When you are playing music, have you ever wondered why there are 12 notes in an octave? I heard that Pythagoras discovered it, but I think some people can discover it by themselves if they understand arithmetic and a little physics.

I’ll explain it here with a few hints, so please give it some thought. If you think “Oh! I see, 12 is the best!”, then you understand completely.

## Start Question

First of all, as a prerequisite knowledge, pitch is determined by the frequency of sound waves. Sounds that have a simple integer ratio of these frequencies will harmonize with each other. Then, we start with the following two theorems.

- When the ratio of frequencies is 1:2, they harmonize very well.
- When the frequency ratio is 1:3, it is quite harmonious.

Only these two conditions will lead to the 12-tone scale. Do you understand? If you can figure it out with just this, I think you have the same IQ as Pythagoras.

## Hint 1

Let’s start with the first hint: If we apply the above theorem to the current 12-tone scale of music, we get the following characteristics.

- When the frequency is doubled, the pitch goes up an octave.
- When the frequency is tripled, it goes up an octave and five degrees.

I describe them in the following figure.

A note with twice the frequency of C is the octave higher C. And a note with 3 times the frequency of C is the G note, which is an octave and five degrees higher. You can see that C and the higher C are harmonized. Since G is the dominant note to C, it is the next most harmonized note. Please try to figure out if you can derive the twelve-tone scale from this clue alone.

## Hint 2

As for the next clue, we will focus on the note of G (5th degree) at 3 times the frequency. If you halve the frequency of the G, you get the octave lower G that is 1.5 times frequency to the standard C. In other words, for every 1.5 times frequency increase, the note goes up by 5 degrees.

If we multiply the frequency of G by 1.5, we get D, which is 5 degrees higher than G. If you multiply the frequency of D by another 1.5, you get A, which is 5 degrees higher than D. (If you have trouble understanding 5 degrees, think of it as 7 half-steps.) Let’s see if we can derive a 12-tone scale with this clue. If you think this is similar to dominant motion, you are almost at the goal.

## Hint 3

From the clue above, let’s think about what happens if we keep multiplying by 1.5. First, if we multiply by 1.5 twice, we get 2.25 times. Now, let’s write the value of the power of 1.5 in the table below. Round off the third decimal place, and write it in two decimal places.

Power of 1.5 | Value |

1 | 1.5 |

2 | 2.25 |

3 | 3.38 |

4 | 5.06 |

5 | 7.59 |

6 | 11.39 |

7 | 17.09 |

8 | 25.63 |

9 | 38.44 |

10 | 57.67 |

11 | 86.50 |

12 | 129.75 |

13 | 194.62 |

14 | 291.93 |

I wrote the value of 12th power of 1.5 in bold. Doesn’t something come into focus? The hint is: “People don’t notice if the frequency is off by a little bit”. Let’s think of an integer that is close.

## Hint 4

The above clue shows that 1.5 to the 12th power is 129.75. If you have the power of 2 in your mind, you may be thinking, “If only it were 128, oh no! Don’t you think? If it were 128, 128 would be 2 to the 7th power, so multiply by 1.5 12 times and go up to the same note 7 octaves higher. 129.75 is about 1.01 times 128, a difference of only 1%. Let’s use our insensitivity and make it the same.

From this diagram, do you somewhat understand that there are 12 different scales? We’re almost at the finish line.

## Explanation

If you raise the note by 1.5 times, the first time you return to the same note is when you have raised it 12 times. In other words, you have to go through 12 different notes, including the first reference note, to get back to the original note. These passed tones can be reverted to the first octave by reducing the frequency by half, quarter, etc., and they all become different tones. So Pythagoras thought of dividing the sounds in the octave into 12 types.

But there is a 1% margin of error. This error is evenly distributed among the 12 types of notes, which is called “equal temperament”, and is used in music today. The idea of tuning only the most frequently used notes (C, E, G, etc.) exactly, and adjusting for errors with other notes, is called “just intonation”. However, if you transpose, the notes you use frequently will also change, so “equal temperament” is widely used because it is easier to use.

Using this “equal temperament”, we divide the octave into twelve equal parts with the same ratio, so the ratio of frequency to the neighboring note is the twelfth power root of two.

The component notes of the C major chord are C, E, and G. The frequency ratio of these component notes is 3:4:5, which is a nice integer ratio. This is the frequency ratio of a major chord, and the reason why it is so comfortably harmonious.

I will also write in the table the values that were multiplied by the twelfth power root of 2.

Power of the 12th power root of 2 | Multiple Value | Sound Intervals (with respect to C) |

1 | 1.0594 | Minor 2nd |

2 | 1.1224 | Major 2nd |

3 | 1.1892 | Minor 3rd |

4 | 1.2599 | Major 3rd（E）approx. 1.25 times |

5 | 1.3348 | Perfect 4th（F）approx. 4/3 times |

6 | 1.4142 | Diminished 5th |

7 | 1.4983 | Perfect 5th（G）approx. 1.5 times |

8 | 1.5874 | Minor 6th |

9 | 1.6817 | Major 6th |

10 | 1.7817 | Minor 7th |

11 | 1.8877 | Major 7th |

12 | 2.0000 | Perfect 8th（C）2 times |

The notes in red are those that are particularly harmonic to the base note. If the base note is C, then E is approximately 1.25 times the base note, and G is approximately 1.5 times the base note, resulting in a C chord (composed of C, E and G) with a frequency ratio of 4:5:6. If we lower the octave of the component note of this chord, so, and consider the component note G-C-E, we also get a beautiful frequency ratio of 3:4:5, since so becomes 6⇨3.

The numbers 3, 4, and 5 are often used in the Pythagorean theorem, which is the condition for right triangles. This 12-note scale was also created by Pythagoras. Is it a coincidence or is there a connection?

The above is an explanation of why the 12-tone scale is used. The story was a mixture of physics, mathematics and music. Once again, I think music is beautiful.