Elsevier

Applied Acoustics

Volume 164, July 2020, 107284
Applied Acoustics

Sound model of an orchestral kettledrum considering viscoelastic effects

https://doi.org/10.1016/j.apacoust.2020.107284Get rights and content

Abstract

The modeling of the mechanical systems that describe musical instruments has long been a topic of interest for acoustic physicists. The orchestral kettledrum, distinguished by a broad spectrum of sounds, has been the subject of multiple investigations related to the goal of generating a realistic digital sound synthesis. In the present work, we apply the Green function method to estimate modal frequencies considering an air-loaded viscoelastic membrane. We propose a method that includes viscoelasticity to accurately predict the sound spectrum of the modeled kettledrum. Results are compared to real sound recordings of an orchestral kettledrum obtained in controlled conditions. The calculated modal frequencies are found to coincide well with the real values with an absolute mean error of 1.25±0.76 Hz and 1.87±1.83 Hz for the A2 and B2 tuned drumheads respectively. The spectral envelope of the synthesized sound spectrum coincides well with the Fourier transform of the real sound. The viscoelastic term was found to generally reduce the amplitude of the sound spectrum and in certain cases, better approximate the modal frequencies and decay times. The modal synthesis method used here is numerically lightweight and can be adapted to be used in real-time applications with low computational resources. To reproduce our experiment, the recorded kettledrum sounds and Python source code of the model are freely available.

Introduction

Drums are considered to be among the first musical instruments ever invented by humans. Throughout history and different cultures, drums have played an essential role in music, religious ceremonies, battles, and social gatherings. Nowadays, drums continue to drive the rhythm in every performance that includes them (See Table 1).

It is well known that the sound produced by most drums, regardless of their size, has no pitch. In the present work, we develop an analytic sound model for a particular type of percussive instrument called the kettledrum, also known as the timpani.

The kettledrum consists of a drumhead (calfskin or synthetic high tensile strength film) stretched over a large empty bowl traditionally made of copper. Its membrane is made to vibrate by the impact of a mallet or drumstick. The interaction of the membrane with the air on the top and beneath its surface generates the kettledrum’s unique pitch and timbre [1]. For this reason, it is considered to be the underlying percussion instrument in the orchestra. Moreover, the kettledrum sound can be tuned over a range of more than one octave by varying the tension of the membrane by using the screws around the circumference or by using a tensioning pedal. Due to the large surface of the drumhead, the striking position plays a crucial role in the resulting sound. As described by Rossing [1], in order to generate the desired harmonic sound, the kettledrum should be struck about a quarter of the way from the edge of the drumhead to the center. If the drum is struck in the center, other inharmonic fundamental modes will dominate the sound.

Although several physical modeling sound synthesis methods of the kettledrum are available in the literature, the primary motivation behind the present work is the lack of a highly realistic, computationally-efficient and freely available model. Our physical modeling approach belongs to the category usually known as modal synthesis where the complex dynamic behavior of a vibrating object may be decomposed into contributions from a set of modes, with each mode oscillating at a single frequency [2].

Related research work has been done in the past. For instance, Christian et al. [3] were the first authors to propose a model for the acoustic pressure using the Green’s function method. Their model was later improved and extended to a composite two membrane case by Sankalp and Anurag [4]. However, none of those models included viscoelastic dependency. In comparison, Chaigne and Kergomard [5], Rhaouti and Chaigne [6] include the viscoelastic damping coefficient, which accounts for average loses, but by employing a numerical model instead of an analytical solution. Previous kettledrum models report unreasonably high decay times for the musically relevant modes Sankalp and Anurag [4] which is mostly attributed to the vibrating membrane that continuously dissipates the energy from it in the form of sound waves. Our model differentiates from former approaches, in addition to including a linear viscoelastic drumhead model, we also incorporate a model sound synthesis technique allowing us to approximate the acoustic pressure field directly.

To verify our kettledrum model, we compared the synthesized sounds with the ones produced by a real instrument. More specifically, we have recorded a set of kettledrum sounds in a controlled environment, then we have thoroughly contrasted the frequency spectrum produced by our model with the spectrum produced by a 32” inch orchestral kettledrum. The similarity between both sounds is remarkable, particularly in the preferred modes.

The remainder of the paper is structured as follows. In Section 2, we formulate the problem and describe the methodology used to obtain the mathematical model. In Section 3, we present the experimental setup, discuss the numerical procedure, as well as the spectral analysis and model parameters sensitivity analysis. Finally, we present concluding remarks in Section 4.

Section snippets

Material and methods

Due to the symmetry of the orchestral kettledrum, it can be abstracted as a cylindrical cavity, as shown in Fig. 1. Besides the mathematical convenience of this abstraction, [7] showed that the difference between the calculated frequency modal ratios obtained using this model and a better-defined kettle-shape model is small and of little musical significance. The acoustical model of the instrument requires solving a coupled boundary problem for membrane vibration and air pressure distribution

Experimental results and discussion

Our results are divided in 5 subsections. First, we describe our experimental setup and then we discuss the numerical procedure. Next, we compare the calculated and measured frequencies of the first 16 modes using the Power Spectral Density (PSD) of the recorded sounds. Then, we analyze the effect of the viscous term in the calculated spectral density and compare both models with the experimental results. Finally, we investigate the effect of the microphone position (the point where acoustic

Conclusions

Among orchestral drums, the kettledrum is generally considered the most important due to its capacity to convey a clear sense of pitch. Hence it is relevant to adequately estimate the sound spectrum. Our results show that the modal frequencies of an orchestral kettledrum can be accurately estimated using the air-loaded membrane formulation defined in Section 2.2. The modal sound synthesis technique introduced here to estimate a synthetic spectrum that closely matches the real sound spectrum.

Acknowledgments

The authors wish to thank Miguel Ángel Cuevas and the Faculty of Arts at the Autonomous University of Baja California for the support during the recording of the kettledrums sounds. The authors also wish to thank the anonymous reviewers for their valuable comments and suggestions that helped us to improve the quality of this manuscript.

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