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Mahillon in Retrospect

Now that a fairly thorough outline of our present-day knowledge of the basic acoustics of the trumpet has been presented, we are in a position to consider retrospectively one of the leading figures in the nineteenth-century development of brass instruments, Victor Mahillon whose book, Elements d'acoustique musicale et instrumentale,23 has been very influential. To begin with, we must remember that Mahillon was an extremely skilled craftsman, possessed of wide experience, great ingenuity, and a perceptive ear. He did not have an extensive formal scientific training but was eager to use whatever scientific knowledge that was available to him in the furtherance of his goal, the improvement of musical instruments. What we must understand, however, is that the established body of science available to Mahillon could not help him much beyond providing guidelines and suggesting the methods of systematic research. Mahillon presents an admirably clear description of the nature of standing waves in a cylindrical tube that is open at both ends and for a pipe closed at one end. He points out that the flute plays at frequencies that are related to the resonances of the doubly open pipe and that the reed instruments and brasses run at or near the frequencies characteristic of the pipe stopped at one end. Reference to Figure 5a shows that for a cylindrical pipe these frequencies favored for collaboration with the lips form the odd members of a harmonic series. In other words the musical interval from the first to the second regime of oscillation of a cylindrical pipe is a musical twelfth. Furthermore the interval from the second to the third regime of oscillation is a major sixth. Clarinetists are completely familiar with this pattern of behavior, and the brass player can observe the same pattern in a moment if he will play on a length of cylindrical pipe (be sure not to use a mouthpiece).

Mahillon was obviously aware of the fact that enlarging the upper (mouthpiece) end of a stopped pipe wind instrument lowers the frequency of the first resonance mode and stretches the musical interval between the playing regimen, and that conversely a progressive enlargement of the lower half of the cylinder raises the first natural frequency and compresses the musical interval between the successive playing regimes. While Mahillon does not state these matters explicitly, the nature of the rules he gives for laying out the tone holes of a clarinet shows that he has allowed for these effects in a real clarinet. Every clarinet maker learns very early how to adjust the twelfth-shrinking effect of the bore enlargement in the lower joint to compensate for the stretching effect of the necessarily compromised register hole position.

Let us digress for a moment to see the implications for brass instrument design of my remarks on the effect of bore enlargements on a cylinder. We have seen that the cylinder gives a series of odd harmonics for its sequence of tones when played with a pressure controlled lip reed. We might conceive of enlarging a pipe at its lower end and reducing its size at the mouthpiece end just the right amount that the original interval of a twelfth between the first and second regimes is contracted to become an octave, with the interval between regimes two and three reduced from a sixth to a fifth. In other words, can we arrange a flaring air column such that when it is closed at the small end, its natural frequencies form a complete harmonic series? The answer is a qualified yes. Figure 5b shows that the addition of a bell to a cylindrical pipe (corresponding to an enlargement at the open end of an extended cylinder) produces changes in the desired direction. Figure 6 and Figure 7 show that the further addition of a proper lead-pipe and mouthpiece will put all but the first resonance peak into a harmonic relation. The first peak, however, remains low by about 30 percent. We have already seen that the instrument plays a harmonic series by ordinary means for all but the pedal tone. We have also seen that the pedal note sustains itself in a different way, but at a frequency that is exactly where our idealized horn would put the first resonance peak.

In the light of the last paragraph, we can see why Mahillon was led (along with many people since) to treat the trumpet as a doubly open cylinder when in acoustical fact it is a modified stopped pipe. In his introductory chapter on brass instruments Mahillon speculated briefly on the nature of a suitable air column shape. He wrote, "[I] have the conviction that [the] proportions follow . . . a geometric curve whose form approaches that of a hyperbola."24 This is in fact an entirely correct surmise, since the hyperbolic shape is that of a Bessel horn with a flare parameter equal to unity. Mahillon thus clearly recognized the basic nature of a non-cylindrical brass instrument air column. He did not comment on the apparent conflict in his discussion when he gives a formula for the frequencies of a brass instrument in terms of its length; this formula is formally identical with that for a doubly open cylinder. In modern notation it may be written,

fn = nc/2[L + 2D]           n = 1, 2, 3, . .

Here fn is the frequency of the n'th member of the harmonic series and c is the speed of sound in free air. The length L is measured from the mouthpiece rim to a somewhat ill-defined point near the end of the bell where the diameter is D. The shape of the formula tells us symbolically what he also said in words: the quantity 2D is taken to be a sort of 'open-end length correction' for the bell. I must emphasize that the formula is based not on mathematical physics but on the results of practical experience. Furthermore, it is important to understand that the relation of this formula to the properties of a doubly open cylinder is purely metaphorical and has no visible connection with Mahillon's undoubted understanding of the stopped air column of varying cross section. Nevertheless, despite its nonexistent mathematical basis, Mahillon's empirical formula has successfully guided instrument makers of several generations in their computations of horn lengths and the calculation of valve crooks.

From a strictly utilitarian standpoint Mahillon's success has been of mixed benefit. It permits the easy calculation of a workable trumpet of conventional design, which is to the good. However, if one takes the metaphor literally, the implied nature of the standing wave within the air column is not in accord with the truth and, hence, the positions of the nodes and anti-nodes are incorrectly predicted. The reader will find diagrams of the actual standing wave patterns in my recent article in Scientific American. On several occasions I have had dealings with craftsmen and musicians who have been led astray by the metaphor (which unfortunately is to be found in many standard books). The metaphor implies that trumpet pitches may be sharpened by making enlargements at either end of the air column, an implication that has led to the considerable bewilderment of more than one instrument maker who finds his expectations reversed when he makes changes of the mouthpiece or back-bore.

Trumpet Acoustics
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
Bibliographic Notes