2C The Legend of Pythagoras
Pythagoras is a figure of legend. He was roughly contemporaneous with three equally legendary teachers: Lao-Tse in China, Gautama Buddha in India and possibly Zoroaster in Chaldea/Persia (present-day Iraq). There is little question that all four of these figures were historical—although there remains considerable uncertainty as to the dates of Zoroaster—and the spiritual traditions that sprang from their teachings (Hellenist philosophy, Taoism, Buddhism and Zoroastrianism , respectively) continue to resonate in the contemporary world. At the same time it is no longer completely feasible to separate what these great teachers may or may not have said in their day, and who they actually were as human beings, from what tradition now assigns to them. Did Lao-Tse really write the Tao te Ching, or does the Tao te Ching represent a long tradition of Chinese philosophical thinking that came to be associated with Lao Tse? Did Buddha teach the full range of spiritual teachings that have flowered into the many forms of Buddhism present in the world today, or did his teachings change and evolve as they mingled with other cultures and later philosophical and spiritual disciplines? Were Zoroaster’s teachings about the duality of the universe in terms of light/dark original to him or were they expressions of older traditions? Similarly, did Pythagoras actually invent the system of thought now associated with his name, or does he represent a long process of assimilation of both musical and spiritual teachings that drew from Egyptian, Persian(Chaldean), Indian and Chinese sources?
However much the Pythagorean tradition is original with Pythagoras, it is nevertheless the case that musical theorists and philosophers, from Plato in the 5th Century B.C.E. to Boethius in the 6th Century C.E. and through to the Renaissance, were deeply steeped in what they believed to be “Pythagorean thought.” Even the more rational theorists of the 18th and 19th Centuries remained aware of his historical legacy, and there is a tradition in 19th Century France that reflected a revival of the so-called “occult” aspects of Pythagorean thought, a tradition that was not lost on the minds of such seminal figures as Claude Debussy and Eric Satie. Two of the central books of 19th and early 20th century music theory — Helmholz’s On the Sensations of Musical Tone (discussed below) and Schenker’s Harmony, assume a Pythagorean stance. And there is a renewed fascination with his teachings among musicians today. The work Harmonic Experience< mentioned in the introduction to this book, is deeply influenced by its investigation of Pythagorean principles. Thus, to the extent I too have been influenced by the Pythagorean tradition through my readings in music history and the history of music theory, this textbook, and now you too, are part of the Pythagorean tradition. Welcome to the club!
Thus, we begin our study of music theory with the man who stands at the source of Western musical thought, who absorbed and codified a prior music theory that was in play for millennia before him throughout the near and far East.
Basic Summary of Pythagorean thought
The study of Pythagorean thought was known to every educated person until recent times. Since the music we are studying is rooted in this assumed knowledge, it is important to understand this thinking in order to understand the roots of music theory.
I quote from the philosophy Arthur Koestler, who in referring to Pythagoras, calls him “…the maestro Pythagoras of Samos, whose influence on the ideas, and thereby on the destiny, of the human race was probably greater than that of any single man before or after him.” This is an extremely hyperbolic statement, one I’m not sure I totally agree with! Yet it does point out the importance to which so many great thinkers in our time grant to the philosopher from Samos.
His thinking clearly derived, in part, from other sources (Egypt, Mesopotamia, Persia, India). One claim is that Pythagoras studied in Chaldea (current Iraq) with Zoroaster, which is fascinating, if debatable, considering that scholars are unable to pinpoint Zoroaster’s dates with any certainty. Other traditions speak of time spent in Egypt, where he learned from their speculative traditions. Even if not factually accurate, it demonstrates that within the Pythagorean tradition, there is an understanding that he claimed his knowledge, in part, through the study of spiritual cultures that preceded him.
He called himself a “philosopher,” meaning a lover of wisdom. Indeed, it is said that he coined the very word. He led a colony of followers and was looked on as a spiritual teacher, not simply a mathematical or musical one.
Like Zoroastrianism and Taosim, his philosophy is based on the concept of dualism – the ordering of the universe into opposing (but mutually inextinguishable) forces: light/dark, limited/unlimited, full/empty, life/death and so forth—all of which are born from and related to the one, the monad, the unity.
This primordial harmony and order of the universe is expressed through number.
