Young's modulus, shear modulus, Poisson's ratio, vibration energy loss, and sound velocity in the wood, are measured using the beam vibration theory (Sobue 1986; Haines et al. 1996; Kubojima et al. 1996, 1997; Matsunaga et al. 1996; Liu et al. 2005, 2007; Rujinirun et al. 2005; Shen 2006), and the vibration mode, dynamic loss, complex Young's modulus, complex shear modulus, and complex Poisson's ratio are measured using the plate vibration theory (Nakao et al. 1985, Tonosaki et al. 1985, Nakao 1996, Liu et al. 2008).

Previous research has focused on the vibration principle of the soundboard. Berthaut et al. (2003) studied the vibrational behavior of the piano soundboard using finite-element models and analytical models, validated their results in experimental modal analysis, and found that the finite-element models accurately describe the dynamical behavior of a given piano soundboard in the low-frequency domain. The new method of electronic speckle pattern interferometry has also been used to study the deflection shapes of a piano soundboard, and the interferometric results have been compared with results obtained using a finite-element model (Moore and Zietlow 2006). Giordano (1997, 1998a) studied the mechanical impedance (Z) of a piano soundboard in the musically important frequency range of 50 to 104 Hz and found that the soundboard ribs have an important effect on the frequency dependence of Z above a few kilohertz; he then constructed a simple finite-element model of the vibration properties of the piano soundboard. The effect of the ribs on the velocity of radial vibration transmission of the soundboards was analyzed by Liu et al. (2009b), who found that the velocity of the soundboard in the radial direction increased by a factor of 2 when ribs were affixed on the resonance board. Giordano (1998b) investigated the generation of sound by a piano soundboard experimentally by measuring the sound pressure, p, and the soundboard velocity, [v.sub.b], produced in response to a force applied at the bridge. To study the effect of the soundboard on the acoustical properties of a piano, Suzuki (1986) investigated the vibration and radiation characteristics of the soundboard for a 6-foot grand piano and found that the radiation efficiency is very low below 80 Hz, fair in the frequency range of 100 Hz to 1 kHz, and high above 1.4 kHz. Adrien et al. (2008) investigated the effect of (down-bearing) string tension on the vibration of a piano soundboard using a finite-element model and evaluated the importance of the crown on the effects of the down-bearing tension on modal frequencies and mobility. Liu et al. (2009a) studied the relationship between the vibration properties of the resonance board and the acoustical quality of a piano as evaluated by piano experts. The piano sound has also been measured and analyzed in many works (Alfredson 1978, Giordano and Jiang 2004, Nishiguchi 2004, Bensa et al. 2005).

Despite the above research, there has been little investigation of the effect of the soundboard on the acoustical quality of a piano. This article investigates the effect of the vibration properties of a resonance board on the piano's timbre after the vibration properties of the resonance board and the timbre of the piano are measured.

Materials and Experiment

Materials

Eight resonance boards (length by width by thickness = 1,408 by 937 by 8 mm) were made from the wood of Picea glehnii, Picea jezoensis, Picea spinulosa, and Picea sitchensis (two boards per species). Eight upright pianos were made using these resonance boards after their vibration properties were measured. The resonance board structure and jointing are shown as Figure 1.

Experiment

Measurement of the vibration properties of a resonance board.--The vibration spectrum of a resonance board was measured using a CF5220 Multi-Purpose FFT (fast Fourier transform) Analyzer manufactured by Ono Sokki following the vibration theory of a thin plate. The flexural rigidities [D.sub.11], [D.sub.22], and [D.sub.12] and torsional rigidity [D.sub.66] can be obtained from the frequency equation for free vibration (Eq. 1) and the resonant frequencies (2, 0), (0, 2), (1, 1), and (2, 2), which were identified from the vibration spectrum (Hearmon 1961, Sobue and Kitazumi 1991, Sobue and Katoh 1992):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [f.sub.r](m, n) is the resonant frequency (Hz); h, a, and b are the height (m), length (m), and width (m) of the sample, respectively; [rho] is density (kg/[m.sup.3]); [D.sub.11], [D.sub.22], and [D.sub.12] are flexural rigidities; and [D.sub.66] is torsional rigidity. The moduli of elasticity of the resonance board in the x and y directions are obtained using Equation 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [[mu].sub.x] is Poisson's ratio in the x direction and [[mu].sub.y] is Poisson's ratio in the y direction.

