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)
