Method Sampling and the emergence of new systems 3/n
“God is a geometer and employs geometry in all his works.”
- Plato (The Republic)
Following the completion of my master’s program, I began to re-examine the relationship between I Ching Harmony and tonal music. By then, I had already used the I Ching Harmony in a number of pieces and the technique had settled in the toolbox of my writing. However, there was always this lingering question — why did these two seemingly disparate systems have such a fundamental commonality? Was it that I invented an answer based on the misunderstanding of the two systems? Or was there some hidden grand unifying truth I happened to uncover?
It seemed to me neither of these hypotheses entirely explained the situation. On the one hand, I certainly was sufficiently familiar with both systems to identify the points of similarities/commonalities in them. On the other, the embedded connection between tertian harmony and I Ching diagrams was my assertion. After a bit of back and forth between the polarity, my current position emerged; it is both. The process of this “sampling” requires the practitioner to be able to distinguish the useful principles in a foreign system and make them fit into one’s own discipline by “reframing” them. Furthermore, because the principles or “methods” one samples from another field almost always fail to make one-to-one translation, they trigger change or innovation in the sampler’s own field.
Even after recognizing the process of how I ended up with I Ching Harmony from chord scales, I Ching cosmology as well as tertian structure — it felt I hadn’t yet fully arrived at the answer. Much of my thoughts were still fragments revealing only partial solutions. However, the intuition that there must be a recognizable pattern within a given system for the kind of synthesis like I Ching harmony to arise, guided me through my inquiries.
During this period, I started a bit of reading on the ancient Greeks, in particular the philosophers before the sophists and Socrates. They’re called, not without disputes, Presocratics and include such thinkers as Thales, Anaximander, Anaximenes, Heraclitus, Pythagoras, and more. Aristotle called them physikoi (physicists) as they were concerned with the structures of the natural world and its arche (first principle). This was precipitated by another “spontaneous insight” which came to me in addition to what I remembered from the early theory books, namely, Le istitutioni harmoniche (Harmonic Institutions, 1558, Zarlino, translated by Guy A. Marco & Claude V. Palisca).
Some of the excerpts that struck me in the volume go as follows —
“Speaking universally, music is nothing but harmony, and we may say that it is that opposition and agreement from which Empedocles proposed all things were generated; it is concord of discords, meaning a concord of diverse things that can be joined together.”
-Gioseffo Zarlino (Le Istitutioni harmoniche, Part I, Chapter 5)
Below, we see the notion of harmony as a pervasive force governing the universe, a distinctly Pythagorean idea.
The universal music, or musica mundana, coordinates the spheres of the heavens, holds together the four elements — fire, air, water, and earth — and organizes time and the seasons. (Le Istitutioni harmoniche, Part I, Chapter 6).
Human music, or musica humana, is the force that knits together the parts of the soul and parts of the body and maintains the harmony of the soul and body (Le Istitutioni harmoniche, Part I, Chapter 7).
This “Pythagorean” theme continues on Gradus ad Parnassum (Steps to Parnassus, 1725, J. J. Fux, translated by Alfred Mann) which dedicates an entire book to the proofs on why and how these harmonies become pleasant or consonant to us with painstaking details of mathematics.
By this time, I was convinced that reading about Pythagoras and other Presocratics would reveal the golden thread that could guide me out of the labyrinth of suppositions. Indeed, the findings on the thoughts of “physicists” were compelling.
Presocratics & Taoists
For instance, Thales concluded that the first principle of the world (arche) was water. In I Ching/Taoist cosmology, all natural things exist as a result of the transformation of water (The Yellow Emperor’s Inner Classic, 黃帝內經). This “transformation of the water” is expressed in the creation cycle of the five elements; water nourishes wood (水生木), wood feeds fire (木生火), fire produces earth (火生土), earth bears metal (土生金), metal creates water (金生水).
The concept of the apeiron, or the unlimited, posed by Anaximander who succeeded Thales at the Milesian School, greatly resembles that of Wuji (無極), the nothingness which is unbound and the origin of all things.
Heraclitus and the doctrine of flux, that “it is not possible to step twice into the same river or to come into contact twice with a mortal being in the same state (Plutarch)”, is in line with the “constant changes” that I Ching describes in nature and human affairs. Furthermore, Heraclitus expounds on how the changes of the contraries or the opposites are linked (unity of opposites).
“As the same thing in us are living and dead, waking and sleeping, young and old. For these things having changed around are those, and those in turn having changed around are these.”
The two opposites, which are parallel to Yin & Yang (陰陽), are originated by Logos (Word), a rational order that governs all things.
