In the Ancient and Medieval periods, the organizing principles of musical sound were seized upon by scholars, philosophers, and theologians as evidence for the existence of rational rules of nature. On the one hand, it was widely understood that music had the power to affect the mood and the bodily comportment of listeners and performers, and on the other, the scholars were fascinated by the mathematical elegance of musical proportion as it related to harmonious intervals. Indeed, many saw the perfect ratios of musical consonance as proof that the power of music was far from coincidental. For modern students, the idea of music as measurement can seem foreign, but if we look at the tools (literally, the instruments) with which Ancient and Medieval music theorists worked to make such claims, the mathematical ratios become clear.

The monochord was far and away the most important technical instrument of musical theory at this time. As the name implies, the mono-chord consists of a single string, placed over a resonating chamber, and provided with moveable bridges that allow the user to change the length of the resonating string. Tension on the string is created by a weight or a tuning peg. There’s a great diagram on wikipedia that shows what I’m talking about, and lots of images of monochords available on the internet.

Those of you who play string instruments (violin, viola, cello, guitar, even harp or piano) are surely familiar with the idea that as the vibrating segment of the string gets shorter, the pitch produced gets higher. What struck early scholars as significant was the mathematical elegance of harmonious intervals. Basically, if you halve the length of the vibrating string, the pitch goes up by an octave. Mathematically, this gives the ratio 2:1 where “2” is the original length of the string and “1” is the length of the short vibrating segment. The ratio 3:2 (stopping the string with a finger or moveable bridge at ⅓ of the way along the string and allowing ⅔ of the length to vibrate) will produce a perfect fifth, and the ratio 4:3 a perfect fourth. After this point, the math gets a little messy--a major third, for example, has a ratio of 81:64, which is nowhere near as elegant!

These “perfect” intervals: the unison (with a ratio of 1:1), the octave, the fifth, and the fourth, are often called the perfect or Pythagorean intervals because of their perfect mathematical ratios, and during the Ancient and Medieval periods they were the only intervals considered to be consonant.

The synchronicity of these intervals and their mathematical perfection were one of the best known and most easily observable scientific laws of nature. They offered early thinkers proof that there were rules that shaped the world around them, that the rules could be expressed in mathematical terms, and that through a process of observation and experimentation, humans could derive and then replicate such rules. Thus music was a science, closely related to measurement and mathematics.