Instrument Design Features

Acoustic phenomena are not always predictable. However, once understood, it is possible to develop methods and techniques that produce desired results. Below, I describe three memorable experiences, which began in moments of complete surprise and ended in contemplating unexpected possibilities.

In 1978, shortly after completing Diamond Marimba I with pernambuco bars, I was perplexed by an unexpected acoustic phenomenon. Although I had tuned the fundamental mode of vibration of the seven bars of the neutral axis to the exact same frequency, G₅ at 784.0 cps, none of the bars’ fundamental frequencies sounded the same. Deeply worried, I sat by the instrument for days trying to determine the cause. Unless a musical instrument requires the inclusion of multiple unisons, this phenomenon is terra incognita to all builders. As if struck by lightning, the answer suddenly came to me: the higher modes of vibration of the bars — similar to yet distinctly different from the harmonics of vibrating strings — were influencing my aural perception of the fundamental frequencies of the bars. From Musical Mathematics, pp. 163–164:

In the G₁–A₃ frequency range, F₂ and F₃ of bars fall well within the span of human hearing. More important, these two modes greatly influence our pitch perception of the fundamental frequency. For example, if we tune F₂ to a “double-octave” plus 25 ¢ above a tuned F₁, then the fundamental will have a tendency to sound sharp even if it is exactly in tune. In this context, the subject of pitch perception should not be confused with the subject of timbre. The former is about tuning, and the latter, about tone color or quality of sound.

Here, F₂ refers to the first mode of vibration above the fundamental mode of vibration F₁, and F₃, to the second mode above F₁. In Musical Mathematics, Chapter 6, I explain in full detail how I managed to solve this problem by methodically removing material from a standard single-arch design, which enables me to simultaneously tune two modes of vibration in a treble bar; and by creating a unique triple-arch design, which enables me to simultaneously tune three modes of vibration in a bass bar.

On the Bass Marimba, I tuned the lowest bar to G₁, G₃, and G₄. And as shown in the graphic below, on Diamond Marimba II, I tuned the lowest bar to G₃, G₅, and G₆.

Cavity resonators are often called Helmholtz resonators, in honor of Hermann Helmholtz (1821–1894) who first used them to hear the faint harmonics of strings and organ reeds. Unfortunately, Helmholtz’s original frequency equation, and many other related equations give reasonably accurate results only if the walls of the resonator are absolutely rigid. Many cavity resonators fulfill this requirement. When we blow across the opening of a glass bottle to produce a musical sound, the walls of the bottle do not vibrate in response to changes in air pressure inside the cavity. Tests show there is fair agreement between the theoretical frequency and the actual frequency of the bottle. However, when the walls of the cavity are not rigid — as in the hollow bodies of violins, guitars, and bass marimba resonators — the actual resonant frequencies are significantly lower than the calculated frequencies.

When I first began to design and build the five cavity resonators for the Bass Marimba, they all sounded much lower than predicted by theory. One day, while I was striking a resonator with a mallet to hear the resonant frequency, my knee accidentally pushed against one of the large sides of the resonator. While my knee contacted the side, the resonant frequency increased by more than a semitone. I quickly began installing 1-inch diameter dowels between the two large sides to inhibit their motion. To my delight, six dowels increased the resonant frequency by more than a minor third.

I also discovered that increases in the stiffness of the sides dramatically increased the amplitude of the resonator. Since less wave energy is spent in vibrating the sides, the amplitude of a tuned resonator with dowels is much greater than an untuned resonator without dowels. To understand this process, imagine riding a bicycle with flexible springs for pedals. Most of the energy supplied by your legs would be lost in compressing and expanding the springs, and very little energy would actually go into turning the front sprocket and driving the chain. Similarly, when a pressure wave encounters a moveable surface, a great deal of the wave energy is lost in bending the surface, instead of compressing and rarefying the air. The graphic below is from Musical Mathematics, p. 220.

Except for the wine glass in the lower left-hand corner of the Glassdance, all the others are large brandy glasses called snifters. The remarkable acoustic properties of these crystal brandy snifters enabled me to tune two octaves from a single kind of glass. Even though the short glasses in the upper rows look completely different from the long glasses in the lower rows, they were all made by a single manufacturer.

One of the most surprising moments in building the Glassdance occurred when I attempted to tune a snifter. Equipped with a precision diamond blade band saw, I sliced a ¾-inch-high ring from the rim of a glass. While absorbed in this delicate operation, I intended to increase the fundamental frequency of the glass. To my complete astonishment, the fundamental decreased by about a minor third. After a further removal of a 1-inch-high ring, the fundamental, as expected, increased by about a fifth. For a second glass, I sliced a ¼-inch-high ring from the rim and the frequency decreased by about a major second; and after a further removal of a ¾-inch-high ring, the fundamental decreased by only about a semitone. All subsequent removals increased the fundamental.

To comprehend why the frequency of the fundamental decreased after reducing the mass of the glass, note that stiffness acts as the only restoring force that returns a vibrating glass to its equilibrium position. Because the walls of a snifter constrict at the opening, the restoring force due to stiffness has an especially high value at the rim of the glass. In the upper portion of the glass, a removal of circular sections has a greater effect on the restoring force than on the mass. Removing rings of glass in this area causes the walls to become less stiff, or more flexible. Consequently, the walls vibrate less rapidly, which in turn decreases the fundamental frequency of the glass. However, after slicing three or four narrow rings from the top, the removal of material has a greater effect on the mass than on the restoring force. This causes the walls in the lower portion to vibrate more rapidly, which in turn increases the fundamental frequency of the glass. The graphic below is from Musical Mathematics, p. 831.