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It arose less than a millennium ago in the music of western Europe and is embraced today only in those musical cultures that trace their origins to that area. The concept of harmony and harmonic relationships is not an arbitrary creation.


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It is based on certain relationships among musical tones that the human ear accepts almost reflexively and that are also expressible through elementary scientific investigation. These relationships were first demonstrated by the Greek philosopher Pythagoras in the 6th century bce. In one of his most famous experiments, a stretched string was divided by simple arithmetical ratios , , ,… and plucked.

By this means he demonstrated that the intervals , or distances between tones, that the string sounded before and after it was divided are the most fundamental intervals the ear perceives. These intervals, which occur in the music of nearly all cultures, either in melody or in harmony, are the octave , the fifth , and the fourth. An octave, as from C to the C above it, encompasses eight white notes on a piano keyboard, or a comparable mixture of white and black notes. A fifth, as from C to G, encompasses five white notes; a fourth, as from C to F, four white notes.

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In other words, a string divided in the ratio yields the octave c of its fundamental note C. Likewise, the ratio or two-thirds of its length yields the fifth, and the ratio , the fourth. These notes—the fundamental and the notes a fourth, a fifth, and an octave above it—form the primary musical intervals, the cornerstones on which Western harmony is built.

The organized system of Western harmony as practiced from c.