# Just intonation redux

This just intonation series has been somewhat tortured so far. I feel like I have a good grasp on this stuff, but clearly explaining it eludes me. I want people who only have a little music theory experience to be able to grasp what makes it so cool and different from the conventional system. But I’m getting bogged down trying to cover all the necessary background knowledge, and I’m struggling to synthesize the details well.

Today, I’m going to take a different approach. I’ll start by defining some terms up front, and then I’ll try to restate some of what I’ve covered in my own language, without too much detail work. This will involve some restating, but hopefully it sets up a better foundation for talking about this stuff. We’ll see how that goes; if I feel it’s necessary to stop and explain something, I will.

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*Elucidanda*

#### Intervals are numbers

As a ground rule, I want to emphasize how intervals can literally be thought of as numbers in just intonation. Specifically, intervals are rational numbers that multiply together. If you multiply two rational numbers, you get another rational number. If you multiply two intervals, that’s the same as “stacking” them to get another interval in conventional music theory. In order to get real pitches from these intervals, you need to multiply them by a base frequency, usually a number given in Hertz.

To emphasize the multiplicative nature of intervals, I’m going to refer to them using numbers preceeded by a ⟨×⟩. Keep in mind that these are interchangeable with their equivalent musical intervals, e.g., “×2” means the same thing as “octave” and “×3/2” means “perfect fifth”. If I am doing math with them, I will take the ⟨×⟩s out to avoid confusion.

#### “Large” intervals

Last time I found it difficult to talk about the intervals being introduced by different multiples of a base frequency. I want to talk about things like how ×3 introduces the “quality” of a perfect fifth, which you can’t get just by multiplying by twos.

But ×3 is actually an octave *plus* a perfect fifth (a perfect fifth is ×3/2, i.e., ×3 minus an octave).
I just think of ×3 as basically the same as a perfect fifth due to octave equivalence.
When looking at just intonation, ×2’s (octaves) are negligible when considering the *quality* of an interval (though necessary for understanding the absolute distance of the interval).
Any intervals that are separated only by octaves can be said to be “congruent”, which is just another way to say “the same under octave equivalence”.

For convenience, I will refer to a ×3 (an octave plus a fifth) as a “large fifth”. You can think of a large fifth (or large third ×5, large seventh ×7, &c.) as “basically a fifth (third, seventh), with some extra octaves attached”. The important thing about these intervals is that they each have a unique harmonic quality that the others don’t.

### Once again, the harmonic series

So the harmonic series is just the series of intervals ×1, ×2, ×3… and so on. Last time, we observed that the primes in this series introduced new interval qualities, whereas composite numbers consisted of combinations of previous intervals. I’ll summarize the first eleven steps of the series in a table using our new nomenclature.

Interval | Prime factors | Constituent musical intervals |
---|---|---|

×1 | ∅ | unison |

×2 | 2 | octave |

×3 | 3 | large fifth |

×4 | 2×2 | octave + octave |

×5 | 5 | large third |

×6 | 2×3 | octave + large fifth |

×7 | 7 | large seventh |

×8 | 2×2×2 | octave + octave + octave |

×9 | 3×3 | large fifth + large fifth (≡ M2) |

×10 | 2×5 | octave + large third |

×11 | 11 | large tritone |

#### Notes

- I mentioned this last week, but notice the equivalence between the prime factor column and the musical intervals column. Notice any time an interval has a prime factor of two, a corresponding octave shows up in its constituent musical intervals. The same applies for large fifths, large thirds, &c.
- Note that ×9 is congruent to a major second (a major second is ×9/8). This is the only composite interval we’ve seen made of two non-octaves (3×3, all the other composites have a 2 in their prime factor list).
- For ×11, “Large tritone” is kind of a weird name, because etymologically
*tritone*implies the stacking of three “tones” (i.e. major seconds), which is not really a good way to describe ×11. There are systems of tuning in which you could consider ×11 to be a consonance, whereas the tritone is conventionally considered a dissonance. Really, at this point you are reaching the limits of how well we can map “traditional interval names” onto our numerically unlimited intervals.

### 5-limit tuning

With the right multiplications and divisions, we don’t have to use anything past ×5 to construct all the intervals we need to navigate conventional twelve-tone music.* I will just share the table for the intervals of a major scale for now, without further explanation, but next time, we’ll look in more depth at constructing many useful intervals.

Interval | Prime factors | Musical interval name | Solfège |
---|---|---|---|

×1 | ∅ | unison | do |

×9/8 | 3×3 / 2×2×2 | major second | re |

×5/4 | 5 / 2×2 | major third | mi |

×4/3 | 2×2 / 3 | perfect fourth | fa |

×3/2 | 3 / 2 | perfect fifth | so |

×5/3 | 5 / 3 | major sixth | la |

×15/16 | 3×5 / 2×2×2×2 | major seventh | ti |

×2 | 2 | octave | do' |

Have you heard any just intonation music?
Are you interested in writing it?
Does the new approach I’m trying work for you?
Let me know your thoughts at my Ctrl-C email: `gome @ ctrl-c.club`

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#### Next post in series

### Just intonation: 5-limit tuning

How do you build intervals out of primes?

* You can also get all the intervals just with combinations of ×2 and ×3, but I think it’s kind of boring compared to 5-limit. Still worth reading about for further learning.