Fearful Symmetry
The Music Master's Lecture
From Dialogue Two, Courses in Music Theory and Composition
by Daniel d'Quincy
[The Music Master appears at a community lecture series on a stormy winter's night, to discuss the natural origins of musical art.]
My dear friends, how it warms my heart to think that you have come together on a night such as this to ponder together with me one of the many mysteries of our beloved music.
Sometimes it may seem that the pace of life, and the material necessities that drive it, have produced in us an indifference to the spiritual values that motivate the creation of an art. Yet, looking into your faces and seeing the earnest and warm devotion reflected there, I know that our world will never see an end to the desire to experience and understand both Beauty and Truth.
Well, then. . . I have been asked many times to talk about the natural origins of our music. That is to say, what are the irreducible natural elements that go into its creation. With sculpture it is wood, stone and metal; with poetry it is human language: with dance it is the body. Just as clearly with music it is sound. But what is the nature of sound? This is in some respects more difficult to grasp than the nature of stone, or words, or even flesh and blood, for many of its features are hidden and may be scrutinized only indirectly.
Moreover, we are accustomed to think of Art as an "imitation of Nature." So that sculpture reproduces the multi-dimensionality of the real objects that we encounter in space; and poetry evokes happenings of all kinds that we encounter in life as it is lived; and dance recreates the movement of physical beings in a realm of gravity and time. What can we say that music imitates in Nature, apart from the occasional bird call depicted by a flute, or the thunder of a storm by drums, or a cataract of rain by sonorously agitated strings? These mannerisms play so small a part in the overall output of the musical soul of humanity; we are left wondering what the greater bulk of musical phenomena may be attempting to depict.
You will have to allow me to digress a bit in order to bring ourselves to a sufficiently elevated perspective on this question. I warn you, it will require an attitude of playfulness on your part. Our approach will aim more at sincerity than seriousness. The attitude I'm aiming at is best expressed, I suppose, in the playing of a unique game, known by its German name as Das Glasperlenspiel, and which we may translate as the Glass Bead Game. [It is a good idea to read the lecture through, and then access the subsidiary links afterwords.]) The object of Das Glasperlenspiel is to make connections between things, not in the sense of metaphors, but as mutually resonating, intra-evocative signs of profound meaning. I am only an indifferent player of this unusual recreation, but I feel a devotion to its aesthetic charm, as a kind of art in itself. It seems to reward my devotion not only with the sensual enjoyment of the beautiful, but also with what can only be described as an enlargement of vision. In the spirit of this game, then, I would like first of all to direct your attention very far-afield, to the world of physical science, and even to particle physics and cosmology. It may seem an odd way to start talking about music, but bear with me, for if, like a fool, I persist in my madness, perchance we may all become wise.
We begin, then, with one concept of science which is both premise and product of all investigation into the physical properties of Nature. It is the concept of symmetry. It has been used in countless contexts, and with diverse shades of meaning. Coming from the Greek symmetros, for "measured together," this word is most familiar to us as a description of the similarity between the two halves of any form separated by a dividing line or plane. The concept is not limited to science, of course. It has been, for example, a mainstay of aesthetics from the beginning. Thus we speak of the symmetry of the human face, and whether we elevate it to an ideal - as in America - or deride it as a sign of banality and mediocrity - as in China - we can all understand what it is and how it looks. Its influence in the arts is pervasive. In music, we hear the symmetry of exposition and recapitulation in the so-called sonata form, where virtually the same music is played and played again. The creation of identities or equivalents is at the heart of the principle of symmetry in music, as in the other arts. It is an emblem of the human predilection for just proportion in all things, and the longing for rational comprehension of anything encountered. In the end, somewhat subconsciously, the idea of symmetry is placed in opposition to chaos, and thereby to ignorance as well.
See how, without symmetry, the entire project of science would lose its meaning. After all, the fundamental use by science of the concept of symmetry is self-reflective on science itself. It is the idea that the so-called Laws of Nature, pertain not only to the here and now, but to everywhere and everywhen. Thus, for example, the force of gravity pervades all the width and breadth of creation. Or, similarly, for right triangles on Earth and for right triangles on Mars, the length of the hypotenuse equals the sum of the squares of the opposing sides.
Science is propelled along its path by the disclosure of the many hidden symmetries that lie about us in unexpected places. Every schoolchild celebrates the simple experiment by which Galileo uncovered the astounding fact that two objects dropped simultaneously from a sufficient height will reach the ground simultaneously regardless of size. How unexpected, and puzzling, was this discovery, and how momentous for science. It was equally momentous when it was successfully demonstrated that magnetism and electricity are different expressions of the same thing. Then, it was explained that matter and energy are also different expressions of the same thing. In each of these instances, we glimpsed the inherent symmetry that imbues Nature in all her dimensions.
