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.

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.

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.

Whew!

:)

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