The vibration transmission velocity of the resonance board is as follows (Rossing and Fletcher 2004):

[C.sub.P] = ([square root of [E/[rho](1 - [[mu].sup.2])]] (3)

where C is the velocity of vibration transmission and p is Poisson's ratio. The vibration transmission velocities in different directions are calculated with values of E and [mu] for different directions.

Measurement of piano timbre.--After the pianos were constructed, sound spectra for 36 typical keys of a piano were obtained using the CF5220 Multi-Purpose FFT Analyzer when the keys were tapped with uniform power. The special hammer was made, and the uniform power was obtained by free fall of the hammer from the same height. These keys cover low-, middle-, and high-frequency regions but exclude the keys having the lowest and highest frequencies. Some keys in the transition region from low to middle frequencies and from middle to high frequencies were also selected.

Analysis method for piano timbre

The timbre of a musical instrument is complicated and difficult to characterize with a parameter. Presently, subjective evaluation is not common because one uses different evaluation terminology according to one's cultural background, language, and professional field (Han 2003). To evaluate timbre objectively, the frequency spectrum is often used. The frequency spectrum can show the number of overtones and the relationship among harmonic energies (Han 2002, Huang and Xie 2005, Liang and Xie 2005).

This article analyzes the timbre for each key using three indices by normalizing the data of the frequency spectrum.

Amplitude mean value

The amplitudes of the fundamental tone and harmonics of each key were taken from the frequency spectrum measured by the FFT analyzer. The amplitude mean value (AMV) of all resonance frequencies was then calculated using Equation 4. The AMV indicates the ability of vibration energy to radiate through the soundboard, and a higher AMV means lower acoustical impedance:

AMV = 1/n [n.summation over (i=1)] [A.sub.1] (4)

where n is the number of modes of the resonance frequency, and [A.sub.i] is the amplitude of the resonance frequency of mode i.

Maximum-to-minimum ratio of amplitudes of the resonance frequencies

The maximum and minimum amplitudes of the resonance frequencies of each key were taken, and the ratio (RV) of the maximum amplitude to the minimum amplitude was calculated (Eq. 5). More vibration energy of the strings is radiated by a soundboard, and full timbre and greater loudness are obtained if the RV is close to 1:

RV = [A.sub.max] (5)

where [A.sub.max] is the maximum amplitude among all resonance frequencies, and [A.sub.mm] is the minimum amplitude among all resonance frequencies.

Mean ratio of successive amplitudes

The peaks of all resonance frequencies form an envelope curve. To enhance and evenly radiate the vibration energy of a soundboard passed from strings in the whole frequency band, the peak shapes of the fundamental tone and harmonics need to change gradually, thus producing a smooth envelope curve (Liu 2004). The envelope curve is smoother if the mean ratio of successive amplitudes (RVS) calculated using Equation 6 is lower. A smoother envelope curve may also make a clearer sound:

RVS = 1/N - 1 [n.summation over (i=1)] [absolute value of [A.sub.i] - [A.sub.i+1]]/ [A.sub.i] (6)

where n is the number of modes of the resonance frequency, [A.sub.i] is the amplitude of the resonance frequency of mode i, and [A.sub.i+1] is the amplitude of the resonance frequency of mode i + 1.

Results and Discussion Analysis of the vibration properties of resonance boards

Following our previous research (Liu et al. 2007), we divided wood strip samples into group No. 1 and group No. 2 according to their density, ratio of the modulus of elasticity to density, ratio of the modulus of elasticity to shear modulus, acoustical radiation, and acoustic impedance. These strips were incorporated into resonance boards No. 1 and No. 2 and subsequently pianos No. 1 and No. 2, respectively.

