February 1, 2015
“Hence arose also that primeval story that the Muses dance day and night before Jove’s altar; and hence comes that ancient attribution of skill with the lyre to Apollo. Hence reverend antiquity believed Harmonia to be the daughter of Jove and Electra, and at her marriage with Cadmus it was said that all heaven’s chorus sang. What though no one on earth has ever heard that symphony of the stars? Is that ground for believing that everything beyond the moon’s sphere is absolutely mute and numb with torpid silence? On the contrary, let us blame our own impotent ears, which cannot catch the songs or are unworthy to hear such sweet strains. But this celestial melody is not absolutely unheard…”
The idea that all matter is connected and directed by the one or various forces of quantum mechanics is not new. A harmony resulting from the pulses of frequency or vibration from the furthest reaches of the universe to the center of our molecular makeup is a concept which may seem a bit removed from our daily experience. But this is exactly what is being explored by today’s leading mathematicians and physicists, building on the basic theoretical principals established by the great philosophers of antiquity.
Some may be surprised to learn of the role music has in all of this. We do not have to look hard to find composers ancient and modern who consciously incorporated into their compositions the same laws of physics that govern the motion of the planets around the sun. And they still do today. In fact, no musician who works with sound can avoid being at a certain level a small picture of the universe.
For one example, musical harmony is not a static collection of intervals chosen because of taste. Although composers have chosen to use harmony in various ways, it has a deep tie to the inner workings of all physical matter. When we look at the way an interval is constructed, we find an amazing, secret world.
According to legend, it started with Pythagoras, the 6th century BC Greek philosopher and mathematician. The legend goes that he discovered the numerical ratios of various sonic intervals by listening to blacksmiths using hammers of differing sizes against an anvil. He supposedly discovered that a hammer one-half the size of another would produce a sound one octave higher. The octave he found to be a 2:1 ratio and the fifth a 3:2 ratio. His measurements proved to be true. If we were to use an oscilloscope to measure the frequency of a pure octave, we would find that the pitch of the upper note of an octave cycles at exactly twice the speed of the lower note and that the upper note of a pure fifth cycles at exactly 1.5 times the speed of the lower note. On a keyboard instrument, if we play twelve contiguous fifths against seven contiguous octaves we land on the same pitch. But in thinking about harmony in such a simple way we run into trouble.
The so called “circle of fifths” is introduced to nearly all students of music theory as a basis for understanding harmony. The idea being that if one were to stack twelve perfect fifths on top of each other, the resulting pitch would be the same as the one which started the series seven octaves higher. However, this is misleading. Taking the ratio of the pure fifth and the pure octave into consideration, we find a problem with the concept of a “circle” of fifths. Ross Duffin provides us with a concise explanation of the fifth/octave discrepancy in his book, How Equal Temperament Ruined Harmony (and Why You Should Care):
A pure fifth has the ratio 3:2, so if we compare twelve of them with seven 2:1 octaves (multiplying the ratios as we add the intervals), what do we get?
[12(3/2)=129.746 and 7(2/1) = 128.0]
That twelfth fifth, the one we expect to complete the circle and bring us back to our starting note, actually overshoots the target pitch and puts us in a ratio of 129.746:128, or 1.014:1 instead of 1:1. In musical terms, it puts us about a quarter of a semitone too high. That may not seem like much, but in fact it’s an excruciating discrepancy called a “comma,” which renders that note unusable as a substitute for a unison or octave.
The fifth (and, therefore, the fourth) are incompatible with the octave. While extensions of the diatonic scale by fifth are close to returning to the same place, they never return close enough to be considered a unison. This means that if intervals are the building blocks of music, the blocks cannot be stacked on each other without falling over.
If your head is spinning, it should be. We have caught glimpse of the infinite “spiral of fifths.” Musicians have been trying to find ways of working with this discrepancy of harmonic ratio since its discovery. Although it is small, this “comma” is the spark that first ignited the art of music, the artful exploitation of attracting and opposing natural forces in a way we can hear.
Johannes Kepler, the German mathematician, wrote about these same musical ratios in his Harmonices Mundi (finished in 1619). However, instead of dealing with hammers, he suggested that harmonic ratios could be measured in the angular velocities of the elliptical orbits of planets around the sun. This directly connects the sounds we hear in music to the motion of the celestial bodies complete with consonances, dissonances, and an astronomical “comma” that keeps things spinning.
Imagine a universe where everything fit together perfectly with no tendency toward motion. Perhaps there would be no universe. There would certainly be no music. Music is a very special part of our human experience because it allows our limited senses to “catch the songs” of Milton’s “celestial melody” in the constant motion of our universe.
Join the Dallas Chamber Music Society as they present the Music of the Spheres Society featuring Stephanie Chase (violin), Jon Manasse (clarinet), and Jon Nakamatsu (piano) performing works by Bernstein, Brahms, Bartók, and Novacek. Monday, February 9, 2015, 8:00PM at Caruth Auditorium.
-Zachariah Stoughton, Contributing Writer