(Caution! Philosophy ahead.)
I'm sorry. I'm going to get a little theoretical today. But I need to get something off my chest. Perfection is a powerful state of being. Overwhelming, I'd argue. But there is a quirk in music in which perfection, stacked upon perfection, stacked upon perfection leads to what? You guessed it! IMperfection.
What?? Pythagoras, the ancient Greek triangle dude, really dropped a whopper on us. Maybe you can help me understand what's going on. The universe is at stake, after all. But before we can get to the heart of the problem, I need to set the stage.
You appreciate well tuned musical instruments, I'll bet. Ozzy Osbourne aside, a well tuned instrument is a must if you don't wish to sour the gentle ear. But what is tuning? Sure, you play into some fancy electronic gizmo, and it tells you what to do. But did you know that modern tuning (or "equal temperament") is based on an artificial compromise? It sacrifices perfection to eliminate an odd anomaly in music. And that anomaly bugs the heck out of me.
Let's keep to the basics. A note is a sound wave oscillating at a certain frequency. The frequency determines the perceived pitch. With me so far?
An octave is created when you double the frequency of any given note. For example, if a note is at 440 hertz, then the same note one octave higher is 880 hertz. Thus, an octave is a ratio of 2:1.
Now when you play two notes at once, sometimes it sounds nice and sometimes it sounds bad. Why? Because the two sound waves interact with one another. When the waves complement, they sound smooth and pleasant--consonant in music theory. When the interaction is dirty, causing some parts of the waves to cancel and some parts to augment (interference), beating is heard--dissonance in music.
After the octave (ratio 2:1), the most consonant of all the intervals in music is the "perfect fifth." Think a C and a G note in the C major scale. The perfect fifth has a ratio of 3:2. With this ratio, no interference or dissonance occurs. The result is harmony. Smiling faces. Beauty.
Yes, perfect fifths are cool. (And I don't mean Jack Daniels! Stay with me now.)
Here's the kicker. Pythagoras discovered a tuning system called the "circle of fifths" or "just temperament." Basically, if you start with a note, go up a ratio of 3:2, then go up another 3:2, and again, and again, eventually after 12 of these "fifths," you will reach a note roughly 7 octaves higher than the starting point. If you bring all those notes down to fit into a single scale, you've just tuned your instrument to the pure mathematics of the universe.
Oops, wait a minute, I said "roughly" seven octaves higher. In fact, after this stacking of perfect fifths, you end up 23.46 cents too sharp (don't get hung up on "cents," it's logarithm mumbo jumbo). That error is called the Pythagorean Comma, and the end result is that your beautifully tuned instrument is now harboring a clunker or a "wolf interval." Basically, certain musical keys will sound great, and some will be unplayably bad. That's a problem for a keyboard instrument which lends itself to playing many keys of music. It's not a problem for many folk instruments (like my bagpipes), which only play in one key.
The problem was so irritating that Bach experimented with artificial adjustments to tune out the error. The result was "well temperament" (hence the musical collection penned by Bach called the "Well Tempered Clavier," which demonstrated the new versatility). Modern tuning is a refinement of well temperament.
Interestingly, the rub between tuning systems can be heard if a piano using equal temperament is played with a guitar tuned by ear. Since ear tuning is based on harmonics and the perfect fifth, the two instruments will diverge and sound out of tune. To remedy the situation, the guitar must be tuned to the artificial, modern system.
What is the source of this mathematical punk, you ask? Simple. No matter how many 3:2 ratios you stack up, you can never equal a 2:1 ratio, which defines the octave. Simple fractions. Just can't do it.
But wait!! Perfect fifths are tidy. They sound nice. They build, logically. Shouldn't there be a reward? If you stack one on another, shouldn't you come full circle? Why do you just dribble over where you expected to be? (A 23.46 cent dribble to be precise.)
The circle of fifths isn't a circle at all. It's a never-ending spiral. To what? Hell? Is 23.46 cents the sign of the devil??
The space-time continuum is ripped! I'm tumbling in! The garden path leads to a swamp.
It just doesn't add up!
And in the midst of all those ear-tuned death spirals, Ozzy's starting to sound pretty damn good to me.