A Thousand Bells: Acoustical Implementation of Bell Spectra Using the Finite Element Method and Its Compositional Realization (DMA dissertation, to be deposited in May 2020)

Download Link: http://hdl.handle.net/2142/108427

        The main purpose of my dissertation research is to assimilate the harmonic spectra generated by using the Finite Element Method technique into compositional processes. The FEM has been widely used in engineering analysis, especially in the analysis of solids and structures (e.g., dam, bridge, etc.) and of heat transfer and fluids. It been also used in the field of Acoustical Physics, especially for the creation and optimization of carillons. Based on the FEM technique, I create an arbitrary number of virtual bells with physically modelled in virtual space, and their spectral profiles are calculated, and finally, these newly generated harmonic parameters will be adopted and realized in a musical composition with acoustical, notational, and practical expression. 

2D Geometry of Bell 018 (just minor third tierce with just minor ninth nominal bell)

        The first chapter provides a brief introduction of pre and post-spectral music that is inspired by or employs bell sounds from which it derives its central materials. This includes music by Toshirō Mayuzumi, Jonatha Harvey, Tristan Murail, Magnus Lindberg, and works by the author.

        The second chapter introduces bell acoustics and the creation of new spectral profiles of optimal bell tone colors based upon just tuning ratios. In this chapter, I discuss how the concepts of consonance and Just Noticeable Difference in psychoacoustics are applied to use the 96 tone equal temperament tuning system for bell harmonic profiles.

Bell geometries with extra fine mesh (3D tetrahedral element)

Excerpts from the dissertation (3-3. Shape Function, Mapping, and Jacobian Matrices)

        The third chapter includes the theoretical basis of the FEM and its application to the isoparametric 2-D quadrilateral elements, which are the fundamental theories of how bell harmonies are mathematically calculated. This includes the central concepts of the FEM, such as the Principle of Virtual Work/Displacement, master to global coordinate transformation, FE shape functions, usages of Jacobian matrices, numerical integration of the stiffness matrix and the equivalent nodal force vector for the element by using the Gauss-Lagrange quadrature.

Nodal circles and nodal meridians /displacement fields of lower four modes: prime [2,1], tierce [3,1], quint [3,1#], and nominal mode [4,1]

       In the fourth chapter, I create bell model geometry by using 2D bell nominal curve and adjustable design variables. Physical parameters, such as the Poisson ratio, Young’s modulus, and material properties are also adopted from previous bell design research. Based upon the aforementioned prototypes, I create 24 different 3-D bell geometries, and analyze the spectra of these virtual bells. These bell models are analyzed, optimized and tuned to create tone colors that are defined in Chapter 2. After a validating process of the bell model, the general backgrounds of optimization theory are also introduced and analyzed for the purpose of creating 3-D virtual bells.

Parametric sweep graph of frequency tracking, by adjusting waist:hip ratio parameter

      By using the gradual transfigurations of virtual bells, in this dissertation I want to present an original musical “passing spectra” by using the physical modeling and its resultant acoustical data. In this example, the beginning harmony is from Bell 018 (perfect fifth prime with just minor 3rd tierce and just minor 9th nominal), and the final arrival harmony is from Bell 013 (just major third tierce with just major seventh prime and harmonic minor seventh nominal).

Transfiguration of bell geometry from Bell 018 to Bell 013.

       To conclude the project, I am currently writing a chamber orchestral piece, Quasi una campana (working title), with a substantial amount of musical material based on generated bell harmonies from the research. This newly generated timbre-harmonic system will be realized by specially tuned acoustic instruments and a microtonal synthesizer, which I had previously engineered and programmed. As such, I intend to further the theory of harmony established by earlier spectral composers, and develop analytical research, focused primarily on the analysis of mathematical and acoustical models of campanology.

The beautiful McFarland Carillon I administered for 3 years (2015–2017) located on the south quad at the University of Illinois. It is 185-feet tall and made up with 49 dutch bells ranging from low C3 to high C7. The combined weight of the bells is about 15.5 tonnes and their combined cost is half-a-million dollars. The carillon is controlled by MIDI.