Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Unit 10

Harmonious Math

10.2 The Math of Time


Throughout history, music has played, and continues to play, an important role in many cultures. In some cultures music is a participatory experience, an active art form in which all are encouraged to partake. In other cultures, music is a form of worship or entertainment, to be practiced by relatively few but appreciated by many. Much of the formal western music tradition falls into the latter category. This relationship with music has its roots in the music of the ancient Greeks.

Music served a number of purposes in ancient Greek society. It was an element of religious ceremonies, sporting events, and feasts, and it was part of Greek theatre. In making their music, the Greeks used techniques that are still commonly used today, employing strings, reeds, and resonant chambers to create and control tones and melodies.

One group, the Pythagoreans, took a particular interest in exactly how instruments could be controlled to make pleasing sounds. In Unit 3, we saw how the Pythagorean obsession with all things involving number led to the development of the idea of irrational numbers. Central to this concept was the notion of incommensurability, which holds that certain quantities cannot be related through whole number ratios. Hipassus, a Pythagorean who is traditionally credited with developing this idea, is said to have been drowned for his heretical ideas.

Heresy is an apt term to describe Hippassus’s ideas, because to the Pythagoreans, the synchrony of numbers and music gave rise to a harmony that was considered among the first guiding principles of the universe. They believed in the "harmony of the spheres," the idea that the motions of the heavenly bodies created mathematically harmonious "tones." This numero-musical mysticism was centered somewhat on the idea of whole number ratios, so Hippassus’s claim was not just an intellectual insult, but also a violation of a fundamental philosophy—even spirituality.

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Why was the idea of whole number ratios so appealing? Pythagoras himself was said to have noticed that plucked strings of different lengths sound harmonious when those lengths are ratios of simple whole numbers. For example, if we pluck a string of length 1 meter, and then we pluck a string of half a meter, we will notice that the tones seem harmonious. The half-meter string sounds higher in pitch, but the tone is "the same." The two tones that come from strings whose lengths are in the ratio of 1:2 represent an interval called the "octave." Other ratios also give aesthetically pleasing results. For example, two strings with a length ratio of 3:2, when plucked, create a harmonious interval called a "fifth."

The Greeks were the first to arrange the individual tones that make up these intervals into sequences, or scales. They named these scales, also known as modes, after local geographic regions: Ionian, Dorian, Phrygian, Lydian, Aeolian, etc. These modes were associated with different mental states. For example, the Dorian mode was said to be relaxing, whereas the Phrygian mode was supposed to inspire enthusiasm. The Greek modes are still important in modern music, though many other basic note sequences have been created throughout the centuries.

The idea of what is considered "musical" has expanded over the years, but western music (i.e., music associated with the western hemisphere—as opposed to eastern music) is still built upon the fundamental idea that tones associated with whole number ratios sound good together. It is indeed a mystery as to why our aesthetic sensibilities should favor this system of organizing musical tones. In any case, this early connection between harmony and math set the stage for centuries of fruitful collaboration. Music, as an academic subject, ascended to a special place in the classical education of both Greek citizens and the learned classes of those cultures that would carry on their intellectual traditions.

For example, the "Quadrivium," composed of music, arithmetic, geometry, and astronomy, represented the curriculum of classical education for centuries. Such was the perceived value of musical education in the classical world that it was made one of the four core subjects. However, the musical studies of the Quadrivium focused mainly on the Pythagorean notion of ratios and scales rather than on the performance of musical compositions. Students learned about harmonics and the proportions that would yield pleasing scales and melodies. This focus on the structure of music is closer to what, in the modern age, would be called "music theory."

The Greeks were some of the first people to apply mathematical thought to the study of music.


This was to be a mere prelude to the understandings that future mathematicians would bring to music. One of the most powerful connections to be discovered was that music, and sound in general, travels in waves. The mathematics of sound, of waves, to which we will now turn our attention, will lead us to powerful ways of thinking not only about music, but about many other phenomena.

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Next: 10.3 Sound and Waves


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