“This world-order, the same of all, no god nor man did create, but it ever was and is and will be: everliving fire, kindling in measures and being quenched in measures.”
The similarities between Presocratic thoughts and I Ching made me realized that the connection between the structure of tertian harmony and the diagrams of I Ching might not be a mere coincidence.
Pythagoras & Harmony of the Spheres
Though Heraclitus called Pythagoras, “the chief of swindlers” or “teacher of lies”, one can’t deny the extensive intellectual influence of Pythagoras in the West. From music to quantum mechanics (string theory), Pythagorean ideas on the first principle of the world, numbers & their expression through mathematics, persisted throughout the history from antiquity to modern world.
One particularly significant Pythagorean idea for our discussion is Harmony of the Spheres.
“That is produced,” Scipio Africanus replied, “by the onward rush and motion of the spheres themselves; the intervals between them, though unequal, being exactly arranged in a fixed proportion, by an agreeable blending of high and low tones various harmonies are produced…. Therefore this uppermost sphere of heaven, which bears the stars, as it revolves more rapidly, produces a high, shrill tone, whereas the lowest revolving sphere, that of the Moon, gives forth the lowest tone; for the earthly sphere, the ninth remains ever motionless and stationary in its position in the center of the universe.”
“Men’s ears, ever filled with this sound, have become deaf to it; for you have no duller sense than that of hearing…. But this mighty music, produced by the revolution of the whole universe at the highest speed, cannot be perceived by human ears, any more than you can look straight at the sun, your sense of sight being overpowered by its radiance.”
Cicero, De Re Publica 6.18–19
It is said that Pythagoras passed by a blacksmith’s workshop and noticed different pitches were generated by different sizes of the hammers as opposed to force of the strikes. When he returned home, he worked out the ratios of these different tones; unison 1:1, octave 2:1, fifth 3:2, fourth 4:3… etc.
Then in a flash of genius, he realized that all things in universe are vibrating in a perfect harmony, including the celestial bodies; some think this model was heliocentric while others think it was geocentric. The details of this “harmony” is described in the above Cicero excerpt, Somnium Scipionis (Dream of Scipio).
A brief recapitulation goes as follows:
A. The distances between celestial bodies revolving around the center (earth or sun) are mathematically proportionate.
B. Some bodies move fast and some slowly thereby creating lower and higher pitches.
C. The notes corresponding to the ratios of the distances between bodies make the resultant sound concordant.
This is a profoundly elegant and inspired idea and many great thinkers following Pythagoras either accepted it as the truth or used it as a starting point in further developing models of planetary motions as well as geometry and music.
Johannes Kepler
Among such great men was Johannes Kepler. Kepler, a brilliant mathematician, was hired by Tycho Brahe, a wealthy astronomer who had the most accurate astronomical data available at the time. Brahe asked Kepler to define the orbit of Mars. Meanwhile, it turned out that he withheld much of the observational data from Kepler. In some ways, Brahe was jealous of Kepler’s aptitude as a mathematician but more importantly, he did not want Kepler to use the data to prove his own geocentric (earth as the center) model wrong.
Then Brahe passed away from a burst bladder; poor Tycho apparently forced himself to stay at a royal banquet for hours fighting the urge to urinate for politeness. This allowed Kepler to finally get a hold of the entirety of Tycho’s data and use them to develop his three laws on the planetary motions.
One of the most significant breakthroughs Kepler had was his realization on the elliptical orbits (the first law). Kepler noticed that planets moved quicker when closer to the Sun and from this, deduced that the orbit of Mars was elliptical. This was a departure from the consensus at the time that celestial motions must be in a circle following the long held belief that circle was the most perfect shape, thus heavenly bodies move in accordance with it.
In 1619, Kepler published Harmonice Mundi, his magnum opus discussing far ranging topics such as geometry, music, astronomy, astrology, and his third law of planetary motion. This was his attempt on a grand unifying theory of the universe, improving upon the Pythagorean idea of “harmony” or congruence of the celestial and terrestrial bodies. Though Pythagorean in its general ethos, in Harmonice, Kepler abandoned Pythagorean tuning and employed geometrically supported musical ratios applying them to each planet to render the “celestial scales”
Kepler proceeded to assign actual notes to each of the planets at aphelion and perihelion and found that when he built with Saturn (the lowest note) at aphelion, the result was a durus scale, a major scale. With Saturn at perihelion the result was a mollis scale, a minor scale. Planetary motion apparently did involve both type of scale. Using other planets as the starting note produced the different modes used in ancient music and church music.
- Kitty Ferguson (The Music of Pythagoras)
He then worked on generating music using the different planetary modes. The idea was to have them played all at the same time much like the constant and simultaneous celestial motions. All the trials ended up in cacophony.
