**UPDATE**

I have been interested in ranking the intervals of just intonation in terms of harmony ever since I began experimenting with different tuning systems and producing temperaments.

I have noticed a pattern that has finally enabled me to do this with conviction.

The following three intervals are usually ranked in the following order from most harmonious to least harmonious:

- Octave (Ratio: 2/1).
- Perfect Fifth (Ratio: 3/2).
- Perfect Fourth (Ratio: 4/3).

These intervals can be ranked in the same order in terms of the lowest common multiple (LCM) of the numbers that form their ratio from lowest to highest:

- Octave (LCM of 2 and 1 is 2).
- Perfect Fifth (LCM of 3 and 2 is 6).
- Perfect Fourth (LCM of 4 and 3 is 12).

Therefore, it seems as if the lower the LCM, the more harmonious the interval.

I suppose this is because the harmonics of the two frequencies that form an interval coincide more frequently as the LCM decreases.

**This relationship only applies to intervals within the Octave**.

For example, if we call 300 Hz the first frequency, the second frequency would be:

- 600 Hz for the Octave.
- 450 Hz for the Perfect Fifth.
- 400 Hz for the Perfect Fourth.

This means that the harmonics of the first frequency and the second frequency would coincide every:

- 600 Hz for the Octave (LCM of 300 Hz and 600 Hz is 600 Hz).
- 900 Hz for the Perfect Fifth (LCM of 300 Hz and 450 Hz is 900 Hz).
- 1200 Hz for the Perfect Fourth (LCM of 300 Hz and 400 Hz is 1200 Hz).

This information indicates that the Octave's harmonics coincide more frequently than those of the Perfect Fifth and the Perfect Fifth's harmonics coincide more frequently than those of the Perfect Fourth.

I will use this pattern to rank all 12 intervals from most harmonious to least harmonious.

**Frequency Ratios**

Semitones |
Note |
Interval |
Ratio |

0 |
C |
Unison |
1:1 |

1 |
C# |
Minor Second |
16:15 |

2 |
D |
Major Second |
9:8 |

3 |
D# |
Minor Third |
6:5 |

4 |
E |
Major Third |
5:4 |

5 |
F |
Perfect Fourth |
4:3 |

6 |
F# |
Tritone |
25:18 |

7 |
G |
Perfect Fifth |
3:2 |

8 |
G# |
Minor Sixth |
8:5 |

9 |
A |
Major Sixth |
5:3 |

10 |
A# |
Minor Seventh |
9:5 |

11 |
B |
Major Seventh |
15:8 |

12 |
C |
Octave |
2:1 |

**Source: https://fundamentals-of-piano-practice.readthedocs.io/en/latest/chapter2/CH2.2.html**

**12 Intervals Ranked by Lowest Common Multiple from Most Harmonious to Least Harmonious**

**Rank** |
**Interval** |
**Ratio** |
**Number 1** |
**Number 2** |
**LCM** |

**1** |
**Octave** |
**2/1** |
2 |
1 |
2 |

**2** |
**Perfect Fifth** |
**3/2** |
3 |
2 |
6 |

**3** |
**Perfect Fourth** |
**4/3** |
4 |
3 |
12 |

**4** |
**Major Sixth** |
**5/3** |
5 |
3 |
15 |

**5** |
**Major Third** |
**5/4** |
5 |
4 |
20 |

**6** |
**Minor Third** |
**6/5** |
6 |
5 |
30 |

**7** |
**Minor Sixth** |
**8/5** |
8 |
5 |
40 |

**8** |
**Minor Seventh** |
**9/5** |
9 |
5 |
45 |

**9** |
**Major Second** |
**9/8** |
9 |
8 |
72 |

**10** |
**Major Seventh** |
**15/8** |
15 |
8 |
120 |

**11** |
**Minor Second** |
**16/15** |
16 |
15 |
240 |

**12** |
**Tritone** |
**25/18** |
25 |
18 |
450 |

------------------------------

Roshan Kakiya

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