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Horns, Strings, and Harmony
By Arthur H. Benade
Dover Publications, Inc.Copyright © 1992 Virginia Benade
All rights reserved.
A Brief Look Around
"The secrets of nature betray themselves more readily when tormented by art than when left to their own course." Roger Bacon (12147-94)
When I was a junior in high school, and a beginner at what my teacher called "blowing the flute" (loud, long, and fast!), I discovered Dayton C. Miller's book, The Science of Musical Sounds. I had been exposed from my earliest days to the fascinations of science by my physicist father, and my fairly new passion for music was beginning to stimulate me to ask questions about how and why my flute behaved as it did. Professor Miller's book crossed my horizon at a most opportune time. In it I read the names of many explorers of physics who had helped to relate the world of musical artistry to the world of scientific understanding, and I learned that Miller himself was a devoted flute player who had given care to the understanding of his beloved instrument. In the years that have passed, I have kept up my amateur interest in the playing of music, and have developed my interest in physics into a professional one chiefly dedicated to the inner workings of nuclei. But off and on I have given considerable time to thinking about how musical instruments work, and it is quite appropriate that I should now find myself at Case Institute of Technology, in the same physics department over which Miller himself presided for many years before his death in 1941. No doubt his spirit is peering over my shoulder now, as I set down the opening words of a small and informal descendant of his classic work. I should invoke his blessing on my labors, and ask his forgiveness of at least some of my heresies!
In writing this book, I have chosen for myself the role of guide and for you, my readers, that of interested travelers in a strange and colorful land. As a guide, I shall not be able to show you all the towering mountain peaks or quiet valleys in this country, for our time will be limited, but I shall try to point out the roads and warn you against some of the dangers which are to be found. Before we begin our travels, however, I must prepare you with a little talk about the sort of place we are to visit, and also provide you with some equipment for the journey.
Music, like the instruments which make it, belongs mainly to composers and musicians who create patterns of sound for our enjoyment, but there are many other people who can lose themselves in fascination as they try to understand how music comes about, and how it can affect us so profoundly. All these people have their own points of view and their own ways of expressing their thoughts, but their words do not make a chaotic babel since they have collected themselves into fairly recognizable groups, each with its own tribal dialect. Let us see what some of these groups are, before we choose one or another for an extended visit. Among musicians the composer's viewpoint is different from that of a performer, while the man who listens to their joint efforts will have his own way of looking at things. There are also those scientifically minded people who can add the excitement of their professional curiosity to the more sedentary enjoyments of ordinary listeners. Three groups of scientists find themselves thinking particularly about music: the physicists, who snoop into the ways in which objects vibrate to set up sounds in the air; the physiologists, who are interested in the way our ears convert these vibrations into nerve impulses; and the psychologists, who trace these impulses into the brain and try to find out how our minds respond to them.
In our travels we shall be able to visit only the regions inhabited by physicists and by musicians, two groups who sometimes think they should look down on each other. Some people hold that musicians have a subjective and artistic way of looking at things, while the scientific folk are supposed to be detached and objective in their work and speech. Once this has been said, it is easy to belittle one group or the other, depending on your own feelings about the importance of being artistic or objective. Such talk is a wonderful way to start a fight, but it has very little use otherwise, since musicians and scientists are both creators, both judge their work by fairly well-established aesthetic standards, and both are human beings, complete with a well-developed set of foibles. Scientists and mathematicians are constantly talking of "elegant" proofs, and "beautiful" theories. They seek out symmetries and contrasts just as artists do, and they have the same joys of creation. The man who speaks of cold, hard science probably has never done any, knows few scientists, and has read almost nothing about their work from firsthand sources!
Having made this little speech on the unity of science and art, I must now go on to admit that real troubles do arise when physicists and musicians try to talk together. They often use the same words for different things, and they do not always consider the same things important. Now, I do not for a moment want to give you the impression that one or the other of these groups is "wrong" in its ways; on the contrary, they both usually have something to contribute, but we must always make sure we know who is speaking at the moment, and what he means. As is usual in a divided country, there is a sort of "no man's land" between the two parts which we have to cross and recross, and we shall need to look for paths laid out by earlier explorers.
