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A New Age in Acoustics

January 24, 2003
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Back in the 19th century, two great scientific figures wrote books on the science of sound. The earlier was Hermann Helmholtz with his On the Sensations of Tone, 1863; the later was Lord Rayleigh in his The Theory of Sound, 1877. Both books are large and heavy tomes. These scientists were both highly skilled and thorough in their investigations. Indeed, these works are for most of us, formidable. Much theory, much mathematics, and much ponderous discussion. Nevertheless, they both set forth a theory of sound that has stood the test of time. As with any such intricate disquisition, the public has taken from the work of these two men what they want and, in a sense, "simplified" the principles. Since those books were written, the science of acoustics has made several substantial and significant advances.

 

Over the last several years, I have encountered two presentations of simple acoustic theory under somewhat formal circumstances: one was a lecture given at the convention of American Institute of Organbuilders held in Virginia, October 1997. The other is in a fine book of the popular press, Music, the Brain and Ecstacy, by Robert Jourdain (Avon). This book is well written and in its later pages examines deeply and interestingly the psychology of the musical experience. But in the earlier pages it gives an explanation of sound, like the speaker at the convention of organbuilders, that reads completely from the old theories.

In brief, the older and still popular theory says that all sound can be analyzed into component harmonics. All the harmonics in this composite sound are integral multiples of the bass frequency. Thus, given an A=440 hertz (cycles per second), its harmonics are 2x440=880, 3x440=1320, 4x440=1760, and so forth. Furthermore, these harmonics, like a pharmacist's compound, are added at various dynamic levels: the second harmonic is, say, 42% of the fundamental, the third harmonic is, say, 86% of the fundamental,and so forth. In other words, if you were to compound a given group of harmonics in their proper proportions to the fundamental, you will, perforce, re-create the original sound.

 Here is the basis on which Hammond constructed his now famous organ. Little sliders above the keyboards--there are nine of them--can be drawn out to one of 8 different positions to represent the various intensities of each harmonic and--voila--there you have the oboe, the violin, or whatever other sound your mind can dream up. Anyone who has played around with a Hammond knows that no such miracle takes place. Rather, the instrument produces some sterile sounds that can beguile briefly but ultimately do not imitate anything but a Hammond organ. The Hammond has been used extensively in the field of popular music. It is portable and not too expensive. And Hammonds do last a long time! Their basic sound-producing mechanism is a whirling shaft--spinning at a constant speed to keep the frequencies constant--with many notched wheels (one for each frequency); electromagnetic pickup heads pointed at each wheel feed the signal from the wheels through the sliders into an amplifier and eventually the loudspeakers. Another fact about the Hammond is that the harmonics are borrowed from the tempered scale of the bass frequencies. Therefore, the pitches of the nonunison "harmonics" are not in tune with the fundamental--not true harmonics--and therefore they cannot form composite sounds because they will beat with the fundamental.

The idea of compounding harmonics to produce composite tones is not new. The major chord is nothing more than the first, third, and fifth harmonics of the basic key note. Intricate harmonies such as in the Bach chorales or the barbershop quartet are merely other combinations of harmonic pitches to make a composite sound. In fact, the major and minor scales we use for our music are composed of the harmonics of a fundamental note. (The C-major scale, by the way, is based on the harmonics of F!) All the notes of the C-major scale can be played or sung simultaneously without producing any beats provided the notes are accurately tuned as harmonics of the frequency of the generating tone, F. The common theory of sound believes that by properly proportioning harmonics, one can make a human being think he hears the strident blare of the trumpet or the soothing coo of the flute. But later research and better understanding of acoustic phenomena have altered these earlier theories.

One of my colleagues, Ralph Showers, at the University of Pennsylvania, way back in 1943, was preparing a master's thesis on sound. I talked with him about synthesizing tones by the old theory. He laughed at me. I exclaimed, "You mean you can't put the proper proportions of harmonics together and recreate sounds of violins, oboes, and others?" He told me that in analyzing the sound spectra of the various instruments, he could find no clearly singular spectrum for any one instrument. "They went all over the place! The spectrum of the violin looks like the spectrum of the oboe; the clarinet looks like a trumpet," he said. I was crushed; I thought it was all so simple and neat. Further conversation with Ralph only made things worse; the whole idea of synthesizing tones by adding proportions of harmonics was nice in theory, but not in practice.

The harmonic synthesis principle, even if true, leaves a whole class of musical sounds out of the "equation." What about the struck instruments, the bell, the cymbal, the drum, the glockenspiel? These are all enharmonic instruments that do not obey the "harmonic" law at all.

Then I took a graduate course in acoustics and learned a fuller approach to the theory of sound. Its premise is that all elements that produce sound vibrate because they have been temporarily deformed. The restoring forces of the deformed bodies are due to one or the other of two forces: tension, or stiffness. The violin string is deformed by contact with the moving resin-laden bow and pushed from its normal position until the restoring force of tension in the string exceeds the grip of the resin on the string, at which point the string flips back. Continued bowing thereby sets the string into vibration. The triangle is struck with a little rod and is thus deformed, as with the violin string, but it returns to normal by virtue of the stiffness in the structure of the triangle. Unlike the violin, whose tone can be maintained as long as the bow keeps moving, the tone of the triangle gradually falls off as the energy of the initial blow is gradually dissipated into the air and the metal of the triangle. All instruments of music are governed by one or the other of the two restoring forces: tension or stiffness.

