Feature Article

Foundations of Modern Western Music

Ties to Greek Philosophies and Mathematics

by Corey Highberg, June 2018

INTRODUCTION

Modern music foundations and principles can be traced back to Ancient Greek philosophy, mathematical theory, and political discourse. The great thinkers of geometry, arithmetic, social debate, poetry, and art in the Axial Age of History give us all the main ingredients that serve for our eyes and ears in the world of harmony, lyrical content, and basic song structure in the modern era. In 1949 the German philosopher Karl Theodor Jaspers coined the phrase “Achsenzeit” (“Axial Age” or “Axis age” in English) to describe a time between approximately 900 - 200 BCE when “The spiritual foundations of humanity were laid simultaneously and independently and these are the foundations upon which humanity still subsists today.”[1]

This topic will navigate a few of the main subjects that some of those thinkers and poets surfaced, which have evolved over time to our present day. The mathematics of Pythagoras and his core deductions of basic fractions and proportions, the dissertation of Plato and his use of Pythagorean’s studies to produce his own ideas about music, and the work of Socrates and his method of inferences as a product of formal argument to organize thought allows us a reasonable starting point to establish this premise.

PYTHAGORAS

(570 BCE - 495 BCE)

One of the fundamental principles of modern western music is the method of tuning. This can be better described as the agreed upon standard of pitch and frequency that the listener hears when music is played. Eastern music often uses a completely different structure for pitch. This is easily recognizable in the use of quarter tones and divisions of frequency that are not considered common practice in western music. While Eastern and Western music both have foundational principles in Greek tetrachordal structure, Eastern methodology diverts from Western concepts of intervals in many ways. The maqam system as described by Professor Marcus Scott of UCSB is a good starting point for further investigations. Western music is based on Pythagorean Tuning, attributed to Pythagoras (ca. 569 BC - ca. 475 BC). This system is one the first documented tuning system known to society today. “Pythagoras calculated the mathematical ratios of intervals using an instrument called the monochord. He divided a string into two equal parts and then compared the sound produced by the half part with the sound produced by the whole string. An octave interval was produced. Thus, concludes that the octave mathematical ratio is 2 to 1.”[2] This fundamental principle seems so common to us, that it is hardly worth mentioning when we discuss general music theory, but during its conception, it caused a great deal of controversy. Aristotle, Socrates, and Plato all developed entire schools based on questions derived from these basic ratios of pitch division.

The Aristotelian Problems are a list of hundreds of such questions that philosophers spent entire lifetimes contemplating. From Problems XIX.32:

“Why is the ‘dia pason’ so called, instead of being ‘di okto’ or correspond with the number, like the ‘dia pente?’ Is it because in old times there were seven strings?”[3]

This, and many other fundamental questions came from basic ideas of acoustics and tonality derived from the system of tuning that Pythagoras studied. Because of the nature of this division of notes, it gave music certain qualities that had inherent similarities to human logic and discourse that other philosophers that followed him would inevitably argue. The ‘Equal Tempered Scale’ as discussed by Plato, reveals the sensation of tones that lead to each other, (leading tones) and the sensation of “falling” and “rising” by the nature of the pitches it produces.

This simple experiment with ratios and its resulting frequency is the foundation from which the entire harmonic series bases all of its tonal qualities. This series is the sequence of ratios derived from the Pythagorean division of pitches and is responsible for the most basic song structures we hear today in popular Western music. The rock and roll, jazz, classical and electronic dance music that is heard is all a product of this system and the tonal qualities it reproduces. Modern mathematics analyzes this core concept in depth and reveals that all notes are available by reproducing the harmonic series. “The series “sum_(k=1)^infty1/k (1)” is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function 1/x. The divergence, however, is terribly slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).”[4] As modern mathematics describes; Elvis may have been a bigger genius than we give him credit for. The modern punk rock coming through your iPad right now has its roots in Ancient Greek mathematics.

Pythagoras believed that these simple principles were not coincidence, and that “the same mathematical laws governing music operate throughout the cosmos in both the visible and invisible world, even the human soul was a composite whose parts were kept in harmony by numerical relationships.”[5] For Pythagoras, numbers were the key to the universe, and music was inseparable from numbers. Modern western music note divisions and scale structure, through which basic harmony and song derive from, are all based on Pythagoras mathematics and fundamental views of the universe itself. Early Christian writers about music transmitted some of these principles in medieval times in their original form. Others were only rediscovered with the advent of Gregorian chant, and more unearthed in the renaissance by historians. These fundamentals of pitch, harmony, and form are some of the key starting points from which modern music evolves.

PLATO

(424 BCE - 348 BCE)

While Pythagoras helped deduce the basic ingredients to tonality, Plato went on to expound on Socrates’ concepts of traditional beliefs. Socrates, Plato’s great teacher, developed new ways of viewing reason, philosophy, art, poetry, and music. He questioned the gods honored in Greek cities, and in or about 399 B.C.E. he was executed for his discoveries and theories[6]. Much of what we know about Socrates comes from Plato, and later his student, Aristotle.