Numbers 1 through 4 were the basis for the entire philosophy. 4 can be used to construct the pyramid, and it forms the tetractys:
X
X X
X X X
X X X X
From the tetractys can be posited the entire universe, represented by the number 10 (1+2+3+4).
Thinking of numbers in this way is somewhat strange to us, because we learn to count numbers in kindergarten. But the explorations of the ancients reveal a far more subtle aspect to number than the simple act of counting things.
Pythagoras’ discovery led to, or codified, a system of harmonic intervals based on the relationship of the numbers 1, 2, 3 and 4. This was based on the concept that out of unity comes diversity and in itself shows affinity with other ancient philosophies, such as the Tao te Ching in China (“From the one, two, from the two, all created things”).
These primary numbers were correlated with geometric forms:
- 1 is the point, but also the monad, the oneness of all things
- 2 is the line
- 3 is the triangle
- 4 is the pyramid or solid.
This has been explained in detail as follows: “When number is imposed upon space and fixed in position, it acquires extension; when number is impressed upon matter, it acquires physicality. Therefore, since the point as concept is correlated with the number 1, it assumes substance when it becomes 1 something— 1 dot in a diagram, or 1 stone, or 1 tree, or 1 person. In this fashion, the monad, infinite and eternal, is placed in relationship to each item in nature. Once the barrier between the conceptual world and the physical world is overcome by establishing the relationship between the monad and the number 1, the rest of multeity can be educed without difficulty. When the number 1 passes from the world of concept to the world of matter, it becomes extended and therefore divisible. 1 becomes capable of 2. Furthermore, two points, though having no dimension themselves, define by their relationship a line, which does have dimension. From there, it is easy to arrive at an explanation for the three-dimensional universe. Three points define a surface, and four a volume. Ecce! A time-space continuum springs from the abyss. The number 4, the final possibility of extension in our three-dimensional world, serves as an ideogram for the entire creation.” (Henniger)
Thus does a contemporary writer look at the meaning behind Pythagoras’ teaching, a teaching that is more directly summarized in the tale of the blacksmith shop.
Here is a basic summary of the blacksmith tale, along with the standard graphic from Renaissance times that depicts his "discovery." This is borrowed from the website on Handel's The Harmonious Blacksmith:
Iamblichus, the fourth-century scholar who wrote nine books about the Pythagorean sect, describes how Pythagoras came to discover the underlying principles of musical harmony: "Once he was engrossed in the thought of whether he could devise a mechanical aid for the sense of hearing which would prove both certain and ingenious. Such an aid would be similar to the compasses, rules and optical instruments designed for the sense of sight. Likewise the sense of touch had scales and the concepts of weights and measures. By some divine stroke of luck he happened to walk past the forge of a blacksmith and listened to the hammers pounding iron and producing a variegated harmony of reverberations between them, except for one combination of sounds." According to Iamblichus, Pythagoras immediately ran into the forge to investigate the harmony of the hammers. He noticed that most of the hammers could be struck simultaneously to generate a harmonious sound, whereas any combination containing one particular hammer always generated an unpleasant noise. He analyzed the hammers and realized that those that were harmonious with each other had a simple mathematical relationship--their masses were simple ratios or fractions of each other. That is to say that hammers half, two- thirds, or three-quarters the weight of a particular hammer would all generate harmonious sounds. On the other hand, the hammer that was generating disharmony when struck along with any of the other hammers had a weight that bore no simple relationship to the other weights. |
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The tale of the blacksmith shop is completely legendary. Indeed, the father of Galileo Galilei actually disproved that this could have occurred in the real world – given the inharmonic nature of vibrating metal objects. Nevertheless, it is a beautiful teaching story that has lasted the ages!
The vibrating string can be thought of as the motion of a pendulum in a grandfather clock. While less obvious to the naked eye, a fixed string also moves back and forth in the manner of a pendulum, creating a repeating frequency that is perceived as a musical tone. If a string is a half the length of another string of equal weight and construction, it will result in a tone on octave higher. More details on the Pythagorean intervals are covered in the next chapter.
Pythagoras thought of himself primarily as a healer and he used music as a remedy for all manner of sickness, mental and spiritual as well as physical.
Here are some further links on Pythagorean thought:
For a very scholarly investigation, separating myth from legend but dealing only a little with music:
Stanford Encyclopedia of Philosophy
Next: 2C: Harmonia Mundi in the Renaissance