Some Trail Breakers
Perhaps the greatest of these explorers was Hermann von Helmholtz (1821-94), a German military doctor who wrote The Sensations of Tone, the foundation book on sound as it is made and heard, and a similar one on light and seeing, carried out some of the most profound mathematical researches into electromagnetic theory, and in addition made a tremendous contribution to our understanding of thermodynamics. After reading all this, you will not be amazed to hear that he was also a talented pianist. A short quotation about this man appears at the beginning of L. S. Lloyd's book, Music and Sound, that is particularly appropriate for me to repeat here. It is taken from an address given in 1878 by the great Scottish physicist James Clerk Maxwell, himself one of the most creative of men:
"Helmholtz, by a series of daring strides, has effected a passage for himself over that untrodden wild between acoustics and music—that Serbonian bog where whole armies of scientific musicians and musical men of science have sunk without filling it up."
Now, I am not in a mood to improve upon the words of my betters, but I may perhaps remind you that Helmholtz had more than daring; he had a thorough, practicing knowledge of his subject from several points of view, and he was gifted beyond ordinary mortals. Because most of us are not brought up to recognize classical allusions of the sort educated Englishmen used so freely seventy years ago, I did a little research and learned that Herodotus described the engulfing of armies in the mud of Lake Serbonis, an Egyptian lake which has since dried up. Since many early musical instruments were made of the kind of cane which grows to this day in swampy places around the Mediterranean, we can see that Maxwell chose his words with skill.
Having named a musical scientist first, I must in fairness introduce now a scientific musician, Theobald Boehm. Boehm was trained as a goldsmith in the family business, but very quickly showed his ability on the flute. He made solo concert tours for several years before beginning to feel that many limitations of the flute of his day could be remedied by someone who really worked at the problem. Between about 1830 and 1850 he performed an extensive series of carefully chosen experiments, guided by what little acoustical theory was available to him, and in the end produced an instrument which was essentially the same as the modem flute. Not only does this instrument show the excellence of his researches into sound, but it also reveals Boehm as a first-class engineer of workable and convenient key-machinery. Most of the improvements which took place in the construction of other woodwinds during the last half of the nineteenth century are directly descended from his ideas and from the stimulus of his success. My own first flute was of a pre-Boehm style still manufactured in England until well into the 1930s. I can well remember the feeling of liberation when I changed to a Boehm flute. Let me assure you from firsthand experience that players of flutes, clarinets, and oboes of the older style have to be pretty expert technicians to be able to play smoothly and in tune. The clarinet concertos of Carl Maria von Weber, for example, require virtuosic finger technique when played on the older instruments for which they were written, yet many a modern high-school clarinetist will play them in public on a Boehm clarinet.
There are many other people who have made large contributions to our understanding of music and musical instruments, but I must pass over them silently to make room for others whose work bears more directly on what we shall be seeing. Dayton C. Miller I have mentioned already, without telling you his part in the exploration. During the first three decades of the twentieth century, before there were any of the electronic instruments that make research easier nowadays, Miller studied the quantitative relations between the sounds we hear and the kinds of vibration which set them up. In Chapter III, I shall show you that it is possible to write down a "recipe" for every sort of sound, and we shall also learn how this recipe can help us to understand what is going on. Dayton Miller was the first to get accurate recipes for all the orchestral instruments, and he did it in a way that has earned him a reputation for experimental skill and inexhaustible patience.
Another more or less contemporary man who did a great deal to clarify the ways in which musical instruments work is the French physicist Bouasse. He is not nearly as well known as he deserves to be, although the cause lay partly within his control. He was a peppery, combative sort of man, who never hesitated to say what was on his mind, and he often said it in a way that made enemies. Because of his controversial approach to things he managed to alienate the editors of several journals, and ended up having to publish all his work in book form, printed in small editions and not widely distributed. These books (more than twenty different titles) deal with many branches of physics. About 1929 he wrote a two-volume work entitled Wind Instruments and another called Pipes and Resonators. These three volumes contain practically everything known at the time about the theory and practice of their subject matter, a great deal of which is his own work or that of his collaborator, Fouché, a skilled musician. It is a real pity that these books are already quite rare, since without them a few people are still going around making meaningless experiments and wrong calculations on things which are clearly discussed by Bouasse. (Bouasse is best known among physicists for his spirited opposition to Einstein's relativity theory. He lost the argument, but he was batting in a big league!)
