Hum a tune alone and it is a melody. Add a guitar strumming underneath and something changes: the same tune can sound hopeful, wistful or unsettled depending on what is played beneath it. That added dimension is harmony.
Chords and triads
Harmony is the sound of different notes heard together and the way those combinations follow one another. A chord is three or more different notes sounded at the same time. Two notes form an interval, as the third level explained; three or more make a chord. The most basic chord in Western music is the triad. A triad is a three note chord built by stacking two thirds: a root, a third above it and a fifth above the root. Play C, skip D, play E, skip F, play G, and you have a C triad. The root is the note a chord is built on and named after, so C, E and G together make a C chord.
Major, minor and others
Remember that thirds come in two sizes. Which size sits at the bottom decides the chord's character. A major triad has a major third on the bottom with a minor third stacked above it: C, E and G. A minor triad reverses the order, a minor third below and a major third above: A, C and E, or C, E flat and G. To most Western ears, major sounds bright and settled, minor sounds darker or more inward. That emotional reading is partly learned, though.
A diminished triad stacks two minor thirds and sounds tight and unstable. Adding a fourth note a seventh above the root makes a seventh chord. Jazz and soul use seventh chords and richer stacks constantly, which is why those styles sound lush compared with a simple folk song.
Chords inside a key
Each note of a scale can serve as the root of a triad built only from notes in that scale. Musicians label these chords with Roman numerals for the scale degree: I is the chord built on the tonic, IV on the fourth note, V on the fifth, and so on, with uppercase for major chords and lowercase for minor ones. In C major, I is C major (C, E, G), IV is F major (F, A, C), and V is G major (G, B, D). In C major the chord on the sixth degree, A minor, is written vi. The great advantage of numerals is that they work in any key: a song that goes I, IV, V in C goes the same way, with different letters, in G or E flat.
What chords do: function
Theorists describe three main functions. The tonic chord, I, is home, where music feels at rest. The subdominant, IV, moves away from home in a relaxed way. The dominant chord, V, creates tension that pulls strongly back to the tonic. Much of that pull comes from one note. The leading tone is the seventh note of the scale, a half step below the tonic, and it tends to rise to the tonic. In C major the leading tone is B, which sits in the G chord and leans hard toward C. Add the note F to the G chord, making a dominant seventh, and the tension grows, because F wants to fall to E.
The notes of a chord need not be stacked in their basic order. A chord is in inversion when a note other than the root is the lowest sounding note. C major with E in the bass is still C major, but it sounds lighter and less grounded. Bass players and arrangers use inversions to make the lowest line move smoothly instead of jumping around.
Harmony beyond the West
This chord-based harmony is a particular European invention that spread worldwide through church, school and recording. Many great traditions, including Indian classical music and much Arabic music, put little weight on changing chords and instead build richness through melody, ornament and a steady drone, a note held underneath throughout. Hearing harmony as one option, not the definition of music, keeps us from mistaking the familiar for the universal.
Worked example: harmonizing "Happy Birthday"
Imagine you are asked to accompany "Happy Birthday" on a guitar in C major, and you know only three chords: C (I), F (IV) and G (V). Start on C, since the tune begins on the dominant note G, which fits comfortably in the C chord as its fifth. As the melody climbs toward "birthday to you," it lands on notes like B and D that belong to G, so you switch to G. The melody then falls back to C on the end of the next "to you," and you return to the C chord: home.
On the third line, where the tune leaps to its highest note for the name of the person being celebrated, many players move to F, the subdominant. The last line then runs C, G, C: home, tension, home again. With three chords and the idea of function, you have harmonized one of the world's most sung songs, and you have also learned the skeleton of thousands of folk, country and rock tunes.