Piano for the Non-Musician

This post will be a bit more elementary, though it’ll be about something that I love to do, and that’s play the piano.

A lot of people don’t play instruments because they say they’re not “musical”. Well, I have one word for you. Okay, it’s a name really. Ludwig van Beethoven. Beethoven is arguably the most well-known pianist and composer that’s ever lived. Most people can pick out one of Beethoven’s compositions, if not by name then by ear. Most people also know, if you remind them, that Beethoven became deaf in his 30s.

Of course, he was already an accomplished musician by then. But I would argue that his success despite his deafness was not remarkable in the least. Beethoven could compose music without hearing it not because of his latent genius (though I would definitely attribute the complexity of his music to that genius, among other things), but because musical technique is inherently mathematical.

Let me give you a simple example. If you walk up to a regular old piano and push and hold a key at random, you can form what’s called a major triad in root position. Here’s how: Move up the keyboard (that is, to the right) exactly four keys, and hold the note you land on along with the first one. Now move up the keyboard exactly 3 keys, and hold the note with the other two. No matter what note you start on, this will form a major triad in root position.

By convention, a major triad in root position is denoted by the capital of the note it starts on. Also by convention, moving up or down a key is called moving up or down a half-step. Moving two keys is called moving a whole-step. Here’s a \mathbf{D^\flat} chord, i.e. a D-flat major triad in root position:

piano.png

There’s a more technical way to talk about how chords are formed, and I’ll stick to \mathbf{D^\flat} to illustrate.

Major chords are formed from two components: a major 3rd, and a minor 3rd. A major 3rd is composed of two notes with three half-steps (i.e. three keys) in between. A minor 3rd is composed of two notes with two half-steps (or a whole-step) in between. So in total, a major chord spans a total of 1 + 3 + 1 + 2 + 1 = 8 keys or 7 half-steps, with the 1’s designating where your fingers should be. (It’s 7 half-steps, and not 8, because from the first note in the chord to the last, you move up 7 times).

Minor chords in root position are denoted the same as above, except with an “m” at the end. \mathbf{D^\flat{m}} is D-flat minor in root position. Minor chords are formed from the opposite configuration as major chords, i.e. a minor third and then a major third. Here’s \mathbf{D^\flat{m}}:

piano21.png

If you look at both chords, you’ll see that \mathbf{D^\flat{m}} is just \mathbf{D^\flat} with the middle note lowered one half-step.

So as you can see, forming chords amounts to configuring how many half-steps are in between each key. If you look closely, you’ll also notice that by alternating major and minor thirds, we’re adding complexity to the chords. If we defined a chord as two major thirds in a row, we’d create overlaps between chords. Look at this image:

piano3.png

Here, we’re just adding major 3rds on. If you start at the first note in the chord, and keep adding major 3rds, you get the same configuration as if you started at the second note in the chord, and kept adding major 3rds. The only difference is the missing first key. This works for the 3rd note in the chord as well. In technical terms, these chords are enharmonic, which basically just means they have the same notes. So we can determine how many unique configurations there are thusly.

If we look at one segment of the keyboard with no repeating keys (i.e. the grey-highlighted part in this image, which we’ll call the suboctave for lack of a better term)…

piano5.png

…we see there are twelve unique notes. Since our “adding major 3rds on” method produces three enharmonic multiplicities, that means there are only \frac{12}{3} = 4 unique configurations. That’s not very many! On the other hand, if we alternate major 3rds and minor 3rds like this–M3, m3, M3, M3, m3, M3–we get the following:

piano4.png

Look at the first three blue keys. Now look at the next three blue keys. You can see that the first three keys are just a half-step up from the next three, except they’re a suboctave down. If this image were lengthened, you would see that the next three (i.e. in the next suboctave up) after that are a further half-step down.

Ah but it gets better. If you look at the 2nd, 3rd, and 4th blue notes, you’ll see we’ve formed a minor chord (if we don’t hold all the rest of the notes). If you look at the 4th, 5th, and 6th, you’ll see another major chord. But if you also look at the 3rd, 4th, and 5th blue notes, you’ll see M3-M3, which is what we formed before. So alternating M3 and m3 in the above manner creates alternating major, minor, and M3-M3 (or augmented major) triads. Repeating this, and assuming we had an infinitely long keyboard, we’d generate all 12 major triads, all 12 minor triads, and all 4 augmented major triads for a total of 27 triads.

But how do we know if a chord sounds “good” or not? If you’ve ever just plopped your hand down on a piano, you’ll notice it doesn’t sound that great. It’s got a lot of dissonance to it. How can we form “good” chords without hearing what they sound like? How can make sure different chords sound “good” together?

It’s difficult to describe what sounds good and why. Here’s a major triad, in MIDI format. And here’s a minor triad. The common feeling is that major chords sound “happy”, while minor chords sound “sad”. Not very objective, but hey.

I urge you to scoop up your old portable keyboard or find a piano and to just experiment some, after reading this post. You’ll find that being “musical” can actually be quite algorithmic.

Next post I’ll go into more about chords and then talk about rhythm.

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