Harmony is the study of the structure, progression, and relationships of chords.
Harmony is also when pitches are in agreement, or consonance.
Harmony is a pleasing combination of two or more notes played together in the background while a melody is being played. Whereas melody represents the horizontal (time progression) of music, harmony represents the vertical principle. Like melody, harmony is derived from scales, but it is concerned with the effects produced by sounding two or more notes at the same time, not one after the other. In other words, it is mainly concerned with chords and chord progressions. Harmony is one of the main characteristics that distinguishes Western music from most other types from around the world.
The mathematical basis of harmony
Two sets of laws govern musical sounds. One stems from the natural properties of acoustics; the other is based on the rules of mathematics.
Every note, regardless of the instrument on which it is made, comprises a sound spectrum or harmonic series. Within this spectrum are contained the tonic, the octave, and the intervals of the triad. A mathematical relationship exists between them. Any two notes an octave apart have a frequency ratio of 2:1. Two notes separated by an interval of a fifth have a frequency ratio of 3:2, and two notes a fourth apart have a frequency ratio of 4:3. The fourth and fifth have an inverse relationship; together, they make up one octave. These ratios and relationships form the basic harmonic structure present in the nature of sound.
Historically, this harmonic structure has been organized in different ways by different cultures. Western music divides the octave into 12 equal divisions of a semitone each. But the octave has also been divided into as few as five and as many as 24 divisions. These steps or increments represent different scales.
It has been suggested that the number 12 (the basis of the Western system) was derived from ancient religions or from astrology. In terms of mathematics, however, 12 is quite simply the lowest common denominator for the fractions of a half, a third, and a quarter. These are the fractions represented by the primary interval ratios – the octave 2:1, the fifth 3:2, and the fourth 4:3. This is one of the logical reasons why the number 12 has a special significance in terms of the natural harmonic structure.
In Western music, every aspect of tonal cause and effect is related to the number 12.
Chords of the 7th, 9th, and 13th
In addition to the normal triad that can be used in harmonic writing, it's possible to add other notes. One of the most important chords is that of the added seventh. This note is usually added to the dominant triad (a triad built on the dominant note of the scale) in the appropriate key. In the key of C major, for example, the dominant seventh chord (V7) is formed by the addition of the minor seventh note to the dominant chord. This chord then becomes a dissonant chord (see dissonance), and must therefore be resolved. The resolution chord is normally the tonic chord, the seventh note always descending to the third of the following chord.
It's also possible to add another third to the dominant seventh chord in which case the resulting chord becomes the dominant ninth (V9). If another third were to be added the dominant eleventh (V11) would be formed, and an additional third would create the dominant thirteenth (V13). These additions of thirds may be written in the major or minor key, and they resolve downward like the seventh notes.