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Intro to Music Theory

2015 December 20. Sunday.

What is sound?

If you strike a spoon hanging from a fishing line, the spoon will vibrate. Some of this energy goes into moving the air around the spoon, and that is how you get a sound.

Frequency and Simple Ratios

The frequency of a sound is how many times it vibrates the air in one second. We call 100 vibrations per second 100 Hz. It only makes sense to consider frequency for simple sounds; you can say that a guitar string sounds high-pitched or flat, but those statements don’t make sense for a tamborine.

Two tones sound good together only if the ratio of their frequencies is a simple fraction. So you could play 150 Hz with 100 Hz because 150/100 = 3/2, and you could play 120 Hz because 120/100 = 5/4, but 173 Hz and 100 Hz makes bad.

A second fact to remember: A bigger thing makes a lower sound. If striking a 1 meter pipe produces 100 Hz, then striking a 4 meter pipe produces 25 Hz.

We can use these two facts to build an instrument: hang a 10 meter pipe, a 9 meter pipe, an 8 meter pipe, … down to a 1 meter pipe. Since any two pipe lengths are a simple ratio, they all sound good together.

We can play songs on this instrument. Striking the fourth pipe and the seventh pipe simultaneously makes a nice chord.

The problem is that this instrument can only be played in one style. It always sounds the same and there is only one way to play a given song. Furthermore, if someone else made their own, but starting with 1 yard then you two couldn’t play together. We’d like an instrument where we can shift our song or style to change the mood or go along with other people’s instruments.

How a Piano Works

In a random freak accident, the powers of the twelfth root of two make several simple ratios:

n 2^(n/12) close fraction
0 1 1
1 1.059 N/A
2 1.122 N/A
3 1.189 6/5
4 1.260 5/4
5 1.335 4/3
6 1.414 7/5 (kinda)
7 1.498 3/2
8 1.587 8/5 (kinda)
9 1.682 N/A
10 1.782 N/A
11 1.888 N/A
12 2 2

To be clear, the 2 and 12 combo yields more simple fractions than any other combo. To see what I mean, look at the 10th root of 3:

n 3^(n/10) close fraction
0 1 1
1 1.116 10/9
2 1.246 5/4
3 1.390 7/5
4 1.552 N/A
5 1.732 N/A
6 1.933 N/A
7 2.158 N/A
8 2.408 12/5
9 2.688 N/A
10 3 3

Look at those fractions, they’re garbage!

So now we build a second instrument. Again the first pipe is 10 meters long. The second pipe is 10/1.116 = 9.443 meters long, down to the last pipe which is 10/2 = 5 meters long. The first pipe (n=0) can be played with the pipes which are bolded in the above list to get good sound like our previous instrument.

Here’s the magic though: you can start on the the second pipe and take the same spacings as before to get all your nice ratios, but in a different scale. So instead of playing the first (n=0) and the fourth (n=3) to get a frequency ratio of 6/5, you’d play the second (n=1) and the fifth (n=4).

If we make our pipe instrument 100 pipes long, then we can play in many different scales and styles by changing our root pipe.

This is how a piano works. Every key (black or white) has a string whose length is 0.9443 times the width of the last string. The C key on a piano has its simple ratios put on the other white keys. That’s the C major scale. To play the A major scale, start on the A key and then take the same distances as before.



Public Domain Dedication.