Theory Thursday: Fifths

Intervals: The Fifth

In the last couple weeks we learned about 2nd, 3rd, and 4th intervals. For seconds and thirds there are major and minor varieties, and with fourths there are perfect and augmented varieties. Today, we're moving right along to the next interval, the fifth.

So from last week, we learned that fourths are the intervals created by skipping over two tones in the scale. This week, we're going to talk about the interval relationship that happens if you skip three scale tones. Using the A minor scale, here's a look at the 5th intervals:

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

Still with me? Fifths are really just the next increment in the pattern we've been exploring for the last few weeks, and again it's just that the interval is getting bigger. So to build a fifth, if we start on A, we simply leap-frog over the B, C, and D to get to E. Then of course it's the same for every subsequent tone in the A minor scale.

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 fifth interval in the A minor scale. We do this by adding all the intervals between our two tones. So for example, A to E is W + H + W + W (using the pattern above), the B to F is H + W + W + H, and so on. So this looks like:

A -> E : W+H+W+W = 7 half-steps
B -> F : H+W+W+H = 6 half-steps
C -> G : W+W+H+W = 7 half-steps
D -> A : W+H+W+W = 7 half-steps
E -> B : H+W+W+W = 7 half-steps
F -> C : W+W+W+H = 7 half-steps
G -> D : W+W+H+W = 7 half-steps

As you can see from the diagram above, most of the fifths are made up of 7 half-steps, and just one of them is 6 half-steps. Similar to 4ths, 5ths that consist of 7 half-steps are called "perfect" fifths (P5) and because the fifth made up of 6 half-steps is just a little smaller, it's called a "diminished" fifth (d5).

If you look back at last week's discussion of fourths, you'll see that an augmented fourth is made of the same 6 half-steps as a diminished fifth. Uh oh! What's worse is that to our ears, there is NO difference between an augmented 4th and a diminished 5th. They sound exactly the same.

So how can we tell them apart?

The answer is actually pretty simple. If you look at the A4 interval F -> B, there are 2 tones in between (G and A). When you skip 2 tones, you have a fourth. If you look at the d5 interval B -> F, there are 3 tones in between (C, D, and E). When you skip 3 tones, you have a fifth. Not so bad, right?

New Terminology

We just learned that even though the interval of a diminished fifth and an augmented fourth sound the same (because they have the same number of half steps), they have different names. There is a special term for these intervals that have different names but the same number of half-steps: enharmonic equivalents. It's kind of a yucky term to remember, but it just means intervals that have the same raw interval but different harmonic names. In a sentence, you'd say "Augmented fourths and diminished fifths are enharmonic equivalents."

In time, we will cover several other enharmonic equivalent intervals, but for now with the A minor scale, the A4/d5 pair is the only one we will encounter.

But wait! There's More!

If you take another look at our B->F and F->B example from above, notice that both of these  intervals use the same two tones, B and F. The only difference is which tone comes first. Intervals that share the same tones but just have a different order are called inversions.

If you start with a fourth interval (C->F for example), then swap the order of the tones (F->C) it becomes a fifth. Any time you invert a fourth it becomes a fifth. Likewise, any time you invert a fifth it becomes a fourth. More specifically, if you start with a perfect fourth, the inversion is a perfect fifth (and vice versa). Likewise, if you start with an augmented fourth, the inversion is a diminished fifth (and vice versa).

Going back to the raw intervals, if you add a perfect fourth (5 half steps) and its inversion, a perfect fifth (7 half steps), the result is 12. If you add an augmented fourth (6 half steps) and its inversion, a diminished fifth (6 half steps), the result is 12 again.

The sum of the half steps of an interval and its inversion is always 12. We'll see a bit more of that when we talk about 6ths next week and 7ths the following week.

Summary

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

• A fifth is the interval that results from skipping 3 scale tones
• A perfect 5th (P5) is 7 half-steps
• A diminished 5th (d5) is 6 half-steps
• Intervals that have the same number of half-steps but different names are called enharmonic equivalents
• When you flip the order of tones in an interval, it is called an inversion
• The sum of the half steps between an interval and its inversion is always 12
• Fourths and fifths are inversions of each other 

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?
• Define the term enharmonic equivalents.
• What interval is A->E?
• What would be its inversion?
• What interval is the inversion?

Reminder

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

Theory Thursday: Fourths

Intervals: The Fourth

In the last couple weeks we learned about 2nd and 3rd intervals, and their major and minor varieties. Today, we're moving right along to the next interval, the fourth.

So from last week, we learned that thirds are the intervals created by skipping over a tone in the scale. This week, we're going to talk about the interval relationship that happens if you skip two scale tones. Using the A minor scale, here's a look at the 4th intervals:

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

None of that seems too bad so far, right? It's really a lot like everything else we've covered so far, it's just that the interval is getting bigger. So to build a fourth, if we start on A, we simply leap-frog over the B and C to get to D, and then we do the same for every tone in the A minor scale.

