Musical Mathematics
on the art and science of acoustic instruments
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© 2000–2019 Cristiano M.L. Forster
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CHAPTER 12: ORIGINAL INSTRUMENTS
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 (
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