My list of great explorers has been very short, and does not contain many of the names that are greatly honored by musicians or by scientists. There are two reasons for this: we are not here to make an exhaustive and exhausting survey, and anyhow I prefer to emphasize the sort of men who were able to provide us with a broad view of the territory of musical physics, and who mapped out its rivers and mountains.
You may wonder why it is that I have apparently given more importance to men who explored among the wind instruments than to those who peered into the habits of violins and pianos. It is not just that I am myself addicted to wind instruments; all stringed instruments are built around a single kind of vibrator, the uniform string, whose understanding came first to mathematicians and whose behavior is simple enough that it was described rather early. The beautifully shaped wood casing upon which the strings of a violin are tightened has such complicated behavior, on the other hand, that we should soon get far beyond our depth if we tried to learn about it in detail. The wind instruments have a much wider variety of shapes and vibrations (a wider variety, at any rate, from the point of view of physics), and mathematically inclined explorers have found much more that they could tackle successfully,
Harmony and melody, which are at the heart of music, are like fields and meadows which can be walked in and enjoyed without quite so much toil as is required for mountain climbing, and people as far back as the Greeks have recognized the way in which science and art have grown intertwined. While there are many beautiful flowers to be seen, and stately trees, there is also much marshy ground that is overrun with weedy growths of pseudo-science and of musical faddism. We will see later on that a great deal of the form that music takes is dictated by the way our ears and our nervous system work, in very much the same way that the form of a building is dictated by the law of gravity and the properties of stone. No artist need be ashamed to bow to such dictation. Every art has a form, arising from nature, which requires knowledge and preparation from the artist and from the enjoyer, whether the art is music or painting or physics. This form challenges the creator and guides his efforts, even when he chooses to depart from it, and it seems to me very foolish to pretend that an artist is so special a creature that he need not understand the materials of his artistic world.
A Plan for the Expedition
In the remaining chapters of this book I plan first to acquaint you with some of the ideas of physics which we will need in our explorations of musical vibrators; we will then pay a very brief visit to our own ears to learn why they are able to pick out certain kinds of sounds as being different from all the others, and so start ourselves on the road through the elementary principles of harmony. Following this, we will travel quickly through the land of the stringed instruments, and see how it is that the piano is a very special and peculiar instrument, as well as some of the reasons for the versatility of the violin family. Before we press on into the realms of the wind instruments, we will have to stop for some more travel equipment and more maps to show us the way past pipes and horns and reeds, and around some famous mudholes.
Those of you who have already visited music in your scientific reading will find that I will often lead you over relatively untraveled paths. You will, in this way, get a wider appreciation of the terrain, even though a repeat tour might show you more detail. Another and more compelling reason for my different route is my belief that a great deal of future progress in musical physics will come from people who approach it along new paths, taking hints from successful explorers in other, more recently developed branches of physics. My hope is that some of you will find yourselves challenged to take up the exploration on your own, or at least that you will recognize that there is a whole continent waiting to be settled.
Let us now proceed with our preparations for the journey by becoming familiar with some of the properties of the common and simple pendulum, which has been guide and friend to physicists since the days when Galileo was a youth.CHAPTER 2
Simple Vibrating Systems
Most people are aware that vibrating objects can set up disturbances in the air, which then act upon a distant ear so that its owner hears a sound. All musical instruments owe their existence to this fact, and before we can safely approach the musician, we need to provide ourselves with some simple ideas about vibrating objects, about the way in which air is set into vibration, and a little about the ear. We also need to set up a suitable language in which we can speak about the properties and behavior of our musical vibrators. These preliminaries will be taken care of in this chapter and in the following two. After this we can ask ourselves what it is about men and their universe that gives music its general shape and why certain kinds of vibrators must be chosen if one wishes to make musical sounds.
Excerpted from Horns, Strings, and Harmony by Arthur H. Benade. Copyright © 1992 Virginia Benade. Excerpted by permission of Dover Publications, Inc..
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