Bridges: Mathematical Connections in Art, Music, and Science (1999)

Pages 123–130

The properties of exponential and logarithmic functions and their historical development are reviewed, with emphasis on application to non-Euclidean metrics in Klein's Erlangen Program. Comparing the roles of angles and logarithms as arguments of circular and hyperbolic tangent functions, linear fractional transformations l.f.t. 's) are introduced which show successive stages of hyperbolic rotations to be projectively equivalent to terms of a geometric sequence. The special role of powers of 2 in human perception of both color and pitch intervals is explored, using music staff paper to yield an aural model of economic growth. A possible explanation for the octave phenomenon is offered in the way it combines properties of both harmonic and geometric spacing, as illustrated by the ancient Egyptian artists' method of rendering human body proportions in murals.

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