The moduli of elasticity ([E.sub.x] and [E.sub.y] for the x and y directions of the resonance board, respectively) and the velocities of vibration transmission ([C.sub.x] and [C.sub.y] for the x and y directions of the resonance board, respectively) were compared between the two resonance boards of the same species and are shown in Table 1, which shows that E ranges from approximately 3.0 to 6.0 GPa for all resonance boards, and C ranges from approximately 3,500 to 4,000 m/s except for P. spinulosa. For the same species, E and C for the No. 2 resonance board are higher than those for the No. 1 resonance board in both x and y directions. The differences in E and C between the two boards of P. spinulosa are higher than those for the other species. Additionally, E and C for P. spinulosa are lowest among the wood species.

Analysis of the timbre of the pianos

Analysis of the AMV.--The timbre was compared for two pianos having resonance boards made of the same species. The ratios of AMVs for corresponding No. 1 and No. 2 pianos, calculated using Equation 7, are shown in Figure 2:

[AMV.sub.L] = 1/n [n.summation over (i=1)] [A.sub.2i]/[A.sub.1i] (7)

where n is the number of modes of the resonance frequency and [A.sub.1i] and [A.sub.2i] are the amplitudes of the mode i resonance frequency for the No. 1 and No. 2 pianos of the same wood species, respectively. Figure 2 shows that, for most keys, the amplitude of the No. 2 piano is higher than that of the No. 1 piano for the same species. The difference in amplitude in the high-frequency region is more obvious than that in the low- and middle-frequency regions. [AMV.sub.L], ranges from 0.955 to 1.270 and has an average of 1.028 for P. glehnii, ranges from 0.939 to 1.188 and has an average of 1.026 for P. jezoensis, ranges from 0.950 to 1.190 and has an average of 1.019 for P. spinulosa, and ranges from 0.972 to 1.299 and has an average of 1.045 for P. sitchensis. Statistically, the fundamental tone and harmonic amplitude of the No. 2 piano are higher than those of the No. 1 piano for all four species for 70 to 75 percent of the keys.

Analysis of the RV.--The difference in the RV between two pianos of the same wood species was analyzed, and the results are shown in Figure 3, which shows that for most keys, the RV for the No. 1 piano is higher than that for the No. 2 piano of the same species. The RVs of pianos with resonance boards of P. glehnii and P. jezoensis are lower than RVs of pianos with resonance boards of the other two species. For the same species, the variation trends of the RV from low- to high-frequency regions of the piano are similar, and the difference in RV between the two pianos for approximately the last 44 keys is more obvious than that for approximately the first 44 keys, especially in the cases of P. spinulosa and P. sitchensis. The mean and range of the RV for all keys of the No. 1 piano are greater than those of the No. 2 piano for all four species.

Analysis of the RVS.--The difference in the RVS between the two pianos of the same wood species was analyzed, and the results are shown in Figure 4, which shows that the difference in RVS between two pianos of the same species is not obvious, but the RVS of the No. 1 piano is slightly higher than that of the No. 2 piano. As in the case for the RV, the variation trends of the RVS from the low- to high-frequency regions of the piano are similar, and the difference in the RVS between the two pianos for approximately the last 44 keys is more obvious than that for approximately the first 44 keys, and the mean of the RVS for all keys of the No. 1 piano is greater than that of the No. 2 piano for all four species. The two pianos of P. sitchensis have the largest difference in the RVS among all species, whereas the two pianos of P. jezoensis have the smallest difference.

The above analysis has shown that E and C of the No. 2 resonance board are higher than those of the No. 1 resonance board for all four species. Correspondingly, the AMV of the No. 2 piano is higher and the RV and RVS are lower than those values for the No. 1 piano. This indicates that a piano has better timbre with higher E and C of the resonance board.

Relationship between the vibration properties of the resonance board and the timbre of the piano

To further explore the effect of vibration properties of the resonance board on the timbre of the piano, their correlation was analyzed. The correlations between E and the AMV, RV, and RVS are shown in Figure 5, which shows that the correlation between [E.sub.x] of the resonance board and the AMV is linear positive to some extent with correlation coefficients of 0.5890 and 0.5925 for the x direction and y direction of the resonance board, respectively. This indicates that the sound volume of the piano increases with increasing E of the resonance board. There is negative correlation between [E.sub.y] and the RV and RVS; however, there is no significant correlation between Ex and the RV and RVS. This means that sound radiation attenuation of the resonance board is slower and that the envelope curve of the frequency spectrum of the piano is smoother as [E.sub.y] increases; that is, the sound radiation of the resonance board and the timbre of the piano improve.

