Tuesday, December 30, 2008

Taking the fun out of Music

I'd imagine everybody who reads this blog knows I play music. If you didn't... well there you have it. I play music.

I stumbled upon something that fascinated me about two years ago, and in discussions I found myself unable to clearly articulate or explain it. Therefore, I will write a bloggings on it to get that group of neurons in order.

We all know there is an underlying physics to music: sound/pressure waves, vibrations, acoustics. Most people have a concept of how plucking one guitar string turns into a Bb in the ear. Fewer people understand that the Bb plucked by a guitar is actually made up of several harmonics (vibrations with different frequencies) happening all at once (the loudest just happening to be a Bb)

The next step is making different pitches/notes. Some of us have noticed that one octave on a guitar means you put a finger right in the middle of the string (cut the length in half). Cut it half again to go up another octave. [if a 12 inch string play a C, a 6 inch string will play a C one octave higher, a 3 inch is yet another octave higher...). But what if you tire of C's? In western music, there are 12 notes in a chromatic scale. And for piano (and guitar) today, what is called an equal temperament is used to decide the length of strings (position of frets). This means the ratio of adjacent strings (or frets) is the same no matter which string you choose. Basically what it comes down to is that if an A is 100 inches long, an Ab is 5.9% longer, 105.9 inches long. If you make this step 12 times... you should come back to an A that is one octave lower and find it to be 200 inches long. So, the difference between each step is 2(1/12), the twelfth root of two (1.05946309...).

Sorry, that last paragraph was a little number heavy, but here's the catch. although pianos are tuned to this fixed interval, the natural order of pitches is based in simple ratios (not 2(1/12)) from the root. The intervals are:
C D E F G A B C
1 9/8 5/4 4/3 3/2 5/3 15/8 2
You may note I did not include the full 12 note chromatic... this is because the 12 not chromatic CANNOT be tuned perfectly. an Ab will be tuned from a C by the relative difference of a major third (5/4)... This is going to be really freaking close... but slightly off. A major 3rd is four steps up the chromatic scale, you'll need three of that interval for an octave... therefore (5/4) * (5/4) *(5/4) should equal 2? Right... You can see it's actually 125/64... or just slightly more than 2. So if you start with a C, then tune an Ab from the C, an E from the Ab, then a lower C from the E... the ratio between Cs won't be two, as it should be... it'll be 125/64 = 1.95.

If you look at the chart above with an analytical eye, you'll find that none of the simple roots work perfectly. But with equal temperament, they all come out to 2 when expected. The downside to equal temperament is that the shape of the sounds waves is such that they interfere with each other. Using "just tuning" music will sound great in some keys (because the physical waves do not interfere and create beats)... but this can cause major problems in other keys.

Here's the skinny. Our modern tuning is extremely versatile, but doesn't exactly fit the nature of the sound waves being emitted. Other systems (which exist) can produce a purer sound, at the expense of versatility. I still think this is fascinating, but it's late. If you still care, start here, where you can hear a lot of what I'm talking about.

3 Comments:

OpenID knitthehellout said...

Explosions. In. My. Mind. I guess you got bored enough to blog, huh?

6:28 AM  
Blogger Brady said...

guess so

6:59 PM  
Blogger charlie said...

You might read "This is your brain on music." It raises a few questions that it fails to answer adequately despite having those questions advertised on the blurbs on the back, but it was heavy on biology once you get past the music theory chapter.

10:04 PM  

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