One of the great mysteries of music is how the renowned violinmakers of past centuries, apparently having had no more than a practical knowledge of the physics and acoustics of their instrument, could turn out violins that are still cherished today for the beauty of their sound. For some 30 years a small but worldwide group of us in an organization named the Catgut Acoustical Society have applied modern methods to the study of the physics and acoustics of violins and other stringed instruments. I described the early stages of the work in these pages nearly 20 years ago [see "The Physics of Violins," by Carleen Maley Hutchins; SCIENTIFIC AMERICAN, November, 1962]. Now the work has progressed to the point where a good deal can be said about the properties of the top and back plates (the "belly" and the "back") of a violin before they are assembled into an instrument. Moreover, it is possible on the basis of these findings to construct violins and other members of the violin family with consistently fine tone and playing qualities.

The two plates are carved traditionally from solid blocks of wood, the top plate from two adjacent pieces of straight-grain spruce (Picea abies) joined down the middle, and the back plate from either a single piece or joined pieces of maple (Acer platanoides) that usually have a "flame," or curl, across the grain of the wood. Yet because of the variations in wood, even between two adjacent pieces from the same tree, it is not possible to reproduce measurement for measurement the parts of a fine-sounding violin and so to create an instrument with the tone and playing qualities of the original. The way to duplicate a fine violin lies not in geometrical measurements alone but must include measurements related to the vibrational properties of the wood.

The long-term investigation described here draws heavily on the experience of violinmakers and provides some new answers to a question asked in 1830 by Felix Savart, a physician and physicist: "What sounds ought the top and back of a violin have before they are joined?" Through the generosity of the eminent French luthier Jean Baptiste Vuillaume, Savart was able to test the disassembled top and back plates of a dozen or so violins that had been made by Antonio Stradivari and Giuseppe Guarnieri. (Imagine!) He applied a measuring machine he had devised, together with a technique worked out by his friend Ernst F. F. Chladni. By the Chladni method the eigenmodes, or normal mode patterns of vibration, of a horizontally mounted flat plate can be demonstrated by sprinkling the plate with a fine powder and causing the plate to vibrate. At certain frequencies (the eigenfrequencies) the vibration bounces the powder into the nonvibrating nodal areas, thereby outlining the nodal and antinodal configurations of the plate at its specific resonance frequencies. These plate resonances, or normal modes, are created by the physical properties of stiffness and mass, which cause standing-wave patterns to be formed in response to vibration at discrete frequencies unique to each plate. In answer to Savart's question he reported: "We have found that the sound varies in good violins between C^# 3 [the 3 indicates the octave] and D 3 for the belly, and for the back between D^# 3 and D ^# 3, so that there is always a difference between them of a half or a whole tone."

Over the years other investigators have made vibrational measurements of violin plates, both free and in the assembled violin, and have assessed the resulting tonal characteristics. Particularly notable is the work of the acoustician and violinmaker Hermann F. Meinel in Berlin during the 1930's, which documents the correlation between the thickness of the plates and the vibrational modes, the volume of sound and the timbre of sound. Meinel also recognized the limits of constructing violins on an empirical basis and noted the effects of the properties of the wood, the arching of the plates and the varnish. He explored the possibility of improving a particular violin in a given frequency range by removing wood from a specific area, following the work of Hermann Backhaus, but concluded that improvement does not always result because it depends on the physical state of the violin. This early work highlights a basic problem in violinmaking: a small change that will markedly improve one instrument may adversely affect another owing to the widely varying configuration of the vibrational modes and the stillnesses throughout the plates.

In 1950 the Harvard physicist Frederick A. Saunders and I began a collaborative study aimed at verifying Savart's findings and at developing other vibrational measurements that relate the unique bending characteristics of each pair of free top and back plates to the particular tone and playing qualities of the assembled instrument. By 1960 the results of some 200 tests on violins and violas in the course of construction had confirmed Savart's main finding: the instrument has good musical qualities when the principal plate tone of the top and of the back are from a tone to a semitone apart. Violinmakers call the principal plate tone the tap tone because it is the principal tone heard when the plate is tapped. Our findings showed in addition that the actual frequencies can vary considerably and that the top plate's tap tone can be higher than the back's or vice versa and still result in an instrument with good tone.

These observations did not go far enough toward explaining our finding that when the pairs of free plates were assembled, the resulting instruments did not always possess the expected tone and playing qualities. Every now and then an instrument proved without apparent reason to be much better than the others or worse. Seeking to explain these inconsistent results, I have continued the research since Saunders' death in 1963, building and testing 160 more instruments of the violin family. (The family includes the traditional violin, viola, cello and bass. Some new and revised in-