The correlations between C and the AMV, RV, and RVS are shown in Figure 6, which shows that the correlation between C of the resonance board and the AMV is linear positive to some extent with correlation coefficients of 0.5883 and 0.5815 for the x direction and y direction of the resonance board, respectively. This indicates that the sound volume of the piano increases with increasing C of the resonance board. The RV and RVS decrease with increasing C to a certain degree, but their correlation is not significant.

Because of their high cost, only eight pianos were constructed in this work. Therefore, we cannot determine whether the correlations between vibration properties in the x direction of the resonance board and the RV and RVS are significant. However, from the above analysis, we know that the correlation between vibration properties in the y direction of the resonance board and the timbre of the piano is more significant than between those in the x direction. This indicates that the timbre of the piano is more affected by the vibration properties in the y direction of the resonance board than by those in the x direction of the resonance board. We expect that this will be investigated further in future work.

Conclusions

We studied the effect of vibration properties of a resonance board on the timbre of a piano and obtained the following results. The timbre of a piano improves with increasing E and C of the resonance board. The AMV increases and the RV and RVS decrease with increasing E and C, especially in the y direction of the resonance board. This indicates that the sound radiation attenuation of the resonance board is slower and that the envelope curve of the frequency spectrum of the piano is smoother with increasing E and C of the resonance board; that is, the sound radiation of the resonance board and the timbre of the piano improve.

Acknowledgments

The authors are grateful for the support by the Fundamental Research Funds for the Central Universities (2572014CB02) and the National Natural Science Foundation of China (31170522) for this study.

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The authors are, respectively, Doctoral Student, Associate Professor, Professor, and Professor, Northeast Forestry Univ., Key Lab. of Bio-Based Material Sci. and Technol. of National Ministry of Education, Harbin, Heilongjiang, People's Republic of China (miao. yuanyuan@foxmail.com, liu.zhenbo@foxmail.com [corresponding author], yxl200488@sina.com, shenjunr@126.com). This paper was received for publication in August 2014. Article no. 14-00079.

doi: 10.13073/FPJ-D-14-00079

Table 1.--Vibration properties of the resonance boards.

                                                [E.sub.x]   [E.sub.y]
Samples                                           (GPa)       (GPa)

A1 (No. 1 resonance board for Picea glehnii)      4.659       4.855
A2 (No. 2 resonance board for P. glehnii)         4.739       6.069
B1 (No. 1 resonance board for P. jezoensis)       4.469       5.114
B2 (No. 2 resonance board for P. jezoensis)       4.809       5.502
Cl (No. 1 resonance board for P. spinulosa)       3.013       3.112
C2 (No. 2 resonance board for P. spinulosa)       3.975       4.468
D1 (No. 1 resonance board for P. sitchensis)      4.881       5.056
D2 (No. 2 resonance board for P. sitchensis)      5.888       6.131

                                                [C.sub.x]   [C.sub.y]
Samples                                           (m/s)       (m/s)

A1 (No. 1 resonance board for Picea glehnii)      3,620       3,696
A2 (No. 2 resonance board for P. glehnii)         3,440       3,893
B1 (No. 1 resonance board for P. jezoensis)       3,527       3,773
B2 (No. 2 resonance board for P. jezoensis)       3,725       3,984
Cl (No. 1 resonance board for P. spinulosa)       2,687       2,731
C2 (No. 2 resonance board for P. spinulosa)       3,072       3,257
D1 (No. 1 resonance board for P. sitchensis)      3,518       3,581
D2 (No. 2 resonance board for P. sitchensis)      3,862       3,941
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Author:Miao, Yuanyuan; Liu, Zhenbo; Liu, Yixing; Shen, Jun
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Date:Jan 1, 2016
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