Plato used many of Pythagoras’ insights to create his theories of human society, harmony, and behavior. His beliefs were representative of Greek conceptions of music, that “The close union between music and poetry”[7] as a measure of this conception and that “for them, the two were practically synonymous.” Plato uses the complex ratios of musical intervals to explain the nature of human unions, the principles of commutativity, a x b = b x a, and associativity, a x (b x c) = (a x b) x c, used as a means to explain “opposite temperaments”. He goes on to say that “Marriages must always be to people of opposite temperaments to preserve framing intervals.”[8] These fractions and mathematics give harmony to these human relationships, and their reproduction through tonality are the embodiment of the human experience.

These ideas are what Plato expresses in his “Musicalized Genetic Theory” and by stipulating double meanings Plato achieves very great mathematical compression whose musical interpretations then functions as a “prelude to the song itself,” the serious study of a philosophy founded on dialectics. This idea of opposing form is what we would today conceive as the basic song structure. The common form of call and response, or ABBA song form, is the evolution of Ancient Greek philosophical study of basic human communication.

To the Greeks, Lycos meant to speak with the lyre, or poetry spoken to music. Plato perceived the general use of human discourse, exchange of thoughts and ideas, and expression of song as a universal foundation of the human experience. From his point of view, “God” is the immovable “1,” the reference point; “2” is “mother” or “receptacle,” symbolized by the undivided circle; and her first “child” is the arithmetic mean between them, the prime number 3 in the octave that has been doubled to avoid fractions. These ideas of concept A, followed by concept B, producing section C, then spoken to music, is the representation of Greek cultural expression and is manifested in modern western music today as our general concept of popular song form. For us, this music has developed into profound expression and arguments that shape our perception of complex society. For the Greeks, as Plato writes, “Our songs have become laws,”[9].

Basic Pythagorean arithmetic explaining pitch, frequency and harmony gave us the ingredients for content, and Plato helped explain how music expresses the common human discourse, marrying pitch and harmony to verse, producing song itself. Plato’s idealism profoundly shaped western philosophy, but Aristotle would be the great authority for many centuries in Europe, second only to the Bible. Aristotle would also have great influence in the Muslim world as well.

ARISTOTLE

(384 BCE - 322 BCE)

To speak of Aristotle is to expound on the teachings of the ones whom he follows. From Pythagoras and the basic ingredients of music, to Plato and his interpretations of harmonic ratios and the relationship to song, come Aristotle. As a student of Plato, “we owe to Aristotle the birth of musicology as a subject in its own right, with the ear - not the number - as the arbiter of events.”[10] He is the Greek philosopher that pulls us from the mundane weeds of mathematical evaluations and emerges us in the poetry and art of music. To Plato, this numerical adventure is where the “games” of his allegories lay for historians to unlock later by means of mathematical analysis. This, he perceived as Plato’s invention, and his intention to encourage study of his arithmetic to those capable in a variety of ways. He modified the Pythagorean notion of soul as harmony and indicated the artistic character of the auditory experience. Music is separated from the intertwining of numbers, ratios, and proportions to flourish in the aesthetic grace of the art that it represents today. When we look under the hood at the physics and properties of tonality, we find the fundamental analysis of the early Greek philosophers, and rising above the rest is Aristotle, shedding the crude edges of literal interpretations to the intuitive sensations of enjoyment and leisure.

Aristotle was a teacher of Alexander the Great. Though his writings were lost after the fall of Alexander’s empire, in the early 12th and 13th centuries, interest in Aristotle's philosophy was revived in Latin translations from Arabic and Greek documents. Only about a third of his original works survive today.[11] Aristotle is responsible for a great many fields of study, but for music, Aristotle cautioned against too much professional training in general music education. He, more than any Greek philosopher recognized the influence and power of song, writing “The right measure will be attained if students stop short of the arts which are practiced in professional contest, and do not seek to acquire those fantastic marvels of execution which are now the fashion in such contests, and from these have passed into education. Let the young practice even such music as we have prescribed, only until they. Able to feel delight in noble melodies and rhythms, and not merely in that common part of music in which every slave or child and even some animals find pleasure.”[12] And even though, as Plutarch writes of him, “Aristotle, the disciple of Plato, regarded melody as something noble, great and divine,”[13] Aristotle recognized that music was no longer intertwined into other schools of thought, but deserved its own field of study. It is Aristotle that pulls the threads of components and theory together to create the school from which modern music gets its true foundations.

CONCLUSIONS

Today, when great messages of cultural importance are conveyed to the masses, they stay in the mind much stronger through song than any other medium. From the American revolution’s “Yankee Doodle Dandy”, to the Beatles songs of revolution in the 60’s, and today’s pop culture expressions of our digital world, music has always been the first place we can look to as a means to understand the cultural times. Western expression of that medium is grown from the seeds of important Greek thinkers like Pythagoras, Plato, and Aristotle. These are the men that gave us the fundamental tools, the scope of human understanding of harmony, and the means to give the gift back to future generations with education. Greek philosophy is truly where the heart of modern western music originates. If you listen closely, you can still tap your feet to its beat.

-CHH, June 2018

References available here