Method Sampling and the emergence of new systems 2/n
Gradual & sudden changes
Natura non facit saltum (nature doesn’t make leaps). As the dictum goes, the aesthetically flat environment of Berklee gradually built my tolerance of non-classical music and elements of popular music started to appear in my compositions. What I thought was a superficial change in my music because of being surrounded by the “raw materials” of various pop genres (i.g. customary harmonic progressions, genre specific rhythmic motives… etc.), was in fact a deeper structural transformation occurring within me — one so incremental that I wasn’t aware of the drastic impact it had in my thinking until it was conspicuous.
Amidst the imperceptible subterranean changes, an unexpected realization took place. As I flipped through the pages of the harmony (yes, the Berklee jazz harmony) textbook looking at the chapter dealing with the four possible qualities of triads*. I noticed that the shape of the triads and construction of their intervals (distance between two tones) were oddly similar to the structure of the bigrams in I Ching (易經). After examining them more closely, I concluded that they weren’t just similar but identical. *triad (a chord built with three notes that are stacked up in 3rds).
Perhaps a little background on I Ching is in order before discussing the unlikely connection between I Ching and the tertian harmony. I Ching is an ancient divination text that is part of the essential corpus of Confucianism called Four Books and Five Classics (四書五經). It was written by multiple authors including the legendary ruler Fu Xi (伏羲) who had the arrangement of the trigrams revealed to him by Yellow River Map (河圖) and Lo Shu Square (洛書) via two mythical creatures (a dragon horse and a tortoise) emerging from Luo River. It is said that Fu Xi realized the five elements’ creation and destruction cycles by looking at the patterns on the dragon horse and the giant tortoise.
The second author was King Wen of Zhou (周文王) who put the eight trigrams together to render 64 hexagrams and wrote judgements on the hexagrams during his three year imprisonment by the last ruler of Shang Dynasty. It is said that King Wen observed the stars through the window of his cell and came up with the decisions/judgements of the hexagrams.
The third and most significant author, in my opinion, was Confucius. He transformed I Ching into a major work of philosophy and cosmology by writing ten commentaries called Ten Wings (十翼). There were two other classical divination texts besides I Ching and the reason I Ching became the relevant source of reference/inspiration to many philosophers and scientists until now, was Ten Wings and their comprehensive explanations on the judgements with detailed annotations as well as exploration into greater laws that are embedded within the “changes”* of the hexagrams. *I Ching is translated as “Book of Changes”
I Ching harmony
Now, the following are details of my discovery:
In the cosmogony of I Ching, the beginning of everything is represented by nothingness, a nothingness that is unbounded and has no shape, color, or sound. This is called Wuji (無極), roughly translated as “without ceiling”. An important aspect of this nothingness is that it’s not actually nothing, for nothing comes out of nothing (Ex nihilo nihil fit). In fact, it’s more like the Pre-Socratic Anaximander’s idea of apeiron where the space is filled with boundless potentiality, perpetually generating fresh materials.
From this “nothingness”, or Wuji, emerges the ultimate creative energy or Taiji (太極). Or, more specifically, Taiji forms within Wuji.
From Taiji, Yin and Yang (陰陽) energies are divided. When Taiji moves, it creates Yang (陽); when it rests, it creates Yin (陰). To put it another way, the act of movement itself is part of Yang, and the act of rest is part of Yin.
From this initial division, more subtle permutations arise by the pairing of Yin and Yang energies into four bigrams.
From these four bigrams, eight unique trigrams or Bagua (八卦) are created by the addition of either Yin or Yang energy to each bigram. These trigrams represent certain natural elements including: Heaven (乾), Earth (坤), Fire (離), Water (坎), Thunder (震), Wind (巽), Lake (兌), and Mountain (艮).
Finally, combinatorial pairing of the eight trigrams creates 64 hexagrams. These hexagrams represent various stages of human and natural affairs.
Amazingly, these very same divisions and multiplications appear within the basic structure of tertian harmony in Western classical music.
If there is no sound, we have a rest.
From this rest, emerges a single tone.
From this single tone, the interval of a minor third or a major third can be created. The third is selected because, unique among the intervals, it confers a definable quality to the sound to which we have an innate reaction. Major thirds convey a sense of light, warmth, and happiness whilst minor thirds evoke dark, coldness, and sadness.
To these minor and major thirds, another minor or major third can be added, creating the four possible triads: diminished, minor, major, and augmented.
The addition of a third minor or major third to each of the triads generates the seven seventh chords. The slight discrepancy here between tertian harmony and the I Ching (i.e., seven seventh chords versus eight trigrams) is due to the repetition of tones. Specifically, the augmented triad with another major third creates a chord where the root is doubled. It is interesting that the structures of the triads, the most basic building blocks of Western European music, recapitulate exactly those of the four bigrams (some of the most foundational energetic components of the I Ching). Indeed, triads and triadic composition were considered to be “the most perfect of all” by many early composers and theorists, among them J. J. Fux, whose Gradus ad Parnassum was revered as a seminal work on the counterpoint by generations of musicians.
