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The Cooperation Needed for Musical Results

For many years acousticians were puzzled and frustrated because their measurements of the natural frequencies in wind instrument air columns did not correlate very well with the pitches played by musicians on these instruments. As I have implied much earlier in this chapter in connection with the water trumpet, we now know that the musician's tone is sustained with the help of several natural vibration modes that form a sort of government-by-vote that we shall formally call a "regime of oscillation." This is a state of steady oscillation in which several air column vibrational modes collaborate with the lip mechanism to generate energy at several harmonically related frequencies at once.

There is abundant evidence that Bouasse was aware of the inadequacy of an oscillation theory based on the assumption that only one of the natural ("sloshing") frequencies of the air column is responsible for collaborating with the lip-valve to produce a tone. In other words, Bouasse recognized the inadequacy of the Weber-Helmholtz formulation of the oscillation problem even while he accepted its basic correctness. Bouasse's interest in the brass player's "privileged tones" (concerning which I will have more to say later) gives the clearest indication of this. Such concerns actually led him to describe the qualitative nature of the true collaborative state of affairs, even though he was unable to work out the quantitative relationships.

It was Bouasse's evident concern in these matters that provided the stimulus for the present author to take up a close study of the subject of sound generation in a system in which several modes of vibration collaborate.15 The first fruits of this study were described in a series of technical reports commissioned in 1958 by Earle Kent of C. G. Conn, Ltd. These studies progressed with the aid of valuable counsel from many people. On the technical side I am particularly indebted to Robert Pyle, John Schelleng, and Earle Kent. In 1968 Daniel Gans and I reported on a more developed form for this theory that could deal with the interaction of several partials in a tone and gave an account of some of its consequences. We were even able to describe the successful design and construction of a nonplaying "tacet horn," which should have been able to sound, according to the Weber-Helmholtz viewpoint.16 Since that time the work has been carried out much further here in Cleveland, particularly by Walter Worman who in 1971 presented a detailed report on it in the form of his Ph.D dissertation.17 For technical reasons his work was focused on clarinet-like systems, but the consequences are of general significance. Robert Pyle has presented results of related studies as his contribution to a symposium on brass instrument acoustics that took place in 1968.18 Worman was able to trace out the ways in which a reed-valve interacts with an air column and showed that the particular 'playing frequency' chosen for the oscillation (along with its necessarily whole-number multiples) is one that maximizes the total generation of energy, which is then shared among the various frequency components in a well-defined way. The steady collaborative vibration belonging to a regime of oscillation is made up of the fundamental frequency component and a set of upper partials whose frequencies are exact whole-number multiples of the fundamental, whether or not the air column's natural frequencies are harmonically related. All that is required is that the natural frequencies are in sufficiently harmonic relation that they can set up a regime. The better that the lower two or three modes are in agreement with one another, the freer the speech of the instrument and the more centered its tone, in agreement with the observations of Bouasse.

It is time now to focus our attention on actual air columns of a musical sort, in order to understand the practical implications of the acoustical theory that we have merely sketched out so far. In the paragraphs immediately following, I will describe briefly certain laboratory measurements on musical instruments which can then be used as a basis for describing the tone that they produce. The ultimate goal of these descriptions is preparation for a meaningful discussion of the tonal similarities and differences to be found between the trumpet of today and of the Baroque era.

We have had hints already that the basic property of the horn that controls the vibration of the lips is the acoustic pressure developed in the mouthpiece cup under the stimulus of a given oscillatory flow of injected air. Let us see how this air column response might be measured in the laboratory independently of the complications engendered by the interaction of the air column with the player's lips. Conceptually, the simplest method would be to have a sort of oscillatory pump that feeds the mouthpiece cavity via a capillary tube such as one might cut from a hypodermic syringe (see Fig. 4). Sinusoidal (pure tone) pressure fluctuations that are produced at the motor's driving frequency in the pump cylinder give rise to a small, well-defined, and perfectly predictable oscillatory flow into the mouthpiece. If we then use a tiny microphone to measure the amplitude of the pressure fluctuations produced in the mouthpiece in response to the oscillatory flow of injected air, we will have the desired response information, and this could be displayed in the form of a graph as a function of pump driving frequency. As a practical matter one uses in place of the pump various cousins of the familiar loudspeaker. Such a driver is controlled by means of an auxiliary microphone that maintains a constant flow stimulus as one sweeps automatically through the interesting range of frequencies. Between 1945 and 1965, Earle Kent and his co-workers at C. G. Conn in Elkhart developed one form of this basic technique to a very high degree of dependability.