The Spontaneous Insight
Let’s now talk about the event that compelled me to read about the aforementioned personalities and their ideas.
While pondering on the cause of commonality between I Ching diagrams and tertian harmony, I looked at the major scale and contemplated on the functions of each chord based on the scale degree. Nothing came to me at first. The next day, however, it became evident that there was an underlying geometric design resembling the solar system that governed the functions of the chords. The way I realized this is presented below; I saw geometric shapes from the known facts regarding functions of the harmonies in tonal music.
My discovery is summarized below:
- The distances between tonic, dominant, and subdominant are all perfect fifths, thus creating a macro orbit in a “perfect circle” which holds the tonality together. Tonic is located in the middle. Dominant is a perfect fifth above the tonic, and subdominant a perfect fifth below. To paraphrase Vincent Persichetti from his book Twentieth-Century Harmony: Dominant and subdominant degrees are a perfect fifth apart from the tonic, and thus serve to balance it.
2. The four remaining degrees (super tonic, submediant, mediant, and the leading tone) are all minor or major thirds apart from the three main degrees (tonic, dominant, subdominant) revolving around them as satellites in “elliptical orbits”. Chords built on these degrees can be used as functional substitutes for the three main chords: tonic, dominant, and subdominant. For instance, a chord built on the seventh scale degree (leading tone) can substitute for a dominant chord and a chord built on the sixth scale degree (submediant) can substitute for a tonic chord.
This revelation combined with the recollections on Zarlino’s and Fux’s works detailing geometric analogies and mathematical proofs, glimmered like Ariadne’s threads. As I pulled them through with the help of Presocratics and Kepler, I started getting more and more convinced that the geometric model embedded in the functional harmony I discovered could in fact be, Harmonia Mundi. This made me think of Alexander Scriabin and his mystic chord. Scriabin was also convinced that he had discovered the chord (“chord of the pleroma”) that held the universe together as well as the totality of our being.
“I am a moment illuminating eternity…I am affirmation…I am ecstasy.”
“I am God! I am nothing, I’m play, I am freedom, I am life. I am the boundary, I am the peak.”
He in addition claimed that he could levitate, which I haven’t yet convinced myself to do so.
With the joke aside, I believe Scriabin had failed to understand that Harmonia Mundi cannot be “a chord”. A chord is a single sound and does not reflect the constant changes of the “multitudes of harmonies” that are being produced by the celestial bodies. One should understand that the model of Harmonia Mundi must use geometry that mirrors the greater universe to which a single sound or planet is connected, much like Kepler’s model. In short, any explanation or proof of Harmony of the Spheres that doesn’t present the relationship of multiple sounds can’t be an adequate answer. Therefore, it makes sense to propose that “a model” of Harmonia Mundi lies within the function of the tonality itself, which deals with the regulated (mathematical) distances of tones that are in motion.
Sampling & Reframing
We shall take as a starting point for the discussion the definition of unison. It is a combination of two or more equal sounds or pitches that do not form any interval but are contained in one point and place [as on a string]. It will be found wherever the proportion is that of equality; that is 1:1, or 2:2, etc. This proportion (as I have said elsewhere) is the basis of inequality. Equality is never found in consonances or intervals, and the unison is to the musician what the point is to the geometer. A point is the beginning of a line, although it is not itself a line. But a line is not composed of points, since a point has no length, width, or depth that can be extended, or joined to another point. So a unison is only the beginning of consonance or interval; it is neither consonance nor interval, for like the point it is incapable of extension.
Gioseffo Zarlino (Le Istitutioni harmoniche Part III, Chapter 11)
The above excerpt, I believe, shows a clear evidence of the geometric rationale that was used to develop Western European harmony in its early stages. For some, this would be a proof that numbers expressed as geometries govern many things in the world including music. For me, it means something entirely different — that in order to develop a system out of something abstract (sound), one needs something concrete (geometry/numbers) to rationalize it.
In other words, to build the harmonic systems or “harmonic institutions” (Istitutioni harmoniche), Medieval and Renaissance theorists have borrowed or “sampled” the principle of geometry.
By this time, a clear pattern emerged. Any kind of significant innovations or system building to take place, there must be an employment of (a) foreign rationale(s) or principle(s) that reframe(s) the existing field.
Slowly but surely, an idea was developing inside of me: a new model of music that utilizes contemporary idioms combined with natural principles of organization and the infrastructure supporting those principles.
A few years after this realization, a culminating event that forced me to break from the aesthetic dogma of concert music occurred in the form of hip-hop, a genre I never even dreamed of listening to.
