More tunes

A new tune, an old favorite, and a medley. And a nerd rant.

I had four wisdom teeth taken out yesterday. Came back to the piano today and played a bunch of stuff.

Here’s a new tune that I’ve been messing around with for a week or so. I actually made it up during a piano gig. I recorded it using the keyboard’s record/playback feature, and then I played it back and improvised fiddle on top of it later in the gig. It ended up being a lot of fun.

Here’s a new rendition of some of My Favorite Things.

Here’s another improv medley. This one goes through Someday, Floating, Daisies in Paris, and Making Memories.

And now for something completely different—this is my rant about equal temperament.

Tuning a piano is complicated. That, in itself, is unsurprising — a piano is a complicated instrument. But the fun part is that the reason it’s so complicated is that music as we know it is inherently messed up and unfixable.

When you pluck a string and it makes a note, it actually produces more than one pitch. The extra pitches that are produced, which are higher and quieter, are called overtones.

There’s a lot of overtones (actually infinitely many, but only the first few are important), so you end up with most of the notes in any one key by filling in the first, say, ten overtones of one note.

And the fundamental reason that two distinct pitches sound good together is that one of them is the same pitch as an overtone of the other one.

This is all great, but the issue is that the overtones of the different notes don’t quite all line up with each other. For example, traveling up a perfect octave interval seven times results in a pitch about 1% lower than what you would get by traveling a perfect fifth interval twelve times—even though they would be the same key on a piano keyboard.

But to explain that, we have to examine where overtones come from.

If you take a string and pull it tight and then pluck it, you get a vibrating wave going up and down, positive and negative. (Think like you’re looking at a jump-rope in action.) In physics we call that a standing wave because the string is anchored on both ends (and so it can’t move up or down at the ends).

If the entire wave goes up and then the entire wave goes down, then that’s the main pitch of the string. I’m not sure how to explain the next bit—it’d be a lot easier to explain visually—but you can also split it into two equal waves that individually go up and down but twice as fast. That’s an overtone. You can also split it into three equal individual waves, three times as fast. That’s the next overtone. And so on. And those whole number splits are the only divisions of the string that will still preserve the stationary ends. So you get very specific extra pitches, and we call them overtones.

If you want to go up an octave, you take the overtone that doubles the frequency (the fastness). So you do *2. If you want to go up a fifth, you take the overtone that triples the frequency. So you do *3. The issue is that going up octaves (doing *2*2*2*2 etc.) will never ever intersect with going up fifths (*3*3*3*3 etc.) because 2 and 3 are prime numbers.

The reason for that: There’s a rule in math that you can take any whole number and express it as just one combination of primes (like, 12 is 2*2*3). So if a number is made of all 3s, then you can never have a 2 in there. And that’s also true for any number made up of all 5s, or all 7s, or all 11s, or any other prime.

So: overtones are all whole number multiples of the main frequency. And overtones OF the overtone pitches result in slightly different ending pitches depending on whether you take the 2^n path or the 3^n path (or the 5^n path, or the 7^n path, or etc.). Thus music is very slightly, consistently, irrevocably broken.

And because of this, tuning any instrument very accurately (especially a piano since they have hundreds of individual strings) is really an exercise in ultra tasteful compromise. You have to tune most everything slightly off from what sounds perfect, in order for nothing to be off enough to actually sound bad. In the case of a piano, people usually tune the octaves perfectly and then compromise for every other interval.

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