## The Monocord

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 octave.

### Hearing Harmonics

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?