# Physics of Baroque Keyboard Instruments: Project Plan – Kachelein

Introduction

– In chapter 6 of the class text, we were first introduced to the motions of waves on strings, even modeling their motion in a few simple cases. In chapter 11, the physics of musical instruments is explored. The guitar and the piano are featured prominently in this chapter, as they have relatively simple mechanics and are stringed, hence the framework for exploring their behavior has been established. Other instruments, each with different physical characteristics, can also be modeled. The goal of my project is to start with information in chapter 11 and build full, working models of the harpsichord, clavichord and, hopefully, the pipe organ, three instruments with which I am very familiar. The final aim of the project is to compare computed data with actual signals from the instruments in question, verifying (or nullifying) the computational models.

Methods

– MATLAB will, of course, be used as the primary, if not sole, computational tool (if a difficult differential equation needs to be solved, Mathematica may be invoked). For the harpsichord, a model for the plucking force will need to be built (harpsichord strings are plucked, rather than struck like those of a piano). Though plucked strings are already discussed in the guitar sub-chapter, I feel that a more complicated but realistic model can be built (for example, by considering the effect of the plectra as the string slides off of it). The string’s motion will be governed, naturally, by the wave equation:

$\frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2 y}{\partial x^2}$

which is computationally realized by eq. 6.6:

$\\y(i,n+1)=2[1-r^2]y(i,n)-y(i,n-1)+r^2 [y(i+1,n)+y(i-1,n)]\\\text{where }r=c\Delta t/\Delta x\text{ and }c\text{ is speed of sound in air.}$

– The clavichord is also a string instrument which has a striking mechanism like that of the piano, but which is physically much different. In particular, the forcing site is also a boundary of the string, so a radically different model from that of the piano/harpsichord may or may not need to be used.

– If the organ proves feasible after addressing the instruments above, I will attempt to model at least one type of pipe and at most two types (round metal and square wooden pipes). Part of chapter 11 addresses a one dimensional pipe, and the methods are claimed to be easily extendable to three dimensions using coupled partial differential equations. For the one dimensional case, we have pressure, p, and velocity, v, in air of density ρ and sound speed c given by:

$\\ \rho\frac{\partial v}{\partial t} = -\frac{\partial p}{\partial x} \\ \frac{\partial p}{\partial t}\ \ = -\rho c^2\frac{\partial v}{\partial x}$

which has a numerical form given by equation 11.37 in the book:

$\dpi{120} v\left(M-\frac{1}{2},n+\frac{1}{2}\right)=\frac{1}{1+\frac{Z\Delta t}{\rho\Delta x}}\left[\left(1-\frac{Z\Delta t}{\rho\Delta x} \right )v\left(M-\frac{1}{2},n-\frac{1}{2} \right )+\frac{2\Delta t}{\rho\Delta x}p(M-1,n) \right ]$

where Z is acoustic impedance, the correct value of which will be of particular importance (material dependent).

– The experimental portion of the project is not yet laid out. I may request assistance from Professor Bradley in taking and analyzing acoustic signals from the aforementioned instruments. Regardless, I intend to compare the Fourier transforms of actual signals to the computed transforms in order to see if the peak frequencies match.

Time Line

– Week 1: Practice by completing a few problems from chapter 11.

– Week 2: Build a working model for the harpsichord, as complete as the piano model in the book, and take data. Begin work on clavichord model.

– Week 3: Complete clavichord model. Begin organ model.

– Week 4: Complete organ model. Record real instruments to compare to models.

– Week 5: Compare experimental data to models. Allow time for write up and possible revision of models.

– Week 6: Further time, submit Wednesday.

Resources:

– Giordano N J, Nakanishi H. “Computational Physics, 2nd Edition.” 2005. Addison-Wesley. ISBN-10: 0131469908.

– MATLAB R2014b by MathWorks®

## 2 thoughts on “Physics of Baroque Keyboard Instruments: Project Plan – Kachelein”

1. David J. Kachelein

Luke, as a point of added information if this fits in with your testing, I think it is important to take into consideration temperature and humidity effects on sound waves, especially on the waves of the pipes, because the density of metal is pretty fixed, were as wood is very variable. Let me know if this info. is worth while. Uncle Dave dkache@msn.com

2. Jenny Magnes

– Use Latex to write equations.
– It will be most interesting to learn how the sound of these three instruments differs and the underlying physics.