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,

**f _{n} = nc/2[L + 2D] n
= 1, 2, 3, . .**

Here **f _{n}** is the frequency of the n'th member of the harmonic
series and

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

Conclusion

Bibliographic Notes