Monday, November 03, 2014
Almost Perfect: Cosmic Music and Mathematical Ratio
by Yohan J. John
Before the scientific age, boundaries between disciplines were not that sharply defined. Many cultures around the world saw art, music, mathematics and theology as reflections of each other. Perhaps the most poetic expression of this idea was the Pythagorean notion of the musica universalis: the harmony of the spheres. According to this conception, there was a deep link between music and celestial motion: the sun, the moon and the planets danced around the Earth to the tune of an inaudible symphony. The heavenly bodies that could be seen with the naked eye were divided into two categories. There were the fixed stars, which were attached to a spherical cosmic canopy that whirled around the Earth, and the wanderers, the sun, the moon and the visible planets, which changed their position with respect to the fixed stars. Each of these wanderers was assigned its own sphere, so the cosmos was a kind of spherical onion, each layer inhabited by a celestial body that contributed its own note to the universal symphony. The tenor for life on Earth was guided by these cosmic vibrations.
Nowadays it seems this conception of the cosmos is only of interest to hippies, mystics and other fringe folk. Compared to the dizzying scale of modern cosmology, the spherical cosmos seems insular, childish, and unacceptably human-centric. The solar system is now viewed with the sun at the center, and the cosmos is recognized as having no center at all. Or rather, the center of the cosmos is everywhere.
Without in any way questioning the importance, power or beauty of the modern scientific worldview, I think it is possible to dust off the discarded image and learn something from it. Not necessarily something about the true nature of the cosmos, but about how we impose notions of beauty and perfection upon reality, and how reality often overturns these notions, leading us to wider and deeper understanding.
Before we get to the idea of perfection, we can pause in order to just look at the geocentric model. Let's just focus on the example of Venus. How many educated people know what the orbit of Venus looks like from an Earth-centric perspective? There is a popular narrative in science that claims that the geocentric model is just plain wrong, and that it is the Earth that moves, not the sun. But if, as Galileo and Einstein established, all motion must be relative to some frame of reference, then you can pick any position as a stable center and see what the motion looks like from there. Now that the heliocentric model has pride of place, we can look back at the geocentric model purely out of curiosity, and see if there is anything of interest to be found.
And there most definitely is! Compared to the heliocentric view (above), the geocentric view offers us a surprisingly beautiful picture of Venus's cosmic dance. For example, in the time it takes Earth to complete 8 revolutions around the sun, Venus completes 5 revolutions. This relationship between orbital periods leads to a striking pattern that you can see below:
This pentagram pattern, an example of five-fold symmetry, has been known for centuries, and perhaps even millenia. No one can say for certain, but the pentagram pattern may have contributed to the ancient cultural association between Venus and beauty. The pentagram is one of the simplest ways to construct a golden ratio. The golden ratio turns up in a variety of ostensibly unrelated places, and to this day it evokes in mathematically-minded people the notion that God is a geometer.
However, the geometries of the cosmos typically confound our notions of sacred perfection. Notwithstanding the claims of New Age mystics (and Dan Brown), the pentagram that Venus traces around the Earth is not perfect. In 8 Earth years Venus completes approximately 5 revolutions. What this means is that after 100 years, the pattern looks something like the figure below. It still has a kind of five-fold symmetry, but it is far from perfect. And over a period of around 1000 years, the pattern becomes essentially random. There are many cosmic cycles and patterns, and all of them seem to have this quality: they initially tempt us with the promise of "perfection": perfect symmetry. But over time we realize that the cosmos does not share our aesthetic preferences: heavenly bodies tend to drift out of "alignment", demanding further investigation .
What conclusions are we to draw from this? That humans should never project their aesthetic conceptions of perfection and symmetry onto nature? I think this might be throwing the baby out with the bathwater. In general we must start with simple patterns before we can uncover complex patterns. Would ancient astronomers have payed such careful attention to the planets if they presented bafflingly random orbits, aligning with no known pattern? Conversely, would patterns that were extremely simple have led people anywhere? What if all cosmic events were synchronized with the Earth year? Sophisticated techniques for observation and calculation may never have become necessary, and humans may simply have ignored the night sky.
