The Golden Mean and Fibonacci Series
Some 20th-century composers and writers have been interested in the
mean" or "golden section," a proportion used for centuries in art and
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
to the whole.
To understand this ratio, consider a line AC with line segments AB
BC. If the proportion of AB to BC is the same as the proportion
BC to the whole line, AC, then AC is segmented according to the golden
mean. This relationship can be expressed as:
Integers (whole numbers) that approximate the golden mean can be
by means of a Fibonacci Series, an
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
of the golden mean.
The most obvious way that this ratio can be applied musically is in the
proportions of a musical form. For example, the beginning of
Seconds, Major Sevenths," from Bartok's
Mikrokosmos, could be subdivided in this way:
.625 .615 .619
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
(Click here to see the
There is some evidence that Bartok (and Debussy)
used the golden mean not only in formal proportions but in other
of his music as well, and this is also true, if to a lesser extent, of
some other 20th-century composers.
Musical frequencies are based on Fibonacci ratios
Notes in the scale of western music have a foundation in the
Fibonacci series, as the frequencies of musical notes have
relationships based on Fibonacci numbers:
The calculated frequency above starts with A440 and applies
the Fibonacci relationships. In practice, pianos are tuned to a
"tempered" frequency to provide improved tonality when playing in
* A440 is an
arbitrary standard. The American Federation of Musicians
accepted the A440 as standard pitch in 1917.
It was then accepted by the U.S. government its standard in 1920 and it
was not until 1939 that this pitch was accepted internationally. Before recent times a variety of tunings were
used. It has been suggested by James Furia and others that A432
be the standard. A432 was often used by classical composers and
results in a tuning of the whole number frequencies that are connected
to numbers used in the construction of a variety of ancient works and sacred sites, such
as the Great Pyramid of Egypt. The controversy over tuning still
rages, with proponents of A432 or C256 as being more natural tunings
than the current standard.