Previously we discussed ninth chords, which are the first in a series of extension chords. Extensions are the notes that go beyond the first octave. Since we build chords by stacking thirds on top of each other, the next logical step, after the ninth chord, is the eleventh and the thirteenth chords. And that’s it: there is no fifteenth chord, because the fifteenth would be the same as the root (albeit two octaves higher).

This strange musical arithmetic is best understood if we translate all intervals into their semitone equivalents in equal temperament. Since we started by constructing the E major chord, let’s work with the E major scale, which consists of the following notes:

`|E |  |F#|  |G#|A  |  |B |  |C#|  |D#|E |`

Let’s chart the chord tones taking E as the root.

We see the clash of several naming conventions. Letter names have their origin is the major diatonic scale, as implemented by the white keys on the piano starting from C.

`|C |  |D |  |E |F |  |G |  |A |  |B |C |`

They go in alphabetical order, wrapping around after G. On the guitar we don’t have white and black keys, so this convention seems rather arbitrary.

The names of intervals (here, marked by digits, with occasional accidental symbols) are also based on the diatonic scale. They essentially count the number of letters from the root (including the root). So the distance from E to B is 5, because you count E, F, G, A, B — five letters. For a mathematician this convention makes little sense, but it is what it is.

After 12 semitones, we wrap around, as far as note names are concerned. With intervals the situation is a bit more nuanced. The ninth can be, conceptually, identified with the second; the eleventh with the fourth; and the thirteenth with the sixth. But how we name the intervals depends on their harmonic function. For instance, the same note, C#, is called the sixth in the E6 chord, and the thirteenth in E13. The difference is that E13 also contains the (dominant) seventh and the ninth.

A full thirteenth chord contains seven notes (root, third, fifth, seventh, ninth, eleventh, and thirteenth), so it cannot possibly be voiced on a six-string guitar. We usually drop the eleventh (as you can see above). The ninth and the fifth can be omitted as well. The root is very important, since it defines the chord, but when you’re playing in a band, it can be taken over by the bass instrument. The third is important because it distinguishes between major and minor modes (but then again, you have power chords that skip the third). The seventh is somewhat important in defining the dominant role of the chord.

Notice that a thirteenth chord can be seen as two separate chords on top of each other. E13 can be decomposed into E7 with F#m on top (try to spot these two shapes in this grip). Seen this way, the major/minor clash is another argument to either drop the eleventh (which serves as the minor third of F#m) or sharp it.

Alternatively, one could decompose E13 into E with DΔ7 on top. The latter shape is also easily recognized in this grip.

I decided against listing eleventh chords because they are awkward to voice on the guitar and because they are rarely used. Thirteenth chords are more frequent, especially in jazz. You’ve seen E13, here’s G13:

It skips the 11th and the 5th; and the 9th at the top is optional.

## The Role of Harmonics

It might be worth explaining why omitting the fifth in G13 doesn’t change the character of the chord. The reason is that, when you play the root note, you are also producing harmonics. One of the strongest harmonics is the fifth, more precisely, the fifth over the octave. So, even if you don’t voice it, you can hear it. In fact, a lot of the quality of a given chord voicing depends on the way the harmonics interact with each other, especially in the bass. When you strum the E chord on the guitar, you get a strong root sound E, and the B on the next thickest string amplifies its harmonic fifth. Compare this with the G shape, which also starts with the root, but the next string voices the third, B, which sounds okay, but not great, so some people mute it.

Inverted chords, even though they contain the same notes (up to octave equivalence) may sound dissonant, depending on the context (in particular, voice leading in the bass). This is why we don’t usually play the lowest string in C and A shapes, or the two lowest strings in the D shape.

In the C shape, the third in the bass clashes with the root and is usually muted. That’s because the strongest harmonic of E is B, which makes C/E sound like CΔ7.

On the other hand, when you play the CΔ7 chord, the E in the bass sounds great, for exactly the same reason.

You can also play C with the fifth in the bass, as C/G, and it sounds good, probably because the harmonic D of G gives it the ninth flavor. This harmonic is an octave and a fifth above G, so it corresponds to the D that would be voiced on the third fret of the B string.

The same reasoning doesn’t quite work for the A shape. Firstly, because all four lower strings in A/E voice the very strong power chord (two of them open strings) drowning out the following third. Also the fifth above E is the B that’s just two semitones below the third C# voiced on the B string. (Theoretically, C/G has a third doubled on the thinest string but that doesn’t seem to clash as badly with the D harmonic of G. Again, the ear beats theory!)

Next: Altered chords.