Monday, December 05, 2005

Did Pythagoras Find a Rip in the Universe?

(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.


Brood Mode said...

good one! i didn't know all these facts... interesting

kyknoord said...

This is why it helps to be vaguely psychotic if you're a musician.

Ultra Toast Mosha God said...

Guitars suffer particularly badly from the traditional tuning method. Only recently did a gentleman by the name of Buzz Feiten develop a more accurate tuning system for the guitar.

Bernita said...

Have you just explained why time travel is possible?

beadinggalinMS said...

WOW! I too didn't know this and found it interesting.

Anonymous said...

Brood Mode, welcome to The Clarity of Night!! So many new faces lately...I love it! I hope to see you back.

Kyknoord, vaguely psychotic musicians--that's a great point. That freedom is empowering.

UMTG, what's the tuning system based on? Is it something like equal temperament?

Bernita, you know, you may be on to something there! Sometimes when I play the bagpipes, I'm transported....

BeadinggalinMS, it's a fascinating subject, isn't it? It really made an impression on me when I first learned of it.

Terri said...

I had no idea. I still don't, but now I have a headache. Thanks, Jason ;-)
Mind you, it makes you wonder how Beethoven did it - he was deaf, wasn't he?
(PS I just googled "deaf dead composer" to check my facts - the results were quite surprising!)

Kara Alison said...

THink about it this way: When two people are trying to sing the exact same note, even the slightest variation sounds like hell (although you don't get that cool wavy effect with vocals in general...they just don't resonate the way two wind instruments do...annnyway). When two singers sing two harmonious notes though (say thirds, fifths, whatever) the sound is pretty clean. Because the vibrations should not match perfectly, the "waawwaaaww" (sorry, technical music term) isn't really discernable by the naked ear. The vibrations compliment one another, but they are not perfect. When the thirds are out of tune though, the clash doesn't appear because again, it wouldn't have anyway.

I guess my point is that two different notes can't possibly be perfectly in tune unless they are octaves, and then...they're essentially the same note (vibration) and will line up perfectly. Try to imagine the oscillations lining up in your head, and you'll see what I mean.

This is why all of the music kids are good at math too! I love this stuff....thanks Jason.

Jeff said...

jason- I've played guitar for thirty-five years and I never realized I might have been holding the answers to the riddles of the universe in my very hands. wow! :)

Sarah said...

The unravelling of the universe is at my fingertips, apparently. I never was good at math... Fascinating entry.

Kara Alison said...

I'm sorry, but I'm an absolute sucker for this type of stuff...philosophy and music together. I have to bite when you bait me that way.

Annnyway, another interesting way of looking at this: 2/3 equals 0.666666...(six repeating to infinity). This is not actually a round number and will never actually end up to 7.0 (The grand total after seven octaves). Tadaaaaa!

Anonymous said...

Terri, sorry I gave you a headache! I know this one was taxing. ;)

Kara, I knew you were going to be the light at the end of the tunnel! For a moment, reading your response, I finally thought I had the answer. Perfect fifths are not perfect at all. But then, something happened. When I was researching for this post, I saw an interference pattern for two diverging tones. Even though the perfect fifth was not actually perfect, it still was a unique consonance on the scale. No other harmonic after the octave was greater in consonance. I still think there's something cosmic in that fact, and I'm still perplexed that tuning based on that moment of highest consonance gets you so close, but still so far.

Jeff, indeed they are there, my friend. They are there. ;)

Sarah, you're so close to the answer. I can feel it....

Kara again, yes, that is the reason mathematically, but I can get past the question WHY (in an order of the universe sense). You can't tune by octaves, since as you point out, it's actually the same note over and over. You need a way to jump from note to note. The perfect fifth gives you that opportunity. Yet, the "perfect" solution eventually falls like a house of cards.... .666, yes, the devil is involved, but he is revealed to be a fraction!

M. G. Tarquini said...

There's a thriller, or a mystery, or a work of commercial fiction in all of this Jason. Titled 'perfect fifth' or 'Circle of Octaves'

And thanks. The best math I ever learned came from Isaac Asimov. He did a book or two of musings on math. I credit him with explaining calculus, ratios and logarithms. He made the mathematics come alive - like you just did now with the music.

Kelly Parra said...

What Brood Mode said. LOL! You're a very smart guy, Jason! =D

Farzad said...

I didn't know any of these... But I'm glad that you put it out there. Something new to learn is what I appreciate.

Kara Alison said...

Hmmm...cosmic meaning. Maybe? I've always felt that music is the one thing that could convince this non-believer of her error. It's so fantastic and it makes me feel things that I can't explain (and we've talked about this). Literally, the right chord will make my eyes well up for no apparent reason. Why is that?

