Music–The Geometry of Sound
What is music, and where do it’s true origins life?
One way that music can be defined is: a structured organization of sounds that express the totality of human emotion. Music evokes emotions in the listener but the emotional content of the music, is but one of the many components that comprise the whole of what we call “Music”.
The word music originates from the Latin word musica, which evolved from the Ancient Greek word “mousike.” Mousike is defined as “the art of the Muses.” In Greek Mythology the Muses were the Greek Goddesses of the Arts and Sciences. The Muse or, Goddess of Music, was a flautist named, “Euterpe”. Euterpe meant, giver of much pleasure. Music is often associated with pleasure and perceived as pleasurable but ironically Euturpe was also known as the Muse of tragedy. The duality and polarity in life is represented here by music, as is comedy and tragedy in Greek Theater.
According to the Ancient Greeks what we aurally perceive as pleasing to the ear is the ratio of two or more harmonics that are vibrating together agreeably or harmoniously. Harmonic ratios are geometrical divisions of sound, approximated by equal temperament intervals that our brains are innately fine tuned to recognize and organize both intervallically and directionally.
The first person in the western world accredited with the discovery of the harmonic series was a Greek mathematician from the 6th century B.C named Pythagoras, well known for the “Pythagorean Theorem”. Pythagoras considered the triangle to be the origin of everything on Earth. He established his famous theorem on this geometrical figure: “The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.” He discovered how ratios of harmonics interact with each other mathematically and experimented with harmonic ratios by dividing a vibrating string into several different divisions. Pythagoras’s then applied a system of mathematics to harmonic ratios.
The Greek Philosopher Plato said, “Mathematics was born from music.”
The mathematics Plato was alluding to were the mathematical relationships that appear in the Harmonic Series. This system included a, geometric series and an arithmetic series.
The octave relationships of the harmonic series use a geometric series of exponentials, ex 1, 188.8.131.52.32.64. This geometric series is also used in rhythm. This mathematical division and movement of sound create music. The main Pythagorean belief was, “all of nature consists of harmony
arising out of number”.”
In the Pythagorean school of thought, the number 5 or the “fifth” represented the division of all universal laws. He created the Pythagorean scale that ascended in fifths up seven octaves. This scale was the first cycle of fifths. It is also quite possible that Pythagoras’s numerological interest in the number five, led to his discovery of the, ‘Golden Ratio” 1.618 or “phi.” (pronounced fee.) The symbol of Phi (1.618) in ancient Greece was actually the Greek number for 5. Because Phi is an irrational number, it is impossible for anything to measure at “precisely” the golden ratio. However, a large number of things in natureapproximate the golden ratio. The golden ratio appears within harmonic ratios, and equal temperament intervals. The golden ratio connects the organization of musical sounds with the perception of it because harmonic ratios, equal temperament intervals and our hearing apparatus used to perceive sound are all constructed from the golden ratio. The Golden Ratio also connects geometrical and harmonic shapes.
In music Pythagoras also considered the numbers 1,2,3,4 to represent the universal laws of life and he used them in geometry and music. These numbers were inspired by the first 4 harmonics that sound harmoniously with the fundamental tone. In geometry the golden ratio is used in the four-sided golden rectangle.
To Pythagoras both the four sided rectangle and the first four harmonics represented
consonance, symmetry, purity, unity, and wholeness. It seems that the importance of symmetry and the golden ratio serve as common denominators of aesthetic evaluation. The ear responds favorably to the symmetry of equal temperament tuning just as it does to the symmetry of a rectangular shape. Both geometric shapes and harmonic shapes adhere to the laws of ratios and proportions, as does all physical matter.
Aristotle:“[the Pythagoreans] saw that the … ratios of musical scales were expressible in numbers [and that]… all things seemed to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number.”
Music indeed is an expression of the many experiences of being alive in every way, thoughts, emotions, expressions, etc. but the creation of music is based in a structured manipulation of the physics of sound that evokes emotion, but has been organized by a system that is largely mathematical in its conception. They are not governed, nor perceived, by the same mental processes. Mathematics and Emotions are left and right brain functions. The right brain perceives the music emotionally, and the left brain perceives the music mathematics. Both the compositional, and perceptual process of music, combine both the creative, and analytical parts of the brain. Compositionally, emotion provides the inspiration, but is the intellect, that forms the emotional content of the music, into cohesive, coherent, musical structures. Scales, chords and arpeggios are the musical structures that constitute the raw material used for composing music.
Emotion provides the inspiration, but is the intellect, that forms the emotional content of the music, into cohesive, coherent, musical structures. It is scales, chords and arpeggios, the musical structures that constitute the raw material used for composing music.
The Interval structures of scales, chords, and arpeggios are rooted in combinations of mathematical ratios. A significant portion of the emotional content of the music, is produced by structured arrangements of Pitches, or intervals that mathematically sound or vibrate together, in a way that sounds agreeable, or disagreeable to the listener. The intervals are derived from our tuning system.
The current system of tuning in Western music is known as Twelve Tone Equal Temperament. The intervals of equal temperament tuning evolved from natural laws of physics. To fully comprehend how Equal Temperament scales came into evolution, one has to look back to some of the common roots that Equal Temperament music shares with acoustic physics. These roots are found in the same natural phenomena known as, “The Harmonic Series”.
Harmonics are a natural phenomenon, generated by vibration. The Harmonic Series is an unchanging, unalterable, pattern of sounds, that are natures physical imprint on the science of acoustics.
Throughout the years Mathematicians and music theorists have assigned various frequencies and ratios the harmonic series to represent musical pitches, and the intervals between the chosen pitches. After considerable experimentation with different combinations of frequencies and frequency ratios, “music” eventually evolved into the twelve pitches and twelve musical intervals, of modern music.
The harmonic ratios produce an audible congruency or agreement between harmonics called ‘harmony.’ The real nature of what we call, “music”, is motion, applied to sound. This motion was generated by using, harmonic ratios. Melody as well as harmony was derived from harmonic ratios. The ratios inspired the distances or intervals between the pitches. A melody moves directionally from one pitch to another, like a connect the dots drawing.
Lines move up and down using different sized lines that connect different points at different altitudes to form a continuous image. A melody is formed in much the same way by connecting together successive pitches directionally with various interval sizes that move up and down in pitch, forming a contoured horizontal line. In this pattern of harmonics, we find the harmonic roots of all of the aural geometries in music.
Music can actually be thought of as the “geometry of sound”.
Even the roots of the words ,”Harmony” and “Arithmetic” had related meanings in Ancient Greece.
“Arithmos” meant units and “Harmonia” meant unity.
Integers represent musical tones and the agreement of musical tones is represented by harmony.
Pythagoras’s discovery of harmonics unified arithmetic and music
The numerical explanation of the universe was a generalization from the discovery made by Pythagoras himself and revealed the numerical ratios which determine the concordant intervals of the scale. Harmonia meant “tuning, scale or octave.” The term, “harmony” is derived from “harmonics.