Tools of Harmony

These means were extensively discussed by Palladio, an Italian architect [3]. Further examples for proportion in traditional architecture have been reviewed by Langhein [4], 

The means have some interesting properties that were known already to the ancient mathematicians. The arithmetic mean has the property of Xi — x = x — X2, the geometric mean has the property of Xi/x = x/x2, and the harmonic mean has the property ofxi/x2 = (xi — x)/(x — X2). The reciprocal of the harmonic mean is 

if the reciprocals are formed, the harmonic mean turns into the arithmetic mean. Such a formula appears in mechanics, e.g., in the vibration of molecules. 

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