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 etc
 W H W W H W W W H  etc

Again, remember, every W is 2 half-steps. So let's now take a look at the raw intervals that make up each fourth interval in the A minor scale. We do this by adding all the intervals between our two tones. So for example, A to D is W + H + W (using the pattern above), the B to E is H + W + W, C to F is W + W + H, etc. So this looks like:

A -> D : W+H+W = 5 half-steps
B -> E : H+W+W = 5 half-steps
C -> F : W+W+H = 5 half-steps
D -> G : W+H+W = 5 half-steps
E -> A : H+W+W = 5 half-steps
F -> B : W+W+W = 6 half-steps
G -> C : W+W+H = 5 half-steps

Remember how there are major and minor seconds and thirds? Well things change a little for fourths. As you can see from the diagram above, most of the fourths are made up of 5 half-steps, and just one of them is 6 half-steps. Fourths that consist of 5 half-steps are called "perfect" fourths (P4) and because the fourth made up of 6 half-steps is just a little bigger, it's called an "augmented" fourth (A4).

So the important things to remember about fourths are 1) a fourth is the interval that results from skipping 2 scale tones, and 2) a perfect 4th (P4) is 5 half-steps, and 3) an augmented 4th (A4) is 6 half-steps.

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?

Reminder

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

Theory Thursday: Thirds

Intervals: The Third

Last week we learned that adjacent tones in a scale have the interval relationship of a second. And we also learned that if the second is made up of a single half-step (or one fret space on a guitar), it is called a minor second. Likewise, a second made up of two half-steps (i.e., two frets on a guitar), it is called a major second. Easy peasy, right?

This week, we're going to talk about the interval relationship that happens if you skip a tone. As you've likely guessed by today's blog title, it's called a third. So let's take a look at what the thirds look like in the A minor scale:

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

None of that seems too bad so far, right? If we start on A, we simply leap-frog over the B to C, and we do the same for every tone in the A minor scale. Let's now take a 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 etc
 W H W W H W W W H  etc

Remember, every W is 2 half-steps. So let's now take a look at the raw intervals that make up each third interval in the A minor scale. We do this by adding adjacent intervals. So for example, A to C is W + H (using the pattern above), the B to D is H + W, C to E is W + W, etc. So this looks like:

A -> C : W+H = 3 half-steps
B -> D : H+W = 3 half-steps
C -> E : W+W = 4 half-steps
D -> F : W+H = 3 half-steps
E -> G : H+W = 3 half-steps
F -> A : W+W = 4 half-steps
G -> B : W+W = 4 half-steps

Remember how there are major and minor seconds? Well the same is true for thirds. A minor third (m3) is 3 half steps and a major third (M3) is 4 half steps.

Over time, you will naturally start to remember that A+C is a minor third. Right now, I don't think it's very important for you to memorize which thirds in the A minor scale are major or minor. What IS important is that you remember that minor thirds are 3 half steps and major thirds are 4. 

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?
• Try to list out from memory the A minor scale and write the intervals between each tone.

Reminder

These lessons are really intended to help you get a grasp on music theory. Things are a little slow moving at first because I'm not assuming ANY prior knowledge of this stuff. It's going to get a lot deeper as we continue. If you have ANY questions or comments, please leave a comment here, on my Facebook page, or tweet at me (@chrisstarkgtr).

Theory Thursday: Gimme a Second

Intervals: The Second

Summing up what we've learned so far about intervals, we know that a) an interval is a relationship between two tones, b) raw intervals measure this relationship in terms of individual half-steps and whole-steps, and c) with harmonic intervals we measure the musical value of this relationship.

Today's lesson is going to focus on one of the smallest harmonic intervals: the second.

From last week, we learned that the interval between any tone in the A minor scale and its nearest neighbor is called a second. But if you play the A minor scale on your guitar (especially the A minor scale all on one string), the raw intervals between some of the tones is just a single fret -- a half-step, while the interval between the rest of the tones is two frets -- a whole-step. Remember our pattern of whole-steps (W) & half-steps (H)?

W H W W H W W

How can it be that the harmonic interval between every neighboring tone in the minor scale is a 2nd when we can see that the raw interval changes between whole and half-steps?

The answer is that both whole steps and half steps can be called seconds. But there's a catch…

When a second is made up of a half-step, it's called a minor second. Likewise, when a second is made up of a whole-step (two half-steps), it's called a major second.

NOTE: It is common to use M for major and m for minor. So if you see M2, it means major second, and m2 means minor second.

So with that in mind, let's have a look at the second intervals in the A minor scale:

A -> B : M2
B -> C : m2
C -> D : M2
D -> E : M2
E -> F : m2
F -> G : M2
G -> A : M2

The important thing for you to remember about the second interval is that it comes in a couple different varieties: major & minor. Remember that a minor second (m2) is ALWAYS a half step, and a major second (M2) is ALWAYS a whole step.

EXTRA: There is one more type of second, but we don't find it in our A minor scale, so we're going to skip it for now. It is called an augmented second, and it is made up of three half steps. We will get to augmented seconds later.

Review

• What is an interval?
• What is a raw interval?
• How is a raw interval different from a harmonic interval?
• What is the difference between a major second and a minor second?
• What are the common shorthand notations for major and minor seconds?