In order to study these phenomena, the physicist, in his formal mathematical way, writes down a differential equation that says: the vibration of a body is proportional to the sum of two restoring forces, tension and stiffness, as a function of time. Fortunately, the physicist finds that the musical instruments fall into either one or the other class: tension or stiffness, but rarely both (see pianoforte later). The equation is then simplified by dropping either the term for tension or the term for stiffness. If tension is the dominant restoring force, the equation reduces to the harmonic series of integral multiples of the bass frequency: one times, two times, three times, the fundamental frequency. The various frequencies are sometimes called upper partials, a good general term. But they can also be called, quite specifically, harmonics.

The class of harmonic instruments is large. The typical tension instruments are all the bowed string instruments, also the guitars, zithers, clavichords, harpsichords and other plucked instruments (the pianoforte is not included here; it's in a class by itself!). Also, it turns out that the wind instruments function also in the harmonic system because a column of air acts like a string under tension. Thus, all the wind instruments are also harmonic devices: the oboe, the flute, the trumpet, the human voice, the french horn, and the organ pipe, for example.

When stiffness is the dominant restoring force, the equation leads to another set of equations known as Bessel Functions. (Friedrich Bessel was a Prussian astronomer, 1784-1846, who developed a series of equations that have been named for him and which are much like the trigonometric equations.) Without going further into the abstruse, the Bessel equations describe the higher frequencies that form the upper partials of the sounds of stiffness-dominated instruments. The frequencies of these upper partials are not integral multiples of the fundamental; the Bessel multipliers are called transcendental numbers. These multipliers are long decimal numbers that have an infinite number of decimal places. They are not harmonic! They should properly be called overtones. These sounds do not blend quietly into the mélange of the harmonic orchestral sounds of the tension-dominated instrumental sounds.

The class of stiffness-dominated instruments is not as large as the tension-dominated. They include the triangle, the chime, the bell, the glockenspiel, the xylophone, the celesta, all the drums, the cymbals, and other struck-type instruments. The triangle with its single ting! sounds out clearly over the entire symphony orchestra. What is the loudest instrument in the football band? Not the trumpets. No, it's the glockenspiel, that brightly plated, lyre-shaped instrument that is played with a little mallet. Its ting sings out over everything else. It is nonharmonic and therefore does not get lost in the rest of the sounds.

So, we can summarize: All instruments are divided into two main classes--harmonic (tension-dominated) and nonharmonic (stiffness-dominated).

Now a special note on the pianoforte. Pianos, as they are commonly called, used to be in almost every American home. Acoustically, the piano is a total hybrid. Did you ever try to replace a piano string? Quite unlikely. But I assure you it is far from easy. All the strings in a piano are stiff and difficult to handle. They are unwieldy in their stiffness. And they must be wrapped around a tuning pin that is about 1/4≤ in diameter! Unlike the thin, delicate strings of the harpsichord, the piano string is definitely a thing of stiffness. And have you heard of the tremendous tension that exists in a piano when it is all strung up and tuned? There are many tons of tension between the tuning block in the front and the terminals at the rear of the cast-iron frame that holds all the strings. With these two facts, yes, the piano carefully sits astride the stiffness-tension systems. It is not quite either; it is both. Thus the piano--like the triangle--can, in a concerto, quite dominate the orchestral scene. At the same time, the pianoforte is not a member of the symphony orchestra. Often one sees pianos as part of high-school orchestras, but these pianos are used to fill in the parts of missing instruments.

Let's change gears rather abruptly. There have been two significant experiments performed in the last 30 years that have made us reconsider the old gospel significantly. One of these experiments might seem to have nothing to do with music. It has to do with language. (But isn't music a language?) A branch of the U.S. military service was curious about the limitations of frequency range on radio communication through earphones. What would the absence of certain frequencies in the communicating system do to the accuracy with which the serviceman would hear the messages? How accurately do the pilots hear the messages through the radios and earphones that guide them?

A test was carried out at MIT in which they had three rooms. Listeners were placed in rooms A and C. An articulator--a person who spoke English clearly--was placed in room B. Room B was connected to room A through an amplifier that passed only frequencies of 400 cycles per second and lower. (The unit, cycles per second, has been replaced with the single word hertz.) Room C was connected to room B by a similar amplifier that passed only frequencies of 400 hertz and higher. The decision on 400 hertz was to separate vowels from consonants. Realize that the vowels, though spoken, are pronounced as if they were sung. Vowels are sustainable; they can be made into continuous sound. These are the tones one makes when singing; they are, for a man's voice, below 400 hertz. The high A of the tenor is 440 hertz. But the fricative sounds--the s's and sh's, the explosives, p's and t's--are all in the upper frequencies, frequencies above 400 hertz.

