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Chords are 3 or more notes played at the same time.  (Sometimes people talk about 2-note "chords", but these are more properly called intervals or double stops.)

Chords are named according to the root note of the chord.  Sometimes the order of the notes gets rearranged so the lowest note is not the root note.  These rearranged chords are called inversions of the natural chord.  Sometimes it's a bit tricky to determine the root note for chord inversions, though often you can tell by looking at the intervals, which are the distances between notes.

First let's look at simple 3-note chords, which are called triads.  Here is the C major scale, with the first 4 notes repeated:
Scale Degree: 1  2  3  4  5  6  7  8  9 10 11
              C  D  E  F  G  A  B  C  D  E  F
Triads are formed by starting somewhere in the scale and then picking every other note, skipping the notes in between.

For example, start at scale degree 1 and pick the note--a C.  Then skip 2, the D, and pick 3, the E.  Then skip 4=F and pick 5=G.  We have 1&3&5, which is C E G.  Since the chord was formed by starting with the C we know it is some type of a C chord.  Here are all the 3-note chords in the C scale formed by using this process:

Scale Degrees
Note Names

How do you know if it's a major or minor or diminished chord?  You have to look at the number of half-steps between the notes.  There are 2 intervals in each triad, the distance between the first and second note in the chord, and between the second and third note in the chord.

Considering a piano keyboard, starting on C count the number of keys (both black and white keys) you need to get to E (each successive key on a piano is one half-step away).  Here they are from C: 1=C#/Db, 2=D, 3=D#/Eb, 4=E.  That's 4 half steps to get there.  The interval is determined by counting note names, not half steps.  The C to E interval is 1=C, 2=D, 3=E, so it's a third.  So the C-E interval is a third consisting of 4 half steps.  A third with 4 half steps is called a major 3rd.  A chord that starts with an interval (i.e. from the root note to the 3rd) that is a major 3rd is called a Major chord.

Now determine the interval from D to F.  The note names are D, E, F, so once again we have a 3rd.  This interval is always a third because of the way we formed the chords.  But now count the half-steps starting on D to get to F. D to D#/Eb is 1 half-step, D#/Eb to E is 2 half-steps, and E to F is 3 half-steps.  So there are only 3 half-steps to get from D to F.  An interval of a third with 3 half-steps is called a minor 3rd.  A chord that starts with an interval that is a minor third is called a minor chord.

When we do the same procedure on the 2nd interval in the triads we find that for C major, E to G is a minor 3rd, and for D minor, F to A is a major 3rd.  So major chords consist of a major 3rd interval followed by a minor 3rd interval, while minor chords reverse the order and have a minor 3rd followed by a major 3rd.

The B D F chord has a minor 3rd (B to D) followed by another minor 3rd (D to F).  A minor chord with the second interval shortened from 4 half-steps (a major 3rd) to 3 half-steps (a minor 3rd) is called a diminished chord.

Another kind of chord consists of 2 major 3rd intervals.  An example is the chord C E G#.  When a major chord has it's second interval extended by a half-step to be a 4 half-step major 3rd instead of the usual minor 3rd it is called an augmented chord.  Augmented chords do not occur naturally in the scale, and are used very much less frequently than major, minor, and diminished chords.

Building Bigger Chords

Chords can be extended by adding intervals above the basic triad.  One of the most frequently used chords in blues, rock, country, and pop music is the dominant 7th chord.   To make a 7th chord you add another note that is a minor 3rd above the top note in the basic triad.  For a C chord, the top note is a G.  Counting 3 half-steps gives us a Bb, so the C7 chord is C E G Bb.  Similarly for the Dm chord, to make it a 7th chord you add a minor 3rd above the A, which is a C.  So Dm7 is D F A C.

A diminished 7 chord follows the same procedure, so for Bdim7 we have B D F Ab.  Another diminished chord is sometimes called "half-diminished" when the 7th is a major 3rd above the top note of the triad instead of a minor third.  In this example we have B D F A.  This chord is also called Bm7b5, which means Bm7 flat fifth.  The Bm7 is B D F# A, so when we flat the fifth, the F#, we get the same notes as the half-diminished chord.  This chord is frequently found in jazz.

Another form of 7th chord is the Major 7, which adds a major 3rd interval to the top note in the base triad.  Normally this is done with Major chords to produce a sweet sounding chord.  The example for C is CMaj7, which has notes C E G B.

Other extended chords are made by adding other notes from the scale.  A 6th chord adds an interval a 6th above the root note of the chord.  For a C chord the 6th is an A and the chord is C E G A.  For a Dm chord the 6th is a B and the chord is D F A B.

Other common extended chords used frequently in jazz are 9th and 13th chords which are built by adding the 7th and also adding the 9th or 13th above the 7th.  These chords are colorful and dissonant and along with diminished and m7b5 chords are characteristic of the cool jazz sound.