Pythagoras of Samos (c.582-c.507 B.C.) originated the idea of music
as a branch of mathematics. The theories of Pythagorean tuning and the
idea of musical intervals expressed as mathematical
ratios were built upon this premise. Although Pythagoras is noted as
the founder of this idea, many Greek mathematicians (including Claudius
Ptolemy) believed in "the universe as being bound together by mathematico-musical
principals" and thought that Nature was founded in numbers. They came
to the conclusion that the universe could be explained in musical terms
(aka, with numbers). This concept was revolutionary for future theorists
on the subject of the Music of the Spheres (the "spheres" part
comes from the ancient Babylonian premise that the cosmos is comprised
of seven spheres). What each sphere represented became the topic of debate
among the Greek mathematicians. Basically, the spheres are the paths and
levels of the planets and the heavens. The theoretical differences came
in exactly how many spheres there were, which changed from dogma to dogma.
The most common changes to the "spherical makeup of the universe"
were the edition of Heaven, the sun and other kinds of "things."
The one theory that stuck was that of Plato, which he sets out in both
Phaedrus and Republic. The following is an excerpt from Republic concerning
the Music of the Sphere:
"...they saw there at the middle of the light the extremities of
its fastening stretched from heaven; ... from the extremities was stretched
the spindle of [the goddess] Necessity, ...And the spindle turned on the
knees of Necessity, and up above on each of the rims of the circles a Siren
stood, borne around in its revolution and uttering one sound, one note,
and from all the eight there was the concord of a single harmony."
Johannes Kepler (1571-1630) is primarily known for discovering the three
basic "laws" of planetary motion. Surprisingly, Kepler discovered
the elliptical shape of the planet's orbit after a long search for musical
harmony in the planets. Starting with the postulate that only eight consonances
exist within the range of one octave, Kepler sought to find these eight
consonances somewhere in the motion of the planets. The eight consonances,
given as the ratios of the frequencies of two played notes, are: 1/1, 1/2,
2/3, 3/4, 4/5, 5/6, 3/5, and 5/8 (certain combinations of these ratios
create harmonic chords). Kepler finally found these rations in the motion
of the planets when he postulated that the planes must move in elliptical
orbits. His reason for doing so was since the speeds of each planet at
it's perihelion and aphelion (the point closest and farthest from the sun,
respectively) could be combined in a few different ways to give the eight
consonaces, the planety motion must be elliptical.
Mathematics are applied directly to the use of time signatures in music.
You might recognize the fraction above as a time signature as it might
appear on sheet music. The numerator represents how many beats will be
in each measure of music; the denominator tells which note will be indicative
of the beat (by placing a one above the bottom number, you will get the
fraction that represents what note it is). In this case, there are four
beats per measure and the quarter note gets the beat. By using this symbol,
the time signature is kept mathematically precise in order to keep the
music moving. Other ways of keeping the rythm precise are by using a metronome,
a device that taps out the beat of the music while a song is playing.
The most prominent form of music that uses mathematics (aside from the obvious use of harmonics and time signatures) is Baroque music. The goal of many of its prominent composers (such as Vivaldi and Handel) was to maintain an stricly measured pulse while keeping the music mathematically precise. Baroque music is said to be a measure of the times in which it was written. This is held true with the fact that the music is loud, obtrusive, and nearly gaudy in style. This is quite similar to the art, architecture, and literature of the time: loud and obnoxious!
It has also been discovered that the precision of the music
creates increased synoptic activity in the brain, which might explain why
some of your teachers have played Baroque music during exams.
The mathematical representation pi has been discovered to carry
certain ratios that can be related to the aforementioned consonances. For
more information on this topic, refer to The
Music of Pi.
and the Fibonacci Numbers
Back to the Arcadia homepage.
Author: Anil Hurkadli
Date: March 6, 1998
Editions and supplementary info. added by: Ann Mogush
Date: April 1, 1999
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