Figure 4
Figure 4: Measuring input impedance

There are several additional methods for measuring the pressure response of an air column to injected air flow. These are more subtle to understand, but they are sometimes freer of complications when making high-accuracy measurements. One such device of great versatility was first described by Josef Merhaut of Prague.19 Another device that is of great utility for the study of brass instruments is an adaptation of an apparatus first constructed by John Coltman for his studies of the sounding mechanism of the flute.20 There is yet another class of air column measuring techniques that is historically much older, being first devised by the Englishman, Blaikley, in the nineteenth century. A modern form of the Blaikley arrangement is easy to set up and involves measurements of the acoustic pressure variations in the mouthpiece, as before. However, the excitation of the air column is done by means of a properly monitored source loudspeaker placed near the open bell of the instrument, instead of through a fine tube leading into the mouthpiece cavity. In my laboratory I find that all of these techniques have virtues that adapt them particularly well to one sort of measurement or to another.

It is time to explore now what sort of pressure response curve we get as a result of a flow stimulus applied at the mouthpiece end of an air column. An acoustician would rephrase the question and ask for the input impedance Z of the horn as a function of frequency. When a piece of cylindrical trumpet tubing about 138 cm is attached to an excitation system, the pressure response curve shows dozens of input impedance (response) peaks whose frequencies are evenly spaced at odd multiples of about 63 Hz (see Fig. 5a).

Figure 5a
Figure 5a: Input impedance of a piece of cylindrical tubing

The nature of this pattern of pressure response peaks shows that they are to be identified with the "natural" frequencies of a cylindrical pipe stopped at one end that are described in every elementary physics textbook. Because the frictional and thermal losses of wave energy taking place at the tube walls increase with frequency, these resonance peaks become less and less tall at higher frequencies. The energy radiated into the room from the open end of such a pipe is, however, only a tiny fraction of one percent as compared with the energy that is dissipated at the pipe wall. If we alter this piece of trumpet tubing by adding a trumpet bell, the input impedance curve changes to one of the sort shown in Fig. 5b.

Figure 5b: Impedance modified by adding a bell

A close look at the frequencies of the response peaks shows that the first peak is hardly shifted by adding the bell, but the frequencies of the other resonances are lowered in a smoothly progressing order because of the way waves move in the bell. The trumpet-bell-plus-pipe system shows a rapid falling-away of the tallness of the peaks at high frequencies because an increasingly large fraction of the acoustic energy supply leaks out through the bell into the room. Above 1500 Hz there is essentially no returned energy from the flaring part of the bell. The small wiggles in the impedance curve at high frequencies are due chiefly to small reflections produced at the discontinuity where the bell joins the cylindrical tubing.

Figure 6
Figure 6: Measured impedance curve for a complete cornet

One need only glance at the impedance curve for a cornet (Fig. 6) in comparison with curves for a pipe or a pipe-plus-trumpet-bell to realize that the presence of a mouthpipe and mouthpiece has a considerable effect on the overall nature of the input impedance. The resonance peaks grow taller up to about 800 Hz, and then fall away in a manner that is only vaguely reminiscent of the falling away of the curves belonging to the trumpet bell plus pipe. The third and fourth impedance peaks of this particular cornet do not follow the smoothly rising trend that proves necessary for a really fine instrument. These irregularities of tallness are associated with irregularities in the frequencies of maximum response. They are caused by slight constrictions and misalignments of the tubing as it connects with the valve pistons, and with the junction of the main bore and the mouthpipe. One finds that irregularities of this sort give rise to difficulties in the tone and response of an instrument which are readily apparent to the player. The cornet whose response curve is shown here was made in 1865 by the respected British craftsman Henry Distin. The original owner of this instrument was Eckstein Case, nephew of the founder of what is now Case Institute of Technology of Case Western Reserve University. He gave it to Dayton C. Miller, also of Case, whose studies in musical acoustics in the early part of this century are well-known.

We have now had an introduction to the nature of the response curves that summarize the acoustics of trumpet-like air columns. We also have dealt in a preliminary way with the interaction of a player's lips with the air column of his instrument. We are finally in a position now to look at the nature of these collaborations between a player's lips and his instrument, as actual tones are sounded on a modern trumpet. First we will see how the tones are generated, and then we will look at the nature of these tones as they are played at various dynamic levels.

Figure 7
Figure 7: Impedance of a modern Bb trumpet

Figure 7 illustrates what goes on within a modern Bb trumpet when the player is sounding the written note C4 and the G4 just above it. The regime of oscillation for the note C4 is based on the second of the impedance maxima of the air column in consort with the fourth, sixth, and eighth of the peaks in the curve. When the tone is sounded at the pianissimo level, the playing frequency closely matches that of the second peak, which is the only contributor to the oscillation. As the loudness level increases, the other peaks successively become influential. A beginner attempting to play this note softly finds it to be quite wobbly because he is unable to maintain a steady lip tension, and the basic resonance of the horn for this note does not have a very large impedance. However, as he plays more and more loudly, the fourth, sixth, and to some extent the eighth peaks enter the regime one by one and add their stabilizing influence to the total oscillation.

