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This page talks about several topics in physics. Although my degree is actually in mathematics from Stevens Institute of Technology, physics has always been a major interest of mine. All aspects of the field interest me, from the classical areas like acoustics and optics to relativity and quantum mechanics. I can often be found with a book about physics at hand, either something from the popular literature like Gribbon's In Search of Schrodinger's Cat, or something more technical. I particularly enjoy reading original works by founding physicists such as Planck or Heisenberg or Feynman. Dover Publications does an excellent job of making these kinds of books available. And of course, Isaac Asimov must be mentioned as the writer who more than any other brought the marvels of science to my attention. I started reading Asimov at a very young age, and still enjoy rereading his books and essays just for the infectious enthusiasm he had. Asimov could explain things in a clear and accessible way no matter what the topic. That's a goal I try to reach in my own science writing, but Asimov was incomparable. When others ask me where's a good place to start learning about physics, either for themselves or their children, Asimov's name is always at the top of the list. Astronomy is a close sibling of physics, and the most accessible branch of this subject. While I would love to have a particle accelerator to play around with, I can nevertheless enjoy my binoculars and telescope. Unfortunately, at the moment I don't get to use it as much as I'd like. In a year or two, when we plant gardens in the back yard, I'm going to set up a permanent pier and possibly a small observatory for this instrument. With the advent of inexpensive CCD cameras and frame stacking, the quality of images a casual amateur observer can achieve at low cost is remarkably impressive, rivaling that of major professional observatories from only a few years ago. But for the moment, I'll stick to occasional visual use, and remain otherwise an armchair astronomer, where it's warmer and the moon and clouds don't interfere First semester freshman year at college I made Deans List, which meant I got to take a free graduate level course in the next semester. Not wanting to take on too much, I asked around for the easiest graduate physics course offered, and "Acoustics of Music" was high on the list. The first day basic oscillators were covered, as typified by a hanging weighted spring, and then made into a sliding spring so that friction could be added into the equations. Since friction causes energy to be lost, this introduced the concept of an envelope bounding the diminishing oscillations. They quickly moved on to solving the one dimensional vibrating string equation, which they immediately generalized using Fourier integrals. I was somewhat familiar with the topic, so I felt encouraged if not confident. Then in the next class they did the two dimensional vibrating membrane equation, which was a rude awakening. But with a bit of coaching from a friendly grad student, and a lot of catch up work, I survived the theoretical first few weeks of the class. After that, they applied this background to the analysis of actual musical instruments. First covered was a simple pipe organ, whose column of air acts like an string unbounded on one end. Similar principles applied to the brass instruments, with complications arising from the changing geometry of the bells, which can be approximated by a cone instead of a cylinder. After this was considered the woodwinds, where the reed is an independent oscillator coupled to the air column. In theory, the equations were solved as if the medium is brought into full vibration instantly. In practice, things don't work this way. When blowing into an instrument, the air is initially still, and the pressure increases over time until it's maximum force is attained. When the note is ended, the higher pressure in the instrument doesn't stop immediately. The breath falls off gradually, and in any case the overpressure gradually dissipates. This brought up the important concepts of attack and decay. For a plucked or struck instrument, the medium begins at it's full vibration very quickly, but the volume of air it moves has inertia, and so again the note starts with a gradual buildup (the attack part) and dissipates more or less slowly at the end (the decay part). This is especially true if the major part of the sound doesn't come from the primary vibrating material, but from a larger, secondary substance (like the body of a guitar or the soundboard of a piano). Even more complicated effects occur when the instrument contains a resonant chamber, like the hollow cavity inside a cello, which will be discussed later. In any case, all of these factors affects the shape of the attack and decay envelope of the notes the instrument produces. These factors get very complicated, and often defy a complete theoretical analysis based on pure principles. A general understanding gives a first approximation, but empirical observation is necessary for better characterization of each type instrument, and often for each individual instrument as well. At first, I'd assumed the harmonics of the note defined it's character. Harmonics, also called upper partials, are integer multiples of the note's primary frequency. Some harmonics