The Sound in Music

Around the world, the beautiful effects caused by music continue to captivate and inspire people to dance, imagine, and view the world through a different frame of mind. Such simultaneously Dionysian and Apollonian delights find their basis in the discovery of physical properties set into place in the world around us. Ancient Greek philosopher and mathematician Pythagoras (±570–±495 B.C.) highlights meaningful human discovery in his creation of the Pythagorean theorem. His contributions to modern music find their basis in a systematic exploration of the way ratios control pitches in music as on a vibrating string. In this essay, I will describe the history of the development of the Pythagorean scale and examine how it operates along with a brief analysis of how the scales are used in tuning. While the captivating nature of music remains irrefutable, the physics of their operation as described the Pythagoras merit careful exploration.

To fully appreciate the work of Pythagoras, one must understand the values held in ancient Greek culture. Fundamentally important to the discoveries made this great thinker is the knowledge of the value of music, standing equal in value alongside arithmetic, geometry, and astronomy as the core subjects of the day (Maor 19). Music was governed by a doctrine of ethos, which stemmed in the belief that music had the power to shape the beliefs of the mind, the development of the body, and the state of the soul. Pythagoras himself viewed music as a tonic boosting overall health when used properly, carrying a potential to sooth anger and malice or even inspire men to war (Stanley 8). As such, it merited a full examination comprising the well-developed research of any talented polymath.

According to modern understandings, Pythagorean scale development extends beyond the history of Pythagoras himself. John Fauvel relates the anecdote in easily comprehendible terms, stating that the Pythagorean theorem came into existence as the result of Pythagoras observing "the harmonious relationships of the sounds produced by the hammer in a blacksmith's forge" (14). From this seemingly serendipitous realization, Pythagoras came to the conclusion that he could change the mass of the hammer based on whole number ratios in order to alter pitches and find consonant sonorities. These discoveries were then perpetuated through the writings of a Roman scholar named Boethius (480–524 A.D.) who proceeded to describe them in mathematical detail.

From the beginning, Pythagorean scales stood as valid models for practical application even as organum modes of singing developed into the Renaissance period, especially in the use of what Pythagoras called the "perfect intervals," of the fourth, the fifth, and the octave (Maor 19). James Barbour describes theorists who insisted upon the Pythagorean model of tuning centuries after tempered methods such as "Equal Temperament" and "Just Intonation" were already in play (3). In fact, even the theoretician Alexander Malcolm of the eighteenth century advocated actions to "tune not only . . . Octaves, but also . . . 5ths as perfectly Concordant as their Ear can judge, and consequently make their 4ths perfect, which indeed makes a great many Errors in the other Intervals of 3rd and 6th" (Barbour 3-4). Today, we can appreciate that Pythagoras's system remained relevant even in the face of evolving ideas concerning proper tuning methods.

The basic functions of Pythagorean scales are most easily conceived in the conceptualization of a string. When affixed between two adjoining points, it produces a certain pitch, p, based on the vibrations of the string. By dividing the string using ratios of 2:1 and 3:1, we obtain basic frequency possibilities, as illustrated below (Fauvel 15):

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These divisions allow us to create the perfect intervals outlined by Pythagoras: the octave, the perfect fifth, and the perfect fourth. The mathematical function of the upper illustration comes in the form of 2t vibrating twice the speed of p and creating an octave above interval while the bottom illustration, 3t, vibrates three times the speed of t. How do these divisions contribute to the formation of a "perfect" scale? With the increase in the basic divisions between 3t and 2t, a new ratio of 3:2 is created, otherwise known as 3/2t. Therefore with t representing our tonic and base octave, 3/2t resonates a perfect fifth above t and perfect fourth below 2t (Fauvel 15). Thus begins the basic framework for modern scalar construction.

As such, Fauvel goes on to provide the basic frequencies and intervals as outlined in the table below (16). The more complex ratios required, especially in the use of the second and seventh intervals, make it clearer to understand why Western cultures have historically valued the octave, fourth, and fifth above all other intervals.

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Although the intervals vary between single notes (C-D, D-E, etc.) in this modern version of the scale, the result of this process of division is completely based off of the 3:2 ratio.

By sticking to a process of powers, we can utilize ratio calculations that respect the intervals of the Pythagorean scale and remain exclusively with octaves, perfect fifths, and perfect fourths. "Thus, each note in the Pythagorean scale can be written simply as 2p·3q where p and q are integers . . . and [we] omit the factor t" (Fauvel 17). In following this method that respects the ratios of two, we find a foundation for the division of the Pythagorean octave on a piano as shown below (Fauvel 17).

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As summarized by Barbour, by "using the ratios of 2:1 for the octave and 3:2 for the fifth, it is possible to tune all the notes of the diatonic scale in a succession of fifths and octaves . . . [thus employing a] rigid, mathematical principle" (1). Through this systematic shortening of strings, the basis for modern tonality as viewed in the history of Western classical music emerges with such composers as Joseph Haydn.

Although Pythagoras's role in generating the system of tuning after his namesake remains reasonably disputed, his systematic mathematical description and contribution to codifying the Pythagorean scale cannot be diminished, with its merits having been defended even millennia after his passing. By basing the ratios of 2:1 and 3:2 in the division of strings, we find a fundamental understanding for tuning that remains in our modern employment of music in spite of alternative temperament methods. By studying this point view, we come to a physical understanding of how music shapes our attitudes and temperaments in the purest of forms of the fourth, fifth, and octave with a historical context that lends weight to our modern understanding of music and its broad gamut of emotions.

Works Cited

Barbour, J. Murray. Tuning and temperament: a historical survey. Mineola, N.Y.: Dover Publications, 2004.

Fauvel, John. Music and mathematics: from Pythagoras to fractals. Repr. ed. Oxford: Oxford Univ. Press, 2006.

Maor, Eli. The Pythagorean theorem: a 4,000-year history. Princeton, N.J.: Princeton University Press, 2007.

Stanley, Thomas. Pythagoras: His Life and Teaching, a Compendium of Classical Sources. Lake Wood, FL: Ibis Press, 2010.