Just intonation: 5-limit tuning
Writing about just intonation is so hard, and I’m pretty sure most of my audience doesn’t care about it, and yet I must keep doing it! In the last just intonation post, I shared a table of intervals for the major scale in just intonation. Today we’ll look at the just intonation equivalents of all twelve tones in a similar fashion. Before the table, here’s some groundwork.
Interval Quality
It’s helpful to talk about two intervals that are the same under octave equivalence as the same interval. I’ll use the term quality to describe the property of an interval that doesn’t change depending on the specific octave it’s in.
In 12-tone equal temperament, you can find an interval’s quality by taking its size in semitones modulo 12. For example, a major tenth, (16 semitones), has the same quality as a major third, (4 semitones), because 16 mod 12 and 4 mod 12 both equal 4. In just intonation, you can take the number representing the interval and remove all factors of two from its numerator and denominator (i.e., by multiplying by some number 2n, which is equivalent to transposition by n octaves).
The table below will include a column representing the quality of the interval derived in this way.
Multiple intervals for a single name
Traditional interval names such as “major second” or “augmented fourth” don’t always map cleanly to specific intervals. The name “major second” only maps to a single 12-tone interval, i.e., 2 semitones, but in just intonation, both ×9/8 and ×10/9 are major seconds, with the former just slightly sharper than the latter. Where I think it’s relevant, I’ll include both variants.
Constructing the intervals
If we didn’t have a table that already told us, how would we go about constructing the intervals we need? Our basic building blocks for this exercise are the primes up to 5: ×2 the octave, ×3 the large fifth, and ×5 the large third. We need to construct at least twelve interval qualities, to match the twelve tones of equal temperament. Since we are looking at qualities, we can keep ×2’s out of the picture and talk about everything in terms of qualities, treating octaves as equivalent. In the table, these qualities will be normalized into the range between ×1 and ×2, so, within a single octave.
So with just a major third and a perfect fifth, how would we construct the rest of our intervals? Well, let’s start by just combining them. A fifth plus a major third makes a major seventh. In equal temperament, you can get the interval by adding semitones: 7 + 4 = 11. In just intonation, the equivalent calculation, 3 × 5 = 15, yields our just major seventh, ×15. Now that we have a major seventh, we can invert it to get a minor second: ×1/15.
What if we subtract a major third from a fifth? That gives us a minor third quality, ×3/5, and inverting that gets us a minor sixth, ×5/3. How would we make a major second? Well, we can stack two fifths for ×9, or we can go down two fifths and up a major third, yielding ×5/9.
All the intervals in the table below can be constructed in the same manner. If you want to verify any of them for yourself, you can reference the prime factors column. Starting from middle C on a keyboard, move an octave for every 2 you see, move by a perfect fifth for every 3 you see, and move by a major third for every 5 you see. If the number is in the numerator, move up by the corresponding interval; if it’s in the denominator, move down. This will land you at the correct interval relative to middle C. The same process can be followed with the quality column, but it won’t land you in the correct octave, since interval quality ignores octaves.
| Musical interval name | 12-tone equivalent | Interval | Prime factors | Quality |
|---|---|---|---|---|
| unison | 0 | ×1 | ∅ | 1 / 1 |
| minor second | 1 | ×16/15 | 2×2×2×2 / 3×5 | 1 / 3×5 |
| major second | 2 | ×10/9 | 2×5 / 3×3 | 5 / 3×3 |
| major second | 2 | ×9/8 | 3×3 / 2×2×2 | 3×3 / 1 |
| minor third | 3 | ×6/5 | 2×3 / 5 | 3 / 5 |
| minor third | 3 | ×32/27 | 25 / 3×3×3 | 1 / 3×3×3 |
| major third | 4 | ×5/4 | 5 / 2×2 | 5 / 1 |
| perfect fourth | 5 | ×4/3 | 2×2 / 3 | 1 / 3 |
| augmented fourth | 6 | ×25/18 | 5×5 / 2×3×3 | 5×5 / 3×3 |
| augmented fourth | 6 | ×45/32 | 3×3×5 / 25 | 3×3×5 / 1 |
| diminished fifth | 6 | ×64/45 | 26 / 3×3×5 | 1 / 3×3×5 |
| diminished fifth | 6 | ×36/25 | 2×2×3×3 / 5×5 | 3×3 / 5×5 |
| perfect fifth | 7 | ×3/2 | 3 / 2 | 3 / 1 |
| minor sixth | 8 | ×8/5 | 2×2×2 / 5 | 1 / 5 |
| major sixth | 9 | ×5/3 | 5 / 3 | 5 / 3 |
| major sixth | 9 | ×27/16 | 3×3×3 / 2×2×2×2 | 3×3×3 / 1 |
| minor seventh | 10 | ×16/9 | 2×2×2×2 / 3×3 | 1 / 3×3 |
| minor seventh | 10 | ×9/5 | 3×3 / 5 | 3×3 / 5 |
| major seventh | 11 | ×15/8 | 3×5 / 2×2×2 | 3×5 / 1 |
| octave | 12 | ×2 | 2 / 1 | 2 / 1 |
I don’t really have time to write much else, but it’s useful to look at pairs of inverted intervals (those across the bold center line from each other) like ×4/3 and and ×3/2. Looking at their qualities, they are always inverses of each other, and looking at their prime factors, they’re always off by just a factor of ×2. Why might that be?
Are you learning anything from the just intonation series?
Have you listened to any just intonation music?
Did you notice a mistake in the big chart?
Let me know your thoughts at my Ctrl-C email: gome @ ctrl-c.club.