[During a lecture] I invite someone who knows how to play the guitar to the front, and then give that person a monochord. A monochord is an old device which is hardly known today. When my friend Carl Mitcham tried to find me one at Penn State, people asked, What the hell is that? He did find an old professor in the music department who remembered using one as a student, but none could be found, until at last he hunted one up in the physics department, where it was stored for possible later use in the museum. It’s an elongated wooden box, about the span of a man’s arms, with single string mounted on it. This string can be stopped at any point along its length in order to demonstrate musical relationships. So in my class the person serving as my assistant demonstrated various divisions of the string, halving it to sound an octave and then stopping it at one third of its length. Suddenly, in this class of 150, sometimes 200 people, a number of faces lit up. They could hear the harmony produced by what musicians call the fifth or the quint. If the finger stopping the string was moved by even a fraction of an inch, it didn’t sound the same to those who could perceive it. Arrangement of these perceptibly harmonious sounds defined music throughout most of its long history.
But music, like human nature, has a glitch. If you repeat the fifth, that is, if you take the longer part of the string and again divide it in a ratio of 1:2, you take the first step in what is called the circle of fifths. If you take repeating this operation, taking the fifth of the fifth of the fifth and so on, you will finally return to your original note, sounded several octaves higher. Except you don’t quite arrive at your starting point. There is a small discrepancy, which the old Greeks called a comma. The circle of fifths doesn’t come out quite right. This has always been a problem, and a point of discussion among musicians, but it was only around the time of Bach that its solution became a serious task. They wanted to see if they couldn’t rearrange the circle of fifths in such a way that they made each step slightly disharmonious – in effect averaging out the comma – in order that the circle of fifths should end where it began. This was in order to prevent individual instruments, or instruments playing together, from getting more and more out of tune if they ventured into keys remote from their starting point. The process was called tempering the scale, and it turned out to be a very difficult task, only fully achieved in the nineteenth century, when it became possible to measure the vibrational frequencies of musical pitches and to use logarithms to make the complex calculation of how much to shave off the tail of one quint and how much from the nose of the other in order to make the scale come out right while still giving the untrained ear the impression of being in a harmonic, rather than a tempered, scale. The key figure was the eminent German physicist and physiologist Hermann Helmholtz [1821-1894].
Source: Cayley, D., & Taylor, C. (2011). The Rivers North of the Future: The Testament of Ivan Illich. New York: House of Anansi Press. Pages 134-135.