Text: http://www.bikexprt.com/tunings/fibonaci.htm A musical equal temperament is most simply described as a mathematical series, or one-dimensional matrix, whose terms are as Fm = 2m/k FR, where F is the fundamental frequency of a musical tone, k is a positive integer constant; m is an integer variable, which may be positive, zero or negative; and FR is a standard reference frequency such as A, 440 Hz. In standard 12-tone equal temperament, k is 12, and so Fm = 2m/12 FR. We may limit the frequency range of musical pitches described by this series either directly, Fm = 2m/12 FR, FL < F < FH or by setting limits to the values of m, describing, for example, the number of keys below and above the reference pitch on a musical instrument keyboard: Fm = 2m/12 FR, mL < m < mH. The following table describes these relationships for a two-octave range centered around the reference pitch of A, 440 Hz (highlighted in yellow). Numbers used in deriving the musical pitches mathematically are shown in green. The cells of the table which represent musical pitches are colored black and white like the corresponding keys of a musical keyboard. Think of the mathematical formula as inside the instrument, behind the keyboard, manifesting itself in the pitches which the keys play. The values in musical cents in this and following tables are with relation to the next lower A, 220, 440 or 880 Hz.

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