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