 Welcome to linear theory lesson 9. In today's lesson, we're going to continue to expand our understanding of intervals or the distance between two different notes. Up till now, we've worked hard to be able to identify the general name of intervals as seconds, thirds, fourths, fifths, sixths, sevenths, and octaves. And more specifically identify those intervals by counting half steps to be able to determine if they are perfect intervals in the case of fourths, fifths, and octaves, or major, minor, diminished, or augmented in the case of the rest of them. So today we're going to take that concept of the interval and we're going to learn about inverting an interval and what the rules are of inversion. So think about what an inversion is. If we invert something, we literally flip it over and that's what we're going to be doing with intervals. So let's take a look and go to work here. If we start with an interval and let's start with one that we've used a lot, the perfect fifth. And we're going to take the interval from G to D. And I'm just going to tell you that from G to D is a perfect fifth. You can see visually that it's a fifth and if we were to count those half steps, take my word for it, it would be a perfect fifth. Now to invert that interval, we take the bottom note and we move it up an octave so that now instead of the G being on the bottom and the D being on the top, now the D is on the bottom and the G is on top. And as you look at that interval from the new interval of D to G, you can see that quickly that it's not a fifth. In fact, that interval is a fourth. And I'm going to go so far as to tell you if we were to count those half steps that it would be in fact a perfect fourth. Let's go over here and take a look. Here we had G to D and it is one, two, three, four, five, six, seven half steps or a perfect fifth. Now we took the G and we moved it to the top and now from the D to the G, one, two, three, four, five half steps or a perfect fourth. Now there are a couple rules here that we're going to learn. First of all is this, that fifths always invert to fourths and fourths always invert to fifths all the time. So that's your first rule. The other rule is that perfect intervals always invert to other perfect intervals all the time. So we really can gain two things from the example that I've shown here. Let's look at a couple other examples and see if we can find some more rules. Let's go with a third and we'll go with a major third from G to B is a major third. Now let's invert that interval and go from B to G. First of all, what's the interval when we make that inversion? You can look at it very quickly and see that that's some kind of sixth. Let's go over to the keyboard and let's take a look at this. We've got G to B which is a major third or one, two, three, four half steps. Now we're going to invert it, B's on the bottom, G's on the top and it is one, two, three, four, five, six, seven, eight half steps. So we had a major third which inverted to eight half steps, some kind of a sixth. What kind of a sixth is eight half steps? A minor sixth. So let's go back over here. What did we learn? Well, and we can bank on this, that a third will always invert to a sixth. So a sixth always inverts to a third and vice versa and a major interval will always invert to a minor interval and vice versa. So anytime we have a major interval it inverts to minor, anytime we have minor it inverts to major. Now one more to do here. Let's look at an interval of a second. Let's start on E and look at the interval between E and F. Here's E, here's F, it's one half step or a minor second. Now we'll take that, we'll invert it, we'll put the E on top and now we have this distance from F to E. Well you can look at that and see already that it's some kind of a seventh. We put that E on top, it's some kind of a seventh and if we count that, one, two, three, four, five, six, seven, eight, nine, ten, eleven half steps. So that minor second inverts to a major seventh. Again, minors invert to majors so that holds true and we know that a seventh inverts to a second and vice versa. One last one to do because we want to take a look at an augmented interval or a diminished interval. I bet you can already guess what the rule is going to be. We take a look at, let's start with the note D and let's go from D to, in fact let's start with the D flat to E which is, let's go down here, from D flat to E is one, two, three half steps or an augmented second. Now we'll flip that interval over, we'll put the D flat up here. We can see visually that it's going to be a seventh and we're going from E to D flat. So now let's go to this D flat and here we go. One, two, three, four, five, six, seven, eight, nine half steps which is a, what do we have here? We have an augmented second and that's going to invert to a diminished seventh, okay? So there's our seventh second relationship and an augmented interval is going to invert to a diminished interval. These rules are hard and fast. Fifths always invert to fourths, fourths always invert to fifths. Six invert to thirds, thirds invert to sixths. Sevenths to seconds, seconds to sevenths. Miners invert to major, majors to minor. Perfects always invert to other perfects, hence the name perfect. And augmenteds invert to diminished and diminish, inverts to augmented all the time. Now let's do a little test here and see how well you've followed. What's the inversion of a minor third? Thirds invert to sixths, minors invert to majors. Minor thirds invert to major six all the time. What's the inversion of a diminished fourth? Fourths invert to fifths, diminished inverts to augmented. What's the inversion of a minor seventh? Sevenths invert to seconds, minor inverts to major, okay? What's the inversion of a perfect fourth? Always a perfect fifth. Get the idea? We could go on and on with this but it's always that way. We can also look at it this way on the staff. If we take from D to A and we invert it and then we put that lower note up on the top, this perfect fifth always inverts to a perfect fourth. If we take a G sharp to an A, that would be a minor second and then we put that lower note all the way on top, G sharp, trust me, that will always be a major seventh. And we can continue in that way across the board. It's now time for you to go ahead and take a look at the assignment for this week. Get to work on that and hopefully this will all make sense as you move through this assignment on inversions.