A power chord is a basic, three note chord consisting of a base note, a note one fifth above the base, and a note one octave above the base note. Power chords are played mainly on the guitar, and they are more often than not distorted.

A power chord can also be thought of as the lower three notes of a 5 or 6 note barre chord. They are often the first thing a beginning electric guitar player learns to play. They allow for great, thick sounding melodies to be played on the guitar, but they do not contain much texture or dissonance.

Although you can find them all over modern music, a famous example is Nirvana's Smells like Teen Spirit. The introduction consists of 4 power chords, repeated. First it is clean, then later distorted.

Some variety can be used with power chords, although it is not done often. In Tool's Sober, the main rhythm is a power chord hit twice in a row quickly, a rest, then the same chord/rhythm repeated, except the middle note is a minor sixth instead of a fifth.

A slang term used among musos for a distorted two-note diad played on the guitar, usually but not always containing the root and the fifth. In heavy metal it often includes the root and the third, and very occasionally the third and the fifth. People often call any fifth chord with any number of notes a power chord, and really there's no problem with this as it's just a slang term anyway, but if someone was to say: "just playing power chords" it would be understood that it was diads being asked for.

Power Chords are used extensively in today's modern punk rock music. (Blink 182, NoFX, The Youth Ahead, A New Found Glory) Without getting into the theory of why a power chord sounds the way it does and diads and such, here is what a power chord is.
|(R+2)-|
|(R+2)-|
|-R----|
In this example, the root note is indicated by R, and then, on the two strings above it, you merely play two frets higher on the fretboard. The actual fretboard looks something like this.
 |--------------------|B
E|  |  |1 |  |  |  |  |R
A|  |o |  |o |4 |o |  |I
D|  |  |  |  |4 |  |  |D
 |--------------------|GE ;)
In this diagram, the fretboard is shown, and the numbers tell you what finger to place where.
  1. Index Finger
  2. Middle Finger
  3. Ring Finger
  4. Pinky Finger

The rhythm guitar part to Blink 182's Dammit is as follows.
e|---------|
B|---------|
G|-5-------|
D|-5-5-7-3-|
A|-3-5-7-3-|
E|---3-5-1-|
Where each chord is strummed in a punk pedal sort of way.
If any of this is wrong, just message me, im only a bassist so be gentle :P.

The reason power chords are used more extensively in rock, usually with distortion, is this:

A note put through a distortion unit will sound like a major chord.

Let's look at why this is. We need to cover harmonics, scale theory and the properties of distortion units to do this. Here we go...

Harmonics

Harmonics are the multiples of any fundamental note. For example, an open A string at 110Hz will have a second harmonic at 220Hz, a third at 330Hz, a fourth at 440Hz, a fifth at 550Hz and so on.

On the guitar, you can produce the nth harmonic by lightly touching the string at 1/n of its length and plucking it. Touch a string very lightly above the twelfth fret and pluck it - you get the second harmonic, which is exactly one octave higher (see below). Doing the same above the seventh fret gets you the third harmonic, which is almost exactly an octave-and-fifth. You can also touch the string at any of the n equally spaced points - so you could touch at 2/3 the length, over the nineteenth fret, and get the same note. Of course, touching at 2/4 of the length will get you the second harmonic not the fourth (as 2/4 = 1/2).

You can demonstrate what is happening with a rope. Get someone to hold one end. You hold the other and start to swing the rope like a skipping rope at the frequency that feels most natural. Now try doubling, tripling or quadrupling that frequency. See the stationary points (nodes)? On the guitar, those nodes are where your damping finger is.

Scale theory

The common scale in Western music is made up of octaves (doublings of frequency) each divided into twelve equally-spaced semitones (although read up on tempering). The ratio between each semitone is therefore the twelfth root of two (hereafter referred to as r) which is about 1.059.

Distortion units

Solid-state distortion units tend to add odd nth harmonics (third, fifth etc), decreasing fairly rapidly as n increases. This distortion tends to sound harsh. Valve distortion tends to boost the even harmonics, which sound warmer.

Relationship between harmonics and notes

OK - so what note does the nth harmonic relate to? We need to find out how many semitones correspond to a multiple of n. This is:

x = log(n) / log(r)

 n      x            interval
 2      12 exactly   octave
 3      19.02        oct + fifth, slightly sharp
 4      24 exactly   two octaves
 5      27.86        two oct + maj third, fairly flat
 6      31.02        two oct + fifth, slightly sharp
 7      33.69        two oct + dom seven, quite flat

...and so on. Usually, anything above the fifth harmonic is quiet enough as to solely form part of the sound of the instrument, distinguishing it from a sine wave, rather than being recognizable as a distinct pitch.

Using this, we see that a distorted power chord of C will contain C and G as fundamentals, G and D as third harmonics and E and B as fifth harmonics. This gives a power chord the feel of an unbalanced Cmaj9 chord. Adding your own third would muddy this further.

This also shows why distorted minor chords are even grungier than major chords.

This writeup is now online at http://tranchant.plus.com/guitar/power-chords

The other reason why power chords are used so often in punk, rock, etc., is that they are not really chords, but only a single voice. Explanation:

When standard four-part harmonies were starting to come around, one of the rules invented was that "two perfect consonances of the same type should never follow one after the other" (according to Johannes de Garlandia, c. 1300). This was because the two voices involved were effectively reduced to a single voice, since they each carried the same tune only at different pitches. Doing so is called parallel fifths or octaves and is to be avoided at all costs in four-part harmonization for that reason.

The reason why this only applies to perfect consonances (i.e. perfect fifths and octaves) is adequately explained by e-troom's writeup - the fifth and the octave are harmonics of the root note.

What does this mean? Punk music usually consists of a singer, guitar, bass, and drums. More often than not, the guitar plays power chords, and the bass plays the root note of those chords. This leaves us with a grand total of two voices - the singer and the guitar power chord/bass conglomeration. Wow. This kind of stuff was last seen in the thirteenth century. The songwriter needs only to harmonize two voices instead of the standard four, and the end product is extremely easy to play.

Of course, some more advanced punk bands (such as Green Day) will add background vocals or spice up the bass line a bit, and then you end up with three or four voices. That is actually better than most other forms of modern non-orchestral music, save Jazz and progressive rock.

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