A constant within change
Once I realized the connection between the symbols of the I Ching (bigrams, trigrams, and hexagrams) and tertian structure, it was evident to me that one can generate music using the energy principles of I Ching. As I discuss below, this system differs fundamentally from previous attempts to use the I Ching as a tool to generate musical materials such as those employed by John Cage and others. While Cage would toss coins to get assigned numbers of the 64 hexagrams that are prefigured with different parameters to “compose” the pieces dictated by chance, my method relies on deeper rationales of the diagrams and their inevitable progressions dictated by the energy principles expounded in Ten Wings (十翼). It is also strictly a harmonic principle as opposed to Cage’s way of coming up with other elements such as meters, rhythms, and tempi.
Bigrams represent the gradation of Yin and Yang energy manifestations in the physical world. One of the simplest ways to visualize this is by the example of seasonal changes.
Lesser Yang (少陽), which is expressed with Yin (- -) line on the bottom and Yang ( — ) line on top, symbolizes spring. It shows that on the basis of Yin, small Yang is growing. Spring is cold but grows warmer with the rising power of the sun and the approach of summer. Greater Yang (太陽), made up of two Yang lines, represents summer as the Yang energy is at its peak. Lesser Yin (少陰), which consists of a Yang line on the bottom and a Yin line on top symbolizes autumn. In autumn, the weather begins to cool. Finally, Greater Yin (太陰), made up of two Yin lines, represents winter. As these diagrams represent seasons, among other things, they could be placed one after the other following the order of seasonal changes. Moreover, since they can be translated into triadic structures, we can create movements of sounds, or chord progressions based on the energy principles of the I Ching. For instance, Greater Yang to augmented triad, Lesser Yin to major, Greater Yin to diminished, and Lesser Yang to minor.
Ancient Chinese philosophers theorized that the natural world is made up of five distinct elements. These include: Wood, Fire, Earth, Metal, and Water (木, 火, 土, 金, 水 ). From this observation they expounded the Theory of Five Elements (Wuxing, 五行) detailing how each of these five elements interact in cycles of creation and destruction.
The Creation Cycle:
Wood feeds Fire (木生火)
Fire produces Earth (火生土)
Earth bears Metal (土生金)
Metal carries Water (金生水)
Water nourishes Wood (水生木)
The Destruction Cycle:
Wood parts Earth (木剋土)
Earth absorbs Water (土剋水)
Water quenches Fire (水剋火)
Fire melts Metal (火剋金)
Metal chops Wood (金剋木)
In Taoist philosophy, “shapes of things (物象)” is the fundamental units of natural and energetic manifestations that has been abstracted from the observable phenomena. Early Taoists expanded the Theory of Five Elements by correlating each of the trigram, which represent natural “shapes”, with one of the five elements.
Correlation of the trigrams with the Five Elements and chords:
Heaven/Metal (乾/金): Augmented triad with the repeated root
Lake/Metal (兌/金): Augmented major 7th
Thunder/Wood (震/木): Dominant 7th
Fire/Fire (離/火): Major 7th
Wind/Wood (巽/木): Minor major 7th
Water/Water (坎/水): Minor 7th
Mountain/Earth (艮/土): Half-diminished 7th
Earth/Earth (坤/土): Diminished 7th
Like the bigrams, all of these symbols can be translated into tertian structures (in this case, seventh chords), and they create musical materials available to a composer that follow the creation/destruction cycles of the five elements as well as seasonal changes inferred by the four bigrams. For example, each season corresponds to a certain five elements: spring to wood, summer to fire… etc.
Translation of hexagrams into tertian structures works exactly the same way as described above for the bigrams and trigrams, the only difference being the number of notes involved. In this case, seven notes are prescribed by each structure (except the augmented and diminished chords), covering the upper extensions of the four triads. These could be treated as thirteenth chords or, perhaps more usefully, as seven note modes.
Another method of translating the hexagrams into musical structures is achieved by regarding each of the lines not as a major or minor third, but as a half tone or a whole tone, with Yin lines representing half tones and Yang lines representing whole tones. This process allows us to create unique seven-note scales. However, this method doesn’t allow the vertical and horizontal unity. The lydian scale in the above example will result in hexagram 9 as opposed to hexagram 63. Given my tendency to optimize, I decided not to pursue this route and opted to focus on further developing the method using connections between vertical harmonic structures and the diagrams.
Once the translation of the symbols to musical tones has been completed, the composer can read Ten Wings (十翼) to gain deeper insight into the energy manifestation represented by each symbol and its relationship to one another. This will allow for the creation of comprehensive progressions and permutations in compositions using I Ching’s cosmological and philosophical rationales. Furthermore, as the system of I Ching relies on analogy rather than formal logic, the interpretations of its principle are inherently personal, which allows the application of the system produce distinctive variations.
As some readers might have realized, the way I made the transition from hexagram to scale was inspired by the principle of chord scales in jazz. The “I Ching harmony” in fact, was an amalgamation of I Ching cosmology, tertian harmony, and chord scales. This way of selecting elements from multiple systems to build a novel structure was something that came naturally to me. However, it took a sudden and almost violent realization to exactly pinpoint the mechanism of this way of thinking; to see such patterns play out in the world at large and to call it, Method Sampling.
To be continued
