The last triad to examine is the augmented triad. At this point, you have seen triads formed from almost every possible combination of thirds; you're only missing one combination. A quick look at FIGURE 7.7 will show you the intervallic formula of a C augmented triad.

If you break down the chord by its third intervals, from C to E is a major third interval, and from E to G# is a major third. At last, you have every possible combination of thirds. The table below will explain what you can do with three notes and two thirds.

As you can see, these are all the possible combinations of major and minor thirds in a triad; these combinations yield the four triads that music uses.

Symmetry relates to intervals being consistent throughout a musical idea or form. Diminished and whole-tone scales are considered symmetrical because they contain either the same intervals (whole tone) or the same repeating interval pattern (diminished). With the introduction of diminished and augmented triads, your understanding of symmetry reaches the world of triads. Since diminished triads are composed solely of minor third intervals, they are symmetric triads. Also, since the augmented triad is composed solely of major thirds, it is also considered a symmetric triad.

An Alternative View of Augmented Triads

Take another look at the augmented triad. This time look at all the intervals from the root of C.

The interval from C to E is a major third. The interval from C to G# is an augmented fifth. This is the second triad that has a nonperfect fifth (the other was diminished). It's also not a coincidence that the fifth is an augmented fifth, and the triad is called an augmented triad (the same with the diminished chords, diminished fifth interval).

Using the derivative approach for forming augmented triads, if you start with the C major triad (C–E–G) and compare it with the C augmented triad (C–E–G#), you come to the following conclusion: To make any major triad into an augmented triad, simply raise the fifth note one half step.

Now, here's a recap of the derivative approach for all of the triads so you can become a triad speller in no time.

The Full Derivative Approach for Forming Triads

The derivative approach presupposes an ability to spell a major triad with ease. If you can't spell a major triad easily, work hard on major scales because much of the theory remaining in this book will derive from the major scale in some way; it's simply an excellent base.

The first slight change is that now you are only going to use numbers. So, instead of talking about the root, third, and fifth, you will simply say 1–3–5.

Here is the full listing of all the triads and how they derive from the major triad:

Since this is a music book, look at all the triads side by side in FIGURE 7.8 so you can see what's happening musically.

Knowing how to build triads on pure interval, from intervals from the root, and by deriving them from the basic major triad are all approaches that should be familiar to you.