The Circle of Fifths
There's a bad meme going around, a musical myth, that's been going around for a few hundred years. It's taught in music schools. Most music students accept it as true without bothering to double check it. Most students who become teachers continue to pass it on, unknowingly poisoning the understandings of even more musicians. Unfortunately, it's a myth that obscures a great deal of musical beauty for the poisoned ones.
That myth is called “the circle of fifths.” Simply put, it proposes that if you start on A and you go up twelve fifths, you land on another A.
Believers might even demonstrate it by playing it on the piano or guitar, but what they misunderstand is that the piano and guitar are not playing fifths. Both instruments are tuned in twelve-tone equal temperament, which means that all the intervals (except the octave and its multiples) have been tempered (slightly changed from true) to force a final condition that all the pitches in the system are equally spaced (on a log scale of frequency). In other words, the melodic distance between any two closest pitches is artificially forced to have the same width as the melodic distance between any other two closest pitches, like inches on a ruler.
The fifth suffers when tempered by being slightly narrowed from its true ratio of 3/2 to an artificial ratio of (2^(7/12))/1. In other words, it's narrowed from exactly 1.5 (a simple rational number) to approximately 1.498307076876681498799280732029... (you can't actually write it in decimal notation, because it's irrational). You can hear that the equal-tempered fifth is out of tune by listening to it played with any simple sustained timbre, such as a clean organ tone. It beats.
If you try the same exercise by stacking “just” fifths (meaning true, natural, real fifths), which have a ratio of 3/2, you'll notice that they never wrap around to the same starting pitch class again. They keep generating new pitch classes indefinitely. To demonstrate that, you'll need instruments capable of playing un-tempered just intervals, such as violins, trombones, human voices, or audio software.
There's a very good mathematical reason why just fifths don't wrap, which is that no power of any prime equals any power of a different prime (ignoring power zero of course). In other words, fifths are based on prime 3, and octaves are based on prime 2, and no power of 3 equals any power of 2, so no number of fifths equals any number of octaves. The same is true for other pairs of just intervals. No stack of minor thirds (6/5) equals any stack of major thirds (5/4), etc.
Once you start to hear that, you begin to envision the internal structure of natural tuning extending indefinitely in all intervallic directions without looping. Natural tuning provides unending resources for the investigation and composition of audible beauty. As with so many things in nature, the closer we look, the more we find, until we reach our own physiological limits to observe.
In summation, just intervals provide infinite pitch resources for compositional exploration. The circle of fifths is an artificial contrivance of historically recent music theory that intentionally introduces tiny errors to enforce a finite loop of only 12 pitch classes where no such loop exists in nature. We do ourselves an aesthetic disfavor by employing errors to express beauty. We disconnect from our ears when we let ourselves accept any chord as being “in tune” while it beats like crazy.
quote and format by Raxin
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