# Math and Music

## The History of Math and Music

### Ancient Greeks

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." (Plato 501-505)

#### This is the first reference to any kind of celestial music where the paths of the heavenly bodies are described as correlating to a specific musical tone.

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.

# 4

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.

### Styles of Music Containing Mathematics: Baroque

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.

### Other Related Sites

Chaos Club

Pi and the Fibonacci Numbers