 This test setup neatly divided the English language into vowels in room A and consonants in room C. The articulator in room B then read a list of words while the listeners in the other two rooms wrote down what they heard. Results showed that room A, with its vowels, heard about 20% of the words correctly; room C, with its consonants, heard 80% of the words correctly. In other words, 80% of the intelligibility of language lies in the consonants, not the vowels. Choruses and singers in general find it difficult to make the words they are singing intelligible to the listener because they sing mainly on vowels. However, there are a few singers, notably Rosemary Clooney and Bing Crosby, who do (did) an incredible job of pronouncing their consonants so distinctly as they sing that we can all understand what they are singing about.

Another experiment--this in music directly--was performed with only two rooms. Room A contained a number of experienced musicians; room B contained a number of familiar sound sources such as violins, trumpets, oboes, and singers. Room A was connected to room B by a fine, high-fidelity sound system. In this experiment, the various sound sources in room B played notes. In room A, the auditors wrote down what they thought was the source of the various sounds they heard. Their answers were nearly all correct.

Then they did a second phase of the experiment. In each case, now, the microphone was turned off until the source had already started its sound and the microphone was turned off before the sound had stopped. In other words, the auditors in room A heard only the steady-state sound without any initial or terminal transients. The "consonants" were excluded and only the "vowels" were sent to room A. The results were nearly all in error. Experienced musicians could not tell the "vowel-sound" of the oboe from that of the violin! In other words, in music as in language, it is the consonants of the "language" that make for intelligibility, not the vowels. As we grow older, our ears lose their high frequency sensitivity; we drop off the consonants so that we fail to understand what other people are saying.

The significance of these two experiments cannot be overstated. Notice now that the entire sound of those "loudest" instruments--triangle, glockenspiel--lead to an idea that sounds silly when posed, yet is not frivolous. Why does a composer use ties to sustain notes across a bar? One might answer unwittingly; "In order to make the note longer." But wait! May it not be that he wishes to avoid a new transient at that moment? Syncopation is another example; by not accenting the beat and accenting the offbeat note, one achieves syncopation. The force of the syncopation is enhanced by the transient sounding powerfully on the offbeat!

The articulation--the breaking up of the music into its parts and pieces by the presence of transients--is the way to understand the details of a work. Mozart used the trumpet--much to the trumpeter's distress because it made Mozart's trumpets parts so simple‚ largely as a source of juicy transients. The ta-ta-ta-tah of the trumpet is really a percussion effect. The trumpet has long been the alarm, the tocsin, that brings us all to attention. The wonderful cackle of the oboe is the source of its joy and rhythm. The breathy concomitant sounds with the flute are its own joy.

Once I was demonstrating a new organ to a group of curious people. While I was bubbling about the wonderful tones of the new instrument, I picked one stop and put my finger on a single note and held it. I continued to rhapsodize about the beauty of the tone as I held the note ever longer. Suddenly, one of the listeners burst out, "Play it again!" He wanted another transient to savor again the "tone" of the pipe, not listen just to the steady-state tone.

Jazz, the great American music, is known for its rhythm and especially its syncopation. What are the instruments most useful to jazz? The trumpet, the saxophone--which has some wonderful transients--the trombone, the clarinet. Notice, the violins are missing. Why? Because the violins and their cousins have weak transients. Listen to a violinist trying to play jazz and you will hear a musician with an instrument that has weak transients doing his best. He will employ every device known to music to overcome his lack. He will play late (agogic accent), he will play off pitch, he will play with vibrato. These are the devices also used by vocalists, who also have weak transients. How often have you heard a jazz singer arrive at the last note singing well flat of the note and then, as he/she sustains the note, gradually begins to add vibrato and slide up to the pitch? A string quartet is, at best, inept at jazz.

Bach's favorite instrument was the clavichord; he also liked the harpsichord. The sounds of both these instruments consist largely of transients. In fact, they are noted for their lack of sustained sound. The distinctive ting of both these instruments makes the counterpoint brilliantly clear. When Silbermann showed Bach his latest pianoforte, Bach is said to have made some remarks that apparently caused a falling-out between these two men. Perhaps Bach's comments had to do with the less pronounced transients of the new pianoforte compared with those of the clavichord and the harpsichord. The counterpoint was not as easily heard as on the earlier instrument. So too, the "chiff" of the unnicked organ pipe provides a transient that also helps to make the counterpoint clear.

One must conclude from these experiments and from personal experience tha the definition of musical tone and its place within the fabric of music is more dependent on the opening and closing transients of the sounds than on the old theories of composite tone that consists of harmonics or overtones in varying proportions. Yes, there are harmonics; yes, there are overtones; yes, they are there in varying quantities. But between the actual music and the laboratory analysis, there is a long distance. We live on transients for most of our interest in music. Many of us may think that the pear-shaped tones are the sine qua non of music. Certainly, the mellifluous sustained tones of a great choir of strings in an orchestra, or the sustained sound of a large chorus of human voices is a stirring sound, but both are sounds that need to be ventilated frequently with consonants to make them totally meaningful.

We must modify the old credo of sound to bring in these new understandings. Transients form the intelligence in music. Music must have transients to be interesting. It is the consonants, not the vowels, that make for articulate, intelligible, and moving music. Otherwise, we have ululation!

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