Harmony is a series of notes in combination, played simultaneously or at least tangent. Listen to your favorite songs, and pay close attention to the back-up vocals when they're being sung along with the lead. Can you hear how the two vocal melodies differ in pitch? That's harmony, and it's highly possible that those back-up vocalists were chosen because of their strong ability to harmonize. Singing a harmony, or rather picking one out on your own without written sheet music, is an almost inherent musical skill that many singers would die to possess. But harmony isn't just the province of singers; it's found in every single area of music. Any time a sound is layered on top of another sound and those sounds match each other in rhythm and melody (but not pitch), a harmony is created.
Harmony is made of intervals, and as such, it can be considered dissonant (scratchy, uncomfortable, like playing an E and an F at the same time) or consonant (pleasing or smooth). What makes a harmony pleasing or unpleasing, however, is extremely relative. In medieval times, only octaves and perfect fifths were considered harmonious, and any harmony that deviated from that was generally frowned upon. In modern western music, though, nearly everything is considered to be harmonious by someone. Fifths are still very popular in modern harmony but are now used in the most unlikely of places; heavy metal music, for example, frequently uses perfect fifths in the vocal harmony to create an eerie effect when layered on top of the more dissonant instrumentation.
In addition to being consonant or dissonant, can also be subordinate or coordinate. Subordinate harmony, the tonal harmony used most frequently today, is a series of harmonies that are based on each other. The harmony moves in such a way that a resolution is somewhat predictable; you can hear this type of harmony in modern pop music, musical parts that flow very easily into each other and don't leave the listener baffled as to the turn the song has taken. On the other hand, coordinate harmony is a series of harmonies that operate independently of each other. They do have some common relation, of course, but don't typically move toward a goal, or predictable resolution. Renaissance musicians often used this type of harmony, and it's capable of producing rich and moving textures within a piece of music.
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