## The Golden Mean and Fibonacci Series

Some 20th-century composers and writers have been interested in the "golden mean" or "golden section," a proportion used for centuries in art and architecture to obtain aesthetically pleasing designs.  The golden mean is the division of a whole into two unequal parts such that the ratio of the smaller to the larger is the same as that of the larger to the whole.

To understand this ratio, consider a line AC with line segments AB and BC.  If the proportion of AB to BC is the same as the proportion of BC to the whole line, AC, then AC is segmented according to the golden mean.  This relationship can be expressed as:

AB      BC
----  =  ----
BC      AC
Integers (whole numbers) that approximate the golden mean can be generated by means of a Fibonacci Series, an endless series of numbers in which each number is the sum of the previous two.  The farther you go in the sequence, the closer you get to the true value of the golden mean.
Integers:        1        2        3        5        8        13        21         34        etc.
Ratios:               .5     .67      .6     .625   .615    .619     .618
The most obvious way that this ratio can be applied musically is in the proportions of a musical form.  For example, the beginning of "Minor Seconds, Major Sevenths," from Bartok's Mikrokosmos, could be subdivided in this way:
meas. 8   =  Strong cadence; first whole-note chord
meas. 21 =  Strong cadence; first appearance of "glissando"
meas. 34 =  End of long accelerando and of the first main section