I'd like to comment on imperfection. I think the fallacy here is the idea that the circle must be closed in order to be perfect. Perhaps the infinite nature of sound and the (upward?) spiral of the circle of fifths (gosh, this takes me back to high school jazz) is actually the embodiment of perfection. The circle would theoretically start with silence, and each "fifth" is really just representative of an interval that we've designated as pretty. If you're looking at limits (some basic calc for you here), then you realize that if you start at, say, middle C and work your way down, by fifths, you should never actually reach silence (the complete lack of vibration). So how can silence exist? It does though.

The idea here is that certain formulas can take you very close to the limit, but they will never quite reach it. 1/x will take you so close to the x axis that you imagine the graph and the axis are touching, but they are not. This doesn't mean that the X axis does not exist, it just means that the formula you're using isn't the one to get you there. Switch your formula to 1-X and you reach the axis with no trouble at all.

The same applies with our musical conundrum. I'd say that we're just not approaching the axis with the right formula. The circle of fifths isn't is damn near close though.

One more thing and then I'll shut up (maybe...hehe). Fifths are considered pleasing, but disonance is generally cringed at. I'd argue that, in the right context, disonance can be just as heart-breakingly beatiful as harmony. Maybe we should invent a circle of fourths and see how that goes!

Anonymous said...

M.G., that's a high compliment! Thank you!

Kelly, *blushing*

Farzad, I like to kick concepts around now and again. Keeps us from getting stale!

Kara, that's a great response! I really have nothing to add. Well done.

Shesawriter said...


I got a headache just reading through all that. Stop trying to make me think! LOL!


Anonymous said...

Tanya, I do apologize. =D

anne frasier said...

please say we aren't going to be tested on this. ;)

Anonymous said...

Anne, it's not on the final.

Ultra Toast Mosha God said...

Yes. It takes into account non mathmetical elements like string tension and nut placement. Or something. I don't know the technicalities. Apparently Feiten tuned guitars are much more pleasant on the ear. I have yet to try one

Robin Caroll said...

I'm confused...but then again, I just got my pc out of the hospital, so I'm still on the "HIGH" from that! LOL

Anonymous said...

UTMG, interesting. Sounds like he's also affecting the quality of the sound.

Robin, don't worry about it. This point was 73% weird.

Tsavo Leone said...

Excellent post!

I was going to mention Buzz, but someone else already did. For ages Washburn guitars were the only ones to have this set up, but... well, now it's out there for everyone.

Having never delved into the physics/math of music too deeply I never could understand why the bassist in my old band used to constantly harp on about me being out of tune (which I wasn't). I guess it was probably to do with this.

Gonna read the other comments before saying (typing) any more...

Tsavo Leone said...

H P Lovecraft: The Music of Eric Zann.

This post had me scratching my head trying to remember something. Worth a look perhaps?

Mary Louisa said...

Just don't f' around with the devil's fifth. (And I don't mean Jack Daniels.)

Anonymous said...

Tsavo Leone, I've seen you around with Bernita, but welcome here! I'll have to check out that link about guitar tuning. I wonder if part of it is unique to that instrument. Also, I have read The Music of Eric Zann, but it's been a while. Didn't the sound of the violin lead the MC to a place in Paris which he could never find again?

Mary Louisa, I actually found a reference to the Devil's Fifth in early modal music. Now you're taxing MY brain if you make me research the characteristics and origins of the old modal forms. =D

Tsavo Leone said...

Re: Mary Louisa.

Are you referring to the interval infernale, as used by Saint Saens in "Danse Macabre"? It's also become something of a standard in a particular genre of music over the last decade or so (the augmented fourth I think it tends to be referred to as).

Continuing in guitar-oriented vein, you might notice that 'closed-chords' tend to sound quite different to their 'open' brethren. Again, it's something to do with harmonics and the interference pattern produced by such closely matched frequencies.

I'm also wondering whether anyone else has thought about the fractal when considering our 3:2 spiral, or the 'magic number' associated with flowers?

Anonymous said...


That's a great point about fractals and the golden ratio. The perfect fifth may be part of a mathematic principle which stretches far beyond music. We might really be on to something here! ;)

Kara may be right. The spiral might actually soar upward into the light, not plummet into darkness.

Mary Louisa said...

Tsavo, yes, that's the one. It's a diminished fifth interval. It was outlawed in sacred music, thus its lovely nick name.

Jason, consider yourself brain-taxed. ;)

Anonymous said...


Well...I better get started on the research, then.