## How Math Fixed Music

### Rational Exponents Sound GREAT

Before we dive in, music is primarily defined by what we hear, not by the analysis and insight provided by math. For example, an octave is a note whose frequency is double that of its *parent* note. The mathematical relationship was discovered after the fact. The following is an exploration of how math is used in music, but I don’t want to put the cart before the horse here. The math supports the music, makes it work. But the math is really fine-tuning what we hear.

Pythagoras developed a musical system that over the years evolved into what we have today. (At least Pythagoras is often credited for it.) Not until “recently,” however, has one of the major problems with music been resolved (see what I did there with resolve?).

The problem the ancients had is that their octaves didn’t line up. An octave, as I mentioned early, is a note that has twice (or half) the frequency of another note. Octaves, in modern western music, share the same names, too. The note A, at 440 Hz, has an octave at 880 Hz, and also 220 Hz. (There are infinitely many octaves, in theory, though our ears have a limited range of things we can hear.) The ancients, however, had a problem because after a few octaves, well, they were no longer octaves.

In western music we have 12 semi-tones, A, A# (or B-flat), B, C, C#, D, D#, E, F, F#, G, G# and then A again. It’s cyclic, repeated infinitely both higher and lower. Each semi-tone in the next series of 12 notes is an octave of our first series of notes. And the relationship between notes is what makes them, well, musical, not just sounds.

The problem is defining that relationship. You see, because each note is slightly higher (has a higher frequency), and each note’s octave is double that frequency, what happens is the notes get further and further apart (the differences in their frequencies increases).

Let’s take a look at the frequencies:

Note | Frequency |

A | 220.00 |

A^{#} |
233.08 |

B | 246.94 |

C | 261.63 |

C^{#} |
277.18 |

D | 293.66 |

D^{#} |
311.13 |

E | 329.63 |

F | 349.23 |

F^{#} |
369.99 |

G | 392.00 |

G^{#} |
415.30 |

A | 440.00 |

A^{#} |
466.16 |

B | 493.88 |

C | 523.25 |

C^{#} |
554.37 |

D | 587.33 |

D^{#} |
622.25 |

E | 659.25 |

F | 698.46 |

F^{#} |
739.99 |

G | 783.99 |

G^{#} |
830.61 |

A | 880.00 |

As you can see, the differences between consecutive notes is increasing, at an increasing rate! This is not a linear relationship. Because of this, the ancients had a very hard time defining what was an A and what was a D, especially when you started moving around between octaves. Things got jumbled, and out of tune.

It is tricky to find the proportion and rate of change between consecutive notes, any two consecutive notes that is. That’s where math comes in to save the day. Let’s build the rate of change, shall we.

First, note that the rate is increasing, at an increasing rate, so we cannot add. I show that in the video below. We have to multiply. When we repeatedly multiply, we can use exponents. Since we need a note and it’s octave to be doubles, our base number is 2.

Since there are twelve notes between a note and its octave, we need to break the multiple of two into twelve equal, multiplicative parts. That’s a rational exponent, 1/12.

The number we need to multiply each note by is 2^{1/12}. Each note is one-twelfth of the way to the octave. It is pretty cool indeed.

For a more in-exploration, visit this page.

For a great read on this topic, consider the book Harmonograph.