There is no way to be certain of this, but it may be that without a prior expectation of cosmic pattern — cosmic meaningfulness — humans may never have arrived at the scientific revolution. The astronomer Johannes Kepler, for instance, initially attempted to understand planetary motion with the help of classical notions of sacred geometry. He believed in the idea that the planets moved in "perfect" circles, and for some time attempted to show that the orbits of the planets could be predicted using the five "perfect" Platonic solids. He later abandoned such notions of perfection, discovering that the planets move in ellipses, not circles. This in turn allowed him to discover the three laws of planetary motion. Perhaps the geometric ideas were a necessary stepping stone on the road to a truer picture of nature.
To modern readers, sacred geometry can seem like a relic of our irrational past. Ironically, it may also be a relic of an older conception of rationality. The word "rational" is closely linked to the idea of calculation, and is etymologically linked with the word "ratio". Rationality can be seen as the sense of proportion. Some of the earliest mathematical insights came from the study of ratios between small whole numbers. The Earth-Venus pentagram that we just looked at emerges because the two orbital periods have an approximate ratio of 8:5.
Pythagoras and his acolytes were among the first to discover that musical notes depended on simple whole number ratios. The length of a harp string, for instance, determines the fundamental frequency of the note produced. For a given type of string, the shorter it is the higher the note it produces. The musical interval between two notes depends on the ratios of the lengths. If the ratio is 2:1 then the musical interval between the notes is an octave. If the ratio is 3:2 then the interval is called a perfect fifth. In the key of C, the note G is the perfect fifth.
Following Pythagoras, many classical and medieval thinkers assigned musical intervals to particular heavenly bodies, but there was no universally accepted system for doing so. In the 17th century Johannes Kepler attempted to provide one by systematically listing all the musical ratios he could find latent in celestial dynamics. Kepler found that the ratios between planetary orbits did not result in musical intervals, but this did not dissuade him. Instead, he looked for these ratios in the relationships between the diurnal arcs, as viewed with the sun at the center. And he succeeded in finding them. For example, in the relationship between Mars and Earth he found a ratio that was close to a perfect fifth. Despite a heroic effort to reveal the nature of cosmic harmony, Kepler's work ultimately led to the demise of the notion, paving the way for Isaac Newton's theory of gravitation. We may see Kepler's belief that "God has established nothing without geometrical beauty" as quaintly primitive, but it may not be all that different from the belief of a modern scientist who insists, even in the face of seemingly random experimental data, that the universe is fundamentally intelligible, and that the theories that account for reality must not only by true and useful, but also elegant.
Only by giving up on cherished notions of celestial perfection could humanity move beyond the limitations of classical and medieval astronomy. As we shall see, a similar phenomenon occurred in the history of western music. The concept of ratio bridged the gap between numbers and notes, and served as a powerful impetus for the development of musical theory, but eventually it had to be abandoned in order to pave the way for further innovations in music.
To understand how all this unfolded we will need to review some basic music theory. Humans produce musical sounds with their vocal chords, and also with strings, membranes, and a menagerie of cylindrical and conical contraptions. A tone is a sound vibrating with a particular fundamental frequency. Even though every source of musical sound has a mimimum and a maximum frequency, we still have a potentially infinite number of tones with which to construct musical patterns. The human voice, for instance, can move smoothly between frequencies. So can fretless stringed instruments. Musical modes and scales restrict our options to a discrete subset of the continuum between the lowest and the highest available frequencies. These sounds are the building blocks of music: the musical notes. For reasons that are still being explored by musicologists, psychogists and neuroscientists, ratios between fundamental frequencies are the key to understanding human musical perception.
The most basic ratio in music is the octave: it represents an interval in which one tone has twice the frequency of the other. The A above the middle C on a piano, for instance, is typically tuned to 440 Hz. So the next A, an octave up, is tuned to 880 Hz. Two tones that have such a relationship sound similar, so we say they belong to the same pitch class. The subjective similarity between these two different frequencies is called octave equivalence.
