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Phi Φ / phi φ

The 21st letter of the Greek alphabet. Its upper-case form, Φ designates the "golden mean" = (1 + 5^1/2) / 2 = ~1.61803398875. Its whose lower-case form, φ designates the reciprocal of Φ.

φ is a unique constant. It is:

• the ratio of two quantities such that the smaller is to the greater, as the greater is to the sum of the greater and the smaller, where a < b, thus: b / a = (a+b) / b
• one larger than its own reciprocal; thus by definition: φ = ( 1 / Φ ) = ( Φ - 1 ) ;
• the mean of quotients of adjacent elements in a series.

The quotient of two adjacent members of the fibonacci series approches Φ increasingly closely as one ascends higher in the series.

Φ and φ play an important role in the comparison of different parts of many natural objects (such as the spacing of plant limbs, the growth of a nautilus shell, the proportions of the human body, etc.), they appear in relationships thru-out the universe, and they have been employed by many human artists, architects, and craftsmen in calculating the dimensions of their creations. They have been employed in various ways in musical tuning calculations -- one significant example is golden meantone.