Theory Thursday: Sixths

Intervals: The Sixth

For the last several weeks we've been learning about a variety of intervals, starting with seconds up through fifths last week. Today, we're moving right along to the next interval, the sixth.

So from last week, we learned that fifths are the intervals created by skipping over 3 tones in the scale. This week, as you've probably guessed, we're going to talk about the interval relationship that happens if you skip 4 scale tones. Using the A minor scale, here's a look at the sixth (6th) intervals:

A -> F
B -> G
C -> A
D -> B
E -> C
F -> D
G -> E

You got this, right? Sixths are really just the next increment in the pattern we've been exploring for the last several weeks, and again the only thing new here is that the interval is getting bigger. So to build a sixth, if we start on A, we simply leap-frog over the B, C, D, and E to get to F. As you've guessed, for the rest of the tones in the scale, it's the same pattern of skipping over 4 tones to get to the sixth.

Counting the Half-Steps

Let's now take another look back at our pattern of whole and half steps that make up our A minor scale (I included the scale on top and the interval between each tone below...):

A B C D E F G A B C D E F G A ... etc
 W H W W H W W W H W W H W W  ... etc

Again, remember, every W is 2 half-steps.

So let's now take a look at the raw intervals that make up each 6th interval in the A minor scale. We do this by adding all the intervals between our two tones. So for example, A to F is W + H + W + W + H (using the pattern above), the B to G is H + W + W + H + W, and so on. So all together, this looks like:

A -> F : W+H+W+W+H = 8 half-steps
B -> G : H+W+W+H+W = 8 half-steps
C -> A : W+W+H+W+W = 9 half-steps
D -> B : W+H+W+W+W = 9 half-steps
E -> C : H+W+W+W+H = 8 half-steps
F -> D : W+W+W+H+W = 9 half-steps
G -> E : W+W+H+W+W = 9 half-steps

As you can see from the diagram above, most of the sixths are made up of 9 half-steps, and a few of them are 8 half-steps. Going back to our 2nds and 3rds, a larger 6th made up of 9 half steps is a major 6th (M6), while a smaller 6th made up of just 8 half steps is a minor 6th (m6).

Inversions, Revisited

Recall from last week, in addition to learning about fifths, we also learned the concept of interval inversions. We learned that 4ths and 5ths are inversions of each other, and we also learned that if we add the number of half steps from a 4th and its 5th inversion, the sum is always 12.

Just as the 4ths and 5ths have an inversion relationship, 6ths and 3rds share that same type of relationship. For example, A->C is a 3rd (a m3 to be precise), while C->A is a 6th (a M6 to be precise). So unlike 4ths & 5ths, where the inversions are both "perfect" (as in P4 inverts to P5), with 3rds and 6ths, the inversion takes on the opposite quality, i.e., major inverts to minor, and minor inverts to major. Here's an illustration:

6th         3rd
=====================
A->F = m6 : F->A = M3
B->G = m6 : G->B = M3
C->A = M6 : A->C = m3
D->B = M6 : B->D = m3
E->C = m6 : C->E = M3
F->D = M6 : D->F = m3
G->E = M6 : E->G = m3 

Just as the sum of the half steps of a 4th and its 5th inversion equals 12, the same is true for 3rds and 6ths. From the chart above, a M6 is 9 half steps, and its inversion would be a m3 which from a few lessons back is 3 half steps, so 9 + 3 = 12. And also a m6 (8 half steps) and its inversion M3 (4 half steps) is the same: 8 + 4 = 12.

Summary

So the important things to remember from today's lesson are:

• A sixth is the interval that results from skipping 4 scale tones
• A major 6th (M6) is 9 half-steps
• A minor 6th (m6) is 8 half-steps
• The inversion of a M6 is a m3
• The inversion of a m6 is a M3
• The sum of the half steps between an interval and its inversion is always 12

Review

• How many half-steps in a minor 2nd interval?
• How many half-steps in a major 2nd interval?
• How many half-steps in a minor 3rd interval?
• How many half-steps in a major 3rd interval?
• How many half-steps in a perfect 4th interval?
• How many half-steps in an augmented 4th interval?
• How many half-steps in an diminished 5th interval?
• How many half-steps in a perfect 5th interval?
• How many half-steps in a minor 6th interval?
• How many half-steps in a major 6th interval?
• What is the inversion of a P4 interval?
• What is the inversion of a m3 interval?
• What is the inversion of a d5 interval?
• What is the inversion of a m6 interval?

Reminder

If you have ANY questions or comments, please leave a comment here, on my Facebook page, or tweet at me (@chrisstarkgtr).