When the player sounds the note G4, the impedance maxima of the instrument that collaborate to form the regime of oscillation are peaks number three, six, and to some extent nine. For the note G4 we observe that the impedance maximum that controls the pianissimo playing is much taller than it was for the note C4, which makes the softly played sound more stable. As one plays somewhat louder, the very tall peak belonging to the second harmonic in the regime adds considerably to the strength and stability of the oscillation. For these reasons G4 is one of the easiest notes to play on the instrument.

Figure 8
Figure 8: Decrease in impedance peak heights for higher notes

In Fig. 8 we show once more the response curve for our trumpet; this time the regimes of oscillation are indicated for the written notes G5, C6, and high E6. Notice that the G5 is what might almost be called a solo performance--the regime of oscillation is dominated by the sixth impedance maximum of the instrument (which is a very tall peak indeed). Because there is only one impedance maximum contributing strongly to this oscillation, it is a note that is very well-described in terms of the original Weber-Helmholtz form of the theory, no matter what the dynamic level of the playing. The same remark applies to the C and the E above the G5. However, these notes are more difficult for the player because the single active Z peak is not very tall. It takes an athletic trumpet player to play the high E and still higher notes. Quite aside from his problems with obtaining adequate lip tension, the player finds that the instrument has begun to turn into a megaphone in the range of such notes, and the energy production is almost completely due to the interaction of the air with the lips themselves in a manner quite analogous to the way the human larynx operates in producing one's voice. (On the Baroque trumpet the design of the bell and mouthpiece is such that the resonance peaks that help sustain these higher oscillations are appreciable and are active to somewhat higher frequencies than is the case on the modern instrument.)

Let us look now at a pair of examples in which the player is able to produce a note on his instrument for a playing frequency that does not correspond to a natural frequency (frequency of maximum response) of the air column. Notes of this sort have been known to brass players since the earliest days, and were a part of the horn player's technique at the time of Mozart and Beethoven. The need for them was, however, reduced as the instrument because more mechanized. In recent years this type of note has returned to use, chiefly by musicians wishing to play bass trombone parts without the necessity for a special thumb-operated valve that is otherwise required. Tuba players also find the technique useful upon occasion. It is tones of this general class that attracted Bouasse's attention, and thence stimulated us to follow up their implications. These are the "privileged tones" referred to earlier. They are also sometimes called "factitious tones" by brass players, and are dealt with in a needlessly apologetic manner, as though there were something immoral about this manifestation of the complexity of nature! Figure 9 shows the regimes of oscillation for two examples of these privileged tones.

Figure 9
Figure 9: Regime of oscillation for two unvalved "privileged" tones

The written note C3 in the bass clef, which is known to musicians as the pedal tone of the trumpet, is run as a regime of oscillation such that the 2nd, 3rd, and 4th resonance peaks of the instrument sustain an oscillation that lies at a frequency equal to the common difference between their own natural frequencies. These is actually a loss of energy at the fundamental playing frequency for this note, rather than a gain, because there an impedance minimum rather than a maximum in the response curve of the horn, which makes it possible to play in a stable manner only at a fairly loud dynamic level. Also one finds that there is a relatively small amount of fundamental component generated in the tone. This pedal tone regime will be recognized as being an almost exact analogue to the compromise frequency situation that we met much earlier in connection with our water trumpet. The situation for the written note G3 is even more peculiar than for the pedal tone, in that the 2nd and 4th components of this new tone are the chief sources of oscillatory energy production. On the other hand the fundamental component of the tone and all the other odd numbered harmonics do not contribute to the oscillation at all, because the air column's impedance is very low at those frequencies. By now we have made a fairly detailed inspection of the ways in which a given air column (the open-fingered trumpet) collaborates with the player's lips to produce a set of tones. This set of tones (aside from the additional, closely related tone a musical fifth above the pedal note) makes up the harmonic series of pitches upon which trumpet music was originally based. The reader may be wondering what happens when any of the piston-valves are depressed on his trumpet. Nothing radically new takes place. The bell, mouthpipe, and mouthpiece design dominate the overall pattern, or the envelope, of a resonance curve--the pattern of peaks getting taller and taller as one goes from low frequencies to about 850 Hz and then falling away and disappearing at high frequencies. Because of this, the simple addition of cylindrical tubing into the middle of the instrument by means of piston valves will merely shift the whole family of resonance peaks to lower frequencies, but will leave them fitting pretty much the same envelope. As a result, my earlier remarks apply essentially unchanged to all the in-between notes that are played using different valve combinations.

Acoustical Preliminaries
The "Water Trumpet"-- An Analog to What Happens inside a Trumpet
The Function of the Player's Lips
The Function of the Pipe and Bell--Inside the Air Column
The Cooperation Needed for Musical Results
The Baroque Trumpet
The 'Internal' Spectrum of the Modern Trumpet
The 'Internal' Spectrum of the Baroque Trumpet
Relation of Internal to External Tone Color Spectrum
The Menke Trumpet
The Problem of Clean Attack
Mahillon in Retrospect
Notes & References