Imperfect intervals

Imperfect intervals, on the other hand, are kind of dualistic in their nature. They have a minor version and a major version, just like the Big Dipper and the Small Dipper.

That doesn’t mean they don’t have a default state though. They’re all by default major, except for the 7th, but, we’ll touch on that later.

The absolute queen of imperfect intervals is the third. As you probably know, it has a major and a minor version.

It’s the queen of imperfect intervals because it’s the one that has the most potency, since it’s the main lever that controls emotional shading in the tonal system. A chord starting out with a major 3rd will be considered major, positive and happy, and a chord starting out with a minor 3rd will be considered minor, negative and sad by association.

The interval spanning 6 note names is the 6th. This one also has 2 versions, the major 6th and the minor 6th. Personally, I calculate 6ths relative to the 5th. So C-G is your 5th, if I’m one half-step above the fifth, this is a minor 6th, if I am 2 half steps, or one full step, above the fifth, this is a major 6th.

The 6th and the 3rd are actually inversions of each other too. It’s good to know when you’re trying to harmonize a melody, which is often achieved by adding a line a 3rd above. If it’s too high for some reason, or if you’re falling out of the range of the instrument, you can always switch the 3rd above to a 6th below.

Now the tricky thing with these imperfect intervals, is that whenever you invert one of them, they also swap their major/minor quality. So inverting a major 3rd is going to give you a minor 6th, and inverting a minor 3rd will get you a major 6th. So we get this dual rainbow with colors cyan and green.

Sevenths and seconds

Harmonically, after the 3rd, the next most important interval is the 7th. And this one is the most confusing since it has a default minor state. So if a chord sheet asks for a C major chord with a 7th, and doesn’t specify anything else, it’s expecting you to add a minor 7th to the chord.

The best way to calculate 7ths is to overshoot the target a little bit by getting the octave, and then moving back down 2 half steps for the minor 7th. If the chord sheet specifies to add a Major 7th to a chord, well then it’s expecting you to add the note that is just a half step below the octave. So those are your 2 sevenths, the minor one, and the major one.

The inversion of the seventh is the second. The second is basically the interval separating 2 consecutive notes. It completely overlaps with the steps and half steps. A major second, the big one, is a step. A minor second, the small one, is a half step. And just like 3rds and 6ths, when you take a minor 7th and you invert it, it gives you a Major 2nd. When you take a Major 7th and invert it, you’re gonna get a minor 2nd.

This is the full set of imperfect intervals. 2 pairs of intervals, that are perfect mirrors of each other, and that flip polarity as they get inverted. The only thing to keep in mind is that the 7th, like in a C7 or a G7 chord, is by default minor, unless Maj7 is mentionned.

Here’s one last representation, that might help make things click. All of those intervals appear when we use the harmonic series to divide the octave over and over again. This is almost a a fractal pattern because we’re applying the same operation at every scale, from the octave down to the half step.