are stronger than others, and all together they define the shape of the wave produced by the instruments oscillator (meaning the string of a guitar, say). Calculating harmonics is the goal for Fourier analysis. But this is a small part of how instruments differ. I was surprised to learn that if only the steady state portion of the note is played, as if a recording of the note was cut apart, it's very difficult to distinguish one brass instrument from other, or one stringed instrument from another. Within each class of instrument, the major perception of it's timbre or "color," the defining character of the instrument, comes from the attack and decay patterns. Without that information, they all sound pretty much the same. Armed with a practical physical understanding of how conventional instruments produce sounds, we were given access to an early computer system to digitally synthesize music, the MUSIC V program developed by Max Mathews of Bell Laboratories. In this software package, the physical characteristics of an instrument could be modeled by piecing together generic modular components such as oscillators, adders, multipliers, band pass filters, and envelope generators. Each component had several inputs, such as the frequency and amplitude of a sinusoidal oscillator, or the distribution type and range of a random number generator. Envelopes were specified by giving vertex points in a graph of amplitude over time, producing straight line segments which approximated the continuous curve of an attack and decay shape. By combining these elements, fairly sophisticated results could be produced. For example, using the output of a low frequency oscillator multiplied with the amplitude input to a note generating oscillator produced a vibrato tone. After the instrument was modeled, it could be "played" under the control of Note statements, which specified the instrument to be used, the time the note was to be started, it's duration, pitch and amplitude. The result of program execution was a file of amplitude numbers, which was then sent through a digital to analog converter and output onto a reel to reel magnetic tape for audio playback. My initial use of this program simulated a flute, which is a fairly simple instrument acoustically. The success of this effort was encouraging, so I next tried to simulate a piano. This proved hard to get right. Pianos are complex instruments, and the attribute most confounding to my efforts was that they are not inherently harmonic in how they produce sounds. That is to say, the strings they use are so large and stiff and under such tension that the upper partials are not integer multiples of the fundamental frequency. This is especially true for the deepest keys on the piano, where the upper partials tend to be sharp. This meant the software package had to be modified. After the course was completed, I applied for and received a small National Science Foundation grant to modify and extend the program to increase it's capacity and add necessary features. Since I was not an experienced FORTRAN programmer, and the program was large and complex, this took some effort. By the end of the summer the task was completed, and two boxes of punched cards were sent to Bell Labs, containing the revised program and a small score of piano music. The run report returned with the audio tape commented that a dedicated IBM 360 ran for just under an hour to produce thirty seconds of music, and while the results were good acoustically, the melody was terrible. Despite that critique, I considered the effort worthwhile. I began work extending the program again to simulate a violin faithfully, which is a most difficult instrument to simulate. For one thing, the bridge acts as a nonlinear resonator and filter, and the exact shape of the cutouts of the bridge affects it's signature behavior. Even more difficult to the software was the effect of resonance within the body of the violin. As the body of the violin vibrates, sound waves repeatedly reflect between the curvaceous sides of the instrument and the top and bottom plates. This difficult geometry could not be solved analytically, and was far beyond my abilities to even approximate. However, by then I was more confident of my computer skills, and I devised a way to store the normal output of the instrument into a time sorted delay buffer, and to multiply the elements of that buffer with a decaying bimodal distribution of random numbers. The summation of these multiplications would then simply be added to the normal violin output, to emulate though not simulate the effects of the resonance. In practice, this method had potential, but it was going to take a lot of trial and error to find the proper amplitudes and distribution of the random resonance array, and how long a time delay the buffer should allow for. Working on the college's timesharing PDP-10, even with special permission to exceed the normal user time and memory limitations of 64K, the processing power available at that time was simply not up to the demands this software placed on the computer center. I handed over the software to the instructor of my course and turned my attentions to the current semester's coursework. Further information on the topics covered in this essay can be found at these links: Musical Acoustics and Instrument Physics. Future Topics
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