The monocord was not original with Vincentio Galilei. It
is described in classical works of music theory, and if it
was not not already known to Vincentio, it was suggested to
him by Girolamo Mei, a humanist scholar of music theory. You
may recall that Vincentio credits Mei with solving the problem
of what Greek music really was like.
We will do three different things with it, (1) hear musical
proportions, (2) hear harmonics, and (3) measure how the
frequency of the fundamental depends on tension. These are
all things that are described by Vincentio in the 1580's and
by Galileo in the Two New Sciences (1638).
To hang weight on the string without breaking it, support the
weight as you gradually lower it to hang freely below the
soundbox. For the first two exercises, use 2 or 3 kg weight.
Hearing Musical Proportions
Vincentio describes how to hear any proportion you like with
a monocord. Let us say we want to hear the Fifth, which
is associated with the proportion 3:2. You add the two numbers,
getting 5, and divide the string into 5 equal parts, and then
put the bridge a distance 2 (or 3) of these parts from one end.
Pluck the string on either side of the bridge and you will hear
two notes separated by a Fifth.
Use this method to hear all the basic intervals of the scale,
according to Pythagorean theory, including the semitone, tone,
minor third, major third, fourth, fifth, sixth, seventh, and
This is not easy to do, and may be impossible if the room
is too noisy. The harmonics are modes of vibration of the
string that have special places called
nodes (places where they don't move) at
regular intervals along the string, the first node being
at one end and the last node at the other end. You can
constrain a vibrating string not to move somewhere by touching
it lightly there as you pluck it somewhere else. If you
carefully do this at a node of a harmonic, the string will
vibrate in that harmonic, and you will hear a high ringing
sound. If you move your finger just a little, it will be
in the wrong place, and you will hear only a dull "plunk."
Pitch as a Function of Tension
In this part, we will change the tension on the string and
see what happens to the pitch (frequency of vibration).
Start with the smallest tension (1 kg weight) and slide
a moveable bridge to match the pitch of the string with
the lowest string on a guitar.
Now add 1 kg weight and pluck the string. How much higher
is the pitch? Compare with the guitar string, and estimate
the interval between the two in well tempered semitones.
If you call the lowest frequency 1 (i.e., all other frequencies
will be in units of this one), then each well tempered
semitone is higher by a factor of about 1.06. Thus, for
example, if the pitch is higher than the lowest frequency
by 3 well tempered semitones, then we should call this
frequency 1.063 = 1.19. Alternatively, we could
notice that this is a minor third, which in the Pythagorean
system would be 6/5=1.20, about the same, of course. Is the
difference between these two assignments, 1.19 on the one
hand and 1.20 on the other (which is less than 1%) --
is this something to worry about? Yes, in
the sense that one should be aware of it. You probably cannot
match pitches better than 1% anyway, so we understand that we
are making a measurement
with an inherent uncertainty of about 1%. This is actually
pretty good, as measurements go, especially when you consider we
are doing it without any kind of scientific instrumentation,
not even something as simple as a ruler!
Make a table with two columns, weight in the first column
and frequency in the second. Add more weight (carefully!)
and get another frequency. Repeat.
Graph the frequency vs. the weight to see how it depends
on weight. Galileo and his father believed
they had found that the frequency goes as the square root
of the weight. A good way to check this idea is to graph
the frequency SQUARED vs. the weight.
Try this: how does it look?