On Math and Music: Pythagorean Tuning

How was the first instrument tuned? On first glance, this question may seem similar to the chicken and egg quandary we learned as children -- simple but circular. When most of us tune an instrument, we generally resort to a mechanical device or another instrument. However, it turns out that the precise mathematical relationships that define musical intervals allow the unaided ear a limited ability to determine relative pitches without any mechanical assistance. As such, the first instrument could have, you might say, been tuned to "itself."

In fact, tuning by ear was quite common in the medieval and Renaissance periods, using a system called Pythagorean tuning. To understand how this works, consider first the octave. Given an arbitrary note, most people could quickly learn to find a pitch that was an octave above or below. The 2:1 ratio between the frequencies of the notes of an octave makes it easy for the ear to pinpoint these pitches, particularly if the two notes are played simultaneously. Therefore, from a starting pitch, the unaided ear could tune the pitches that were at intervals of 2:1, 4:1, 8:1, 16:1, etc. by finding the note an octave above the starting note, followed by the note an octave above that, and so forth. Using the inverse process, one could also identify notes at intervals of 1:2, 1:4, 1:8, 1:16, etc. Unfortunately, that only leaves us with a musical scale that sounds like this: listen.

Fortunately, the human ear can, without too much training, learn to identify another interval, the fifth. At a ratio of 3:2, this interval blends almost as smoothly as the octave, and again becomes easier to identify when the notes are played simultaneously. At first glance, the ability to tune a fifth may seem like a minor improvement, but this development actually gives us a great deal more freedom in frequency space. This fact is easier to see from the mathematical point of view.

Suppose I were to tackle the standard problem of constructing a scale of twelve tones between a pitch of arbitrary frequency and a pitch one octave up (frequency ratio of 2:1). Since I can tune a fifth, I automatically have one additional note at a ratio of 3:2, which gives me the following scale: 1:1, 3:2, 2:1 (listen). However, I can add to this by considering the pitch that is a fifth below the starting pitch: 2:3. Although this particular pitch is not between 1:1 and 2:1 and therefore does not belong in my scale, remember that I can tune intervals of both an octave and a fifth. Therefore, the following set of tunings is allowed:

1:1 -> down a fifth (x 3:2) = 2:3 -> up an octave (x 2:1) = 4:3

I now have a way to tune the following pitches: 1:1, 4:3, 3:2, and 2:1 (listen). Another pitch can be added with the following set of operations:

1:1 -> up a fifth (x 3:2) = 3:2 -> up a fifth (x 3:2) = 9:4 -> down an octave (x 1:2) = 9:8

If I continue performing tunings of this kind, I can construct a full twelve-tone scale with the following intervals from the root pitch: 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 1024:729, 3:2, 128:81, 27:16, 16:9, and 243:128. This technique of tuning is also known as the "circle of fifths," for obvious reasons. Note, however, that the scale doesn't include some of the small-number ratios I discussed in my intervals post, most notably 5:4 and 5:3. Although it includes intervals that are close to these frequency ratios (81:64 and 27:16, respectively), the major third and the major sixth still tend to sound more dissonant in the Pythagorean tuning system than in systems that give them small-number ratios. In fact, it was in part due to the widespread use of this tuning system that medieval composers favored intervals of a fourth and fifth over intervals of a third or a sixth. In the 15th century, as triadic harmony saw more widespread use in compositions, musicians began to favor other tuning methods.

Related Links: Extended discussion (medieval.org); Meantone temperament (wikipedia)

On Math and Music: Intervals

Some believe that the most fundamental aspects of reality -- everything that we are and everything that we perceive -- ultimately come down to mathematics. It is the language of science and perhaps the only surviving bastion of irrefutable truths in the aftermath of the Age of Enlightenment. It is this irrefutable quality, this perfection of sorts, that also lends mathematics a certain beauty; in fact, the origin of all beauty may come down to simple mathematical relationships.

Our brains perceive mathematical relationships in countless ways, but perhaps none are so direct as the way in which we process sound. Suppose I were given two tuning forks, one designed for a frequency of 440 Hz and the other for 880 Hz. When I strike the first, the metal begins to vibrate, moving back and forth at a rate of 440 times per second. This vibration, in turn, causes the surrounding air molecules to oscillate at the same frequency, an oscillation that travels outwards from the tuning fork and reaches my ear. Assuming that the tuning fork continues to vibrate at this frequency, my brain will interpret the oscillation of air molecules as a steady and constant "pitch."

Now suppose I strike the second tuning fork, which vibrates 880 times per second. This new pitch corresponds to a frequency twice that of the first, and if I strike the second tuning fork while the first is sounding, they together produce a sound something like this. Notice how smoothly the pitches blend together. The interval heard here is called an "octave," a musical term reserved for any pair of pitches with a frequency ratio of 2:1. Our ear easily identifies the relationship between these two pitches because their frequencies are in a small, natural-number ratio to one another. By contrast, listen to the major seventh, an interval that corresponds to a frequency ratio of about 15:8. The blend is not nearly so pleasing to the ear.

When two pitches blend together well, like the octave, they are referred to as "stable," or a "consonance." Those that don't blend so well, such as the major seventh, are referred to as a "dissonance." In medieval music, the consonances were the octave (2:1), the perfect fifth (3:2, listen), and the perfect fourth (4:3, listen). In the early 15th century, starting with the Burgundian School, intervals of a major third (5:4, listen) and a major sixth (5:3, listen) began to be treated as consonances, allowing for developments such as triadic harmony.