As mentioned, chords are based on thirds. They can, however, extend PAST just the root, third, and fifth. The next third on top of the fifth would be the seventh.
Let’s build a major C chord in the key of C, adding a seventh. Refer to the keyboard, and let’s see what those notes would be.
We already know that the basic C major chord consists of the notes C, E, and G. When you add another third in this case over the G, you would add a B. So, the notes here are C, E, G, and B.
This chord is called “Cmaj7.” The intervals here are major third, minor third (both to make a major chord), and another major third. Note that in major and natural minor keys, no minor chord will have a major third on top of the fifth. Any third in this case would be minor. However, when you get into harmonic and melodic minor keys, you very well may have intervals of minor, major, major. If this particular chord had C as a root, it would be “Cmin7.” This is NOT to be confused with “Cm7.” What’s the difference? Glad you’ve asked. It shows you were paying attention. It has to do with “dominant” sevenths, which is forthcoming.
Before we get into “Cm7,” let’s back out of the minor chords, and back into major chords. Because of the popular usage of a V-I chord progression to resolve a part of a song (go into the editor, and whip up a quick C-F-G-C progression, and you’ll see how nice the G (or V) chord leads back to the C (or I) chord), the V chord is most often associated with adding sevenths. However, if you look to add a seventh to the G chord while you’re in the key of C, you’ll notice that the seventh is a minor third on top of the fifth - as D to F is a minor third. This makes the interval different than the major-minor-major for the Cmaj7. Instead, this is major-minor-minor.
Now, why is this so popular? One big reason, if you’d notice, is that every note in any particular key is covered by the I, IV, and V chords. Since the V chord leads so well into the I chord, this progression makes for a nice resolution to a particular section of a song. So, most “by-the-book” songwriting will have these three chords, and end on a V-I progression.
In the C-F-G-C progression that you had made (or maybe didn’t, whatever), the “F-G” progression is the harshest, because they have no common notes. If, however, you add the seventh to the G chord, you’ll have an “F” note, which blends well with the chord before it. This is why the V chord is most associated with sevenths. When you have a seventh on a V chord, that seventh is referred to as a “dominant” seventh. In the case of the G chord, it will be noted as “Gdom7,” or more commonly, simply as “G7.”
When you apply this to minor keys and minor chords, the connotation will be similar, but it will explain the difference between “Cmin7” and “Cm7.” Because the dominant seventh is associated with the fifth chord of the key, a Cm7 would most often be the v7 chord. Therefore, Cm7 would be in the key of F minor. As in the case of the major chord progressions, you can go into the editor and make a i-iv-v7-i progression to see how it works. In the key of F minor, that progression would be Fm-Bbm-Cm7-Fm.(Edit: BTW, if you have a V7-I progression, the V7 tends to lead itself more into the I chord than a regular V chord)
From here, more thirds can be added, giving ninths, elevenths, and thirteenth chords. (Seventeen would be back to the root note.) When you get thirds above the seventh, it is worth noting that you’re starting an octave over again. For instance, the “ninth” would be the second, plus an octave.
To be quite honest, I’ve never really dealt much with these chords, so I’m no expert here. I know they exist, and I know I can use them if I decide to. I don’t know, however, what they would be appropriately named, as some are “C9,” “Cadd9,” “C7add13,” etc. Largely, the differences would be skipping some of the thirds, and essentially having something like the root, third, fifth, skipping the seventh, and going to the ninth. Maybe somebody can fill me in on nomenclature, but I think this will suffice for a PB introduction to music theory – as long as you don’t get into many in-depth conversations!