Musical pieces consisting only of octaves don't sound dissonant, but they do sound rather boring. To spice things up, other musical intervals are added to create a musical mode or scale. A mode or scale is a way of dividing up the octave so that the intervals between the notes tend to sound pleasant. The notes that divide up the octave can be thought of as the steps on a spiral staircase. Moving up the staircase means moving to a higher pitch. When you start on one note and climb up by an octave, you arrive at the same pitch class, one "floor" up. The other notes of the same pitch class are directly below and directly above you. Choosing the frequencies of the other pitch classes is like deciding where to attach the steps on the spiral staircase.
Pythagorean tuning, which was used in Europe until the 16th century, involves using only the two simplest ratios — the perfect fifth(3:2) and the octave (2:1). Using these ratios we can divide up the musical octave into 12 pitch classes . This turns out to be the easiest way to tune an instrument by ear, since the fifth and the octave are typically the most consonant sounding intervals. The spiral staircase that results from the Pythagorean division involves uneven steps. But fifths and octaves don't quite line up with each other. When we insist on using only our perfect fifths and octaves, some decidely imperfect intervals crop up as well. These intervals are called "wolf intervals", because they end up sounding like a howling wolf. So a consequence of the desire for "rational" — ratio-based — beauty was that it also necessitated a certain amount of ugliness .
In medieval Europe this wasn't that big a problem. Composers initially just kept things simple by restricting themselves to the seven notes of a particular musical key, avoiding the notes that sounded discordant when played together in that key. But there seems to have been inexorable forces pulling composers towards greater complexity. The first of these forces was the gradual development of polyphony: the playing of multiple melodic lines at the same time. When humans sang together, they could use their ability to smoothly modulate, and thereby avoid dissonant intervals in more complex melodies. But this was a luxury that was unavailable to musical instruments, which had to be tuned to particular keys. So if a composer wanted to create pieces that used multiple keys, he or she would have to use instruments specifically tuned to each key, or re-tune the instruments as needed. Moving between keys within a single composition would have been very tricky indeed, and practical only for the simplest of harmonies.
The "perfection" promised by Pythagorean tuning was starting to become a drag on musical innovation, and by the Rennaissance, musicians were seeking a solution. The first widely accepted solution was well temperament, which involved shifting the fifths slightly — rendering them less "perfect" — in order to facilitate more types of harmony. Johann Sebastian Bach was instrumental in popularizing well termperament in the 17th century.
Musical tempering is a compromise: it avoids dissonant wolf intervals by sacrificing perfect intervals. The modern western tuning system — equal temperament — achieves this by creating a spiral staircase consisting of equally sized steps: the intervals between consecutive notes are all the same. This standardized step size is not based on a whole number ratio at all. In order to divide the octave into equal steps, a more sophisticated mathematical concept was introduced: the logarithm.
Equal temperament has brought music into the modern era . It serves as a common musical language that facilitates harmonious interplay between instruments from all over the world, and allows composers to move between different keys within a single piece of music. So western musical theory arrived at its current form by giving up on perfect intervals. By stepping away from one notion of perfection, western musicians found their way to a revolutionary new conception of musical beauty.
The patterns we discern in nature, whether in celestial oscillations or in terrestrial vibrations, seem to tantalyze us with the possibility of a more perfect understanding. Faith in the intelligibility and beauty of nature can lead us to the boundary of a totally new conception of the world. But in order to go beyond that boundary, we may eventually need to transcend the very notions of truth, beauty and simplicity that led us there in the first place.
 It is very common to divide up the octave into 7 distinct notes, with the octave being the eight note (hence the name). Most western musical instruments divide up the octave into 12 pitch classes: these are the notes labeled from A to G (the white keys on a piano), plus the various flats and sharps (the black keys on a piano). Other cultures may use a larger or smaller number of notes: South Indian classical ragas tend to feature 7 or fewer notes out of a possible 22, while an Arabic maqam consists of 7 notes of a set of 24.
 For more on the mathematics of Pythagorean tuning, see this page.
 The revolutionary transition from Pythagorean tuning to equal temperament is explained in this excellent documentary.
The image at the top is from James Ferguson’s Astronomy Explained Upon Sir Isaac Newton’s Principles, and was found at this web page, which belongs to the Grand Lodge of British Columbia and Yukon.
Other images of the orbit of Venus were simulated by me in Matlab.
Posted by Yohan John at 12:35 AM | Permalink