Musical Mathematics

on the art and science of acoustic instruments


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Stringed Instruments


Section 12.3


          The canon as described in the works of Ptolemy (see Chapter 10, Section 19) and Al-Jurjani (see Chapter 11, Section 52) represents the mathematical embodiment of tuning theory. Although this instrument has a noteworthy history, it did not develop into a precise musical instrument because of a persistent mechanical problem: rattling bridges! When one places a triangular-shaped bridge under a string, and then plucks the string, the applied force causes the bridge to rattle against the soundboard. To avoid this difficulty, it is possible to make a long bridge, so when one plucks a given string the other strings hold the bridge in place. Even so, after much playing such a bridge begins to creep due to the vibratory motion of the strings. To prevent the bridge from moving, it becomes necessary to increase the downbearing force (or downward force) that the strings exert on the bridge. An increase in the height of the bridge increases the strings’ deflection, which in turn increases this secondary vertical force. However, because the downbearing force effectively increases the tension of the strings, all the stopped strings sound sharp.


          To qualify as a precision instrument, a canon must satisfy two mathematical requirements. For example, if a canon bridge stops the right side of a string at length ratio 2/3, then the right section must sound a “fifth” above the open string, or frequency ratio 3/2. Also, since this bridge stops the left side of the string at complementary length ratio 1/3, the left section must sound an “octave and a fifth” above the open string, or frequency ratio 3/1. (See Chapter 3, Sections 11 and 13.) Now, suppose that a canon bridge is too high, so that the “fifth” on the right side sounds 30.0 ¢ sharp, and the “octave and a fifth” on the left side sounds 50.0 ¢ sharp. Under such circumstances, we would be correct to call this instrument a kind of zither (see Chapter 11, Section 3), but incorrect to call it a canon.


          A fundamental principle of canon tuning states that the mass density of the string material, the string diameter, and the string tension must be constant for all strings. Therefore, the only acceptable variable is string length. To achieve this requirement,

an accurately built canon must have bridges that are only slightly higher than the strings above the soundboard.


          If we rule out the downbearing force of the strings above the bridge, then the only alternative is to design a mechanical device that exerts a force onto the soundboard below the bridge. Since a canon should have moveable bridges for the exploration of myriad tuning systems, simply gluing the bridge to the soundboard is not a meaningful alternative. Instead, the force below the bridge should act over a relatively wide surface area to facilitate the unrestricted placement of bridges.


          Figure 12.3 shows a longitudinal cross-section of the infinitely adjustable canon bridge assembly. Ribs (a) provide structural support for the soundboard (b), and anchorage for threaded inserts (c). The latter are called knife thread inserts because on the outside they have sharp threads designed for turning into wood, and on the inside, standard machine screw threads. Above the soundboard, the bridge carriage (d) consists of 1/2 in. square aluminum tubing with rounded edges. In Plate 2, notice a long slot milled into the top side of each carriage. Two socket head cap screws (e) pass all the way through this slot and into two holes in the bottom of the carriage; from here, they go through the synthetic washers (f), the soundboard, the upper portion of a rib, and finally screw into the inserts in the lower portion of the rib. This secures the carriage to the soundboard. The washers consist of a material called E.A.R., which is an acronym for energy absorbing resin. I used this material to dampen the high mode frequencies produced by the metal parts of the assembly, and to prevent the carriage from marring the finish of the soundboard. Two round head cap screws (g), designed to hold the aluminum plate (h) against the carriage, slide back and forth in the long slot. Two nuts (i), which have a close fit inside the carriage, will not turn when one tightens or loosens the cap screws. Finally, a flat head countersunk tapping screw (j) fastens the Delrin bridge (k) to the aluminum plate. By simply securing the bridge assembly anywhere on the soundboard, sliding the bridge to a desired location along the carriage, and tightening the cap screws, a musician may achieve any tuning imaginable. For this reason, I like to refer to the Harmonic/Melodic Canon as a “limited form of infinity.”





          On the H/M Canon, the adjustable bridge assembly works for all strings except the first string. For String 1, the bottom of the bridge slides in an aluminum slot in the shape of a dovetail track. A vertical slit divides the lower portion of the bridge into two symmetrical halves. When one turns the knob, a machine screw widens this slit and, thereby, pushes the bottom sides of the bridge against the angled edges of the dovetail track. This enables one to lock the bridge anywhere under String 1. After tuning the first string, a musician slides the bridge to a location in line with another bridge on the canon soundboard, locks it in place, and then tunes the other string in unison to String 1.


          The H/M Canon is as much a musical instrument as a scientific instrument. Since the bridge assemblies fasten directly to the soundboard, (1) rattling bridges do not exist, (2) every string has its own bridge, and (3) the tuning is accurate on both sides of the bridges. However, a tuning accuracy limit does exist. As strings become shorter than 200.0 mm (8 in.) they sound progressively sharp. The cause for this imperfection is string deflection. Every bridge must deflect its string to some extent; otherwise, the string does not make full contact with the bridge, which results in a weak tone because the transfer of mechanical energy from the vibrating string to the bridge is inefficient. Now, when one moves a bridge closer to either end of the string, the angle of deflection of the string between nut and bridge increases. As the angle of deflection increases, the downbearing force that the string exerts on the bridge increases as well. Consequently, the vertical force has a greater sharpening effect on short strings than on long strings. To minimize this tendency, I intentionally gave the soundboard of the H/M Canon a length of 1000.0 mm plus 4.0 mm; that is, both the left and right sides of the soundboard are 502.0 mm long. For all bridged strings, and especially for short strings with lengths from 250.0 mm to 200.0 mm, this added length — called string compensation — counteracts the sharpening effect of string deflection. (See Chapter 10, Section 31.) But for strings shorter than 200.0 mm, string compensation no longer produces favorable results.