Harmony is used differently around the world and some genres like Indian Classical music do not focus on it as a separate element. Western Classical music and jazz dive deeper into this concept more than any style.
Harmony – the simultaneous blending of notes for background and support of the musical framework
Many folk musics around the world do not go beyond a few chords and similarly pop music only uses a few chords except in the case of jazz rock fusion.
Harmony is built off the relationship between 3 or more notes played simultaneously. You can imply harmony with 2 notes but 3 notes give a complete harmonic foundation. A basic chord (or triad) is built in intervals of a 3rd (the second note is a 3rd away from the first and the last note is a 3rd away from the second) and create a symmetrical foundation much like a pyramid. When the chord is played with these intervals it is said to be in “root position” and the lowest note (or bass note) of the chord is considered the “root.” If you move these notes around you get different intervals and different positions (1st position, second position), but the chord’s identity remains the same.
One of the main aspects of a chord’s identity is the middle note of the chord, or the first 3rd of the chord. If this note is 3 half steps away from the root it is considered a minor 3rd, if it is 4 half steps away it is considered a major 3rd. This small change has an important effect on the way the chord will sound. Generally in Western culture a minor 3rd sounds sad and a major 3rd sounds happy. This relationship creates a curious binary situation which has infinite possibilities.
The notes for more advanced chords come out of what is known as the harmonic series. This series of notes is built off of the mathematical ratios of dividing a string in equal lengths. In mathematics it is known as the divergent infinite series.
In jazz music these notes are called extensions of a chord. If you have a basic triad and begin adding extensions you will get the following chords:
7th, 9th, sus4 or 11, 13th
These chords can also be altered to be sharp or flat as in a C7#11 or a G7b13
One of the main determinants of a chords identity is the bass note. The bass note need not be the root of the chord but most times the bass note needs to be identified.
A C7#11 chord with an E in the bass would be written C7#11/E
An argument could be made that this chord is an E9b13 but it all depends on the chords occurring before and after this one. Context is essential.