THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

532 18. Music and Musical Instrumentsbow is drawn across the strings, the string vibrates as the result of its being draggedwith the bow and then springing back under the impetus of the restoring force.The normal range of the bow’s action on the strings depends on its force actingon the strings and the position of the bow relative to the bridge. Too heavy a forceresults in raucousness of the tone. Too little force results in instability of the stringdisplacement’s saw-toothed curve. According to Schelleng (1974), the maximumbow force depends primarily on the string and on its coefficient of friction. Thisbow force is inversely proportional to the distance of the bow from the bridge; theminimum bow force, on the other hand, is inversely proportional to the square ofthe distance of the bow from the bridge. The maximum and minimum bow forcesare equal when the bow is placed at a point very close to the bridge, and theydiverge as the bow moves away from the bridge. The disparity between these twolimits, as the bow is positioned further away from the bridge, provides the widetolerance that makes fiddle playing possible. As a far distance from the bridge,for a given bow velocity, the volume of the sound is less, the amount of highfrequencycontent lessens, and the timbre possesses a gentle character designatedby composers as sul tasto (“bow over the fingerboard”). In placing the bow nearthe bridge, the required bow force soars dramatically to almost prohibitive levelsand the solidity of the fundamental tone disappears, leaving little more than theremnants of high harmonics to suggest the fundamental tone: this is the eeriersounding sul ponticello (“bow over the little bridge”). In the normal playing area,the tone becomes more brilliant (i.e., the relative harmonic content increases) as thebow moves toward the bridge or as the bow force increases toward the maximum.A vibrating string alone produces almost no sound, because it is so slender thatalmost no air is displaced by its vibration and, moreover, the two diametricallyopposite sides of the string are so close together that when the air on one side iscompressed, the air on the other side is rarified. These two effects are so close toeach other that they effectively cancel each other. In order to avoid the cancellation,one vibration must follow another by a substantial fraction of a phase.In the bowed instruments, under optimal conditions, the top and the back platescan move inward and outward at a given moment that nearly the entire surfaces ofthe instrument act to change the volume of displaced air, thus acting as a “simplesource” in the lower frequency range (Cremer, 1971). In the violin family, thisvolume change, essential to generating sounds in the lower octaves, is renderedpossible by the asymmetric layout of the bass bar and the sound post inside thebody. When the bow is pulled across the string, a rocking motion is set up inthe bridge so that the two feet of the bridge are in a “push–pull” mode, with onefoot of the bridge pressing down and the other foot going up in opposition. Ifthe box (i.e., the body of the instrument) has total bilateral symmetry, the motionof one foot of the bridge would cancel that of the other foot of the bridge, andno volume displacement would occur. But the sound post, essentially a dowelfirmly coupling the back plate to the top plate tends to immobilize the right (withrespect to the player holding the instrument) foot while allowing the left foot tomore freely and causing the body of the instrument to vibrate under the influenceof the left foot. Thus, the sound post’s chief role can be considered as creating