The evolution of physics in the late 20th century, in particular, has underlined the importance of symmetry in the unfolding evolution of the Universe. As the study of physics has gone on from Electromagnetism and the Theory of Relativity to reach for a Theory of Everything, it has assumed the symmetrical identity of all forces and forms of matter. It is even now prosecuting this venture toward a theory of so-called Supersymmetry.
Wow! Supersymmetry! Was this the same thing the poet called "fearful symmetry?" You may remember the lines:
When the stars threw down their spears,
And watered heaven with their tears.
Did he smile his work to see?
Did he who made the Lamb make thee?Tyger! Tyger! burning bright
In the forests of the night.
What immortal hand or eye
Dare frame thy fearful symmetry?
According to the emerging theory of Supersymmetry, it is said that our Universe is the product of a spontaneous breaking of primordial symmetry which happened at the beginning of time. I am reminded of an old tale that tells of a time after God created the Universe, when He stood back and contemplated what He had done. Seeing its perfection and regularity, He brought to His mind a calculation of its entire future evolution. This too was, of course, perfect. But, in the course of time, as He contemplated this perfection, He came to the realization that the perfection was only apparent, for with it He had become bored. Thus, He saw the lack of completion in what He had done. Rubbing his hands together, He created one last thing: a button to press at whim, on which was written a single word: "Surprise."
The creation of the Universe out of a super-symmetrical so-called Singularity is viewed by contemporary physics as the unfolding of a what has been called a "broken symmetry." This is attributed to the possibility that forms of symmetry may be inherently unstable. It is true that The Singularity is described as having been in a state of perfect equilibrium. Yet, the quantum reality that has been uncovered beneath the surface of things teaches the inherent instability of every state of equilibrium, including that of the Singularity.
One can truly be startled to consider the degree of fineness to which modern physics has taken its measurements of Nature and its evolution in time. Picture in your mind, if you can, a quantity of time measured as a part of a single second, and denoted by a period followed by 43 zeros followed by the number one. We have names for the decimal divisions of a quantity. We have a hundredth part, and a thousandth, and millionth, and even a hundred thousand billionth, but we run out of names long before we reach a quantity with a period and 43 zeros followed by a one. Yet this number in the measurement of time does have a unique name, in honor of a great physicist: it is called the Planck Epoch, and it denotes the moment at which matter and energy as we can know them came into existence. In other words, we are speaking of a moment that occurred after the elapse of an unimaginably tiny portion of a single second after the creation of the Universe. Before that moment, it appears that a Singularity existed in such a powerful state of energy, or, in other words, at such a high temperature, that no single particle of matter or energy could condense out of its perfect sameness. Then stretches this minute span of time in which it is believed that the symmetry of the Singularity was broken. The initial break itself may have been as infinitesimal in magnitude as the appearance of a single photon of light out of a texture of so-called virtual particles.
Yet, from this unimaginably small divergence from the perfect equilibrium of perfect symmetry came everything that is and everything that ever will be. Was God was the oyster, and the Singularity the shell; a single photon the grain of sand from which was made the loveliest of pearls, our beautiful home amidst the stars?
From the dawning of the Planck Epoch to this very moment there has been no perfect symmetry left in all the known Universe. Out of eternal constancy came hot and cold, and human beings have been running back and forth from one to the other ever since. Wherever we look, we see only broken symmetries and the incredible creative energies that they unleash. We might say that the breaking of symmetry is at once the fuel and the engine of all that happens in time and space. In our human heart, broken symmetries create movement and desire. For, it seems that the imperfect and broken symmetries we can see serve only to evoke in our consciousness the perfect symmetry from which we came, and to which we devoutly hope, against hope, to return.
I should like to move on to demonstrate how broken symmetry in the nature of sound may have given birth to the universe of music. Before I do, however, I must complete this very inadequate survey of symmetry in physical science with one crucial consideration which you will want to carry with you as we enter into the field of music.
This is the ostensible fact that any Singularity we can imagine is in some sense perfectly and instantly comprehensible. Understanding the reality that developed after the destabilization of the Singularity, by contrast, requires all the brilliance of many, many generations, perhaps an infinite number of scientific geniuses. Understanding the Singularity presents no difficulty whatsoever because there is nothing (no "thing") in it to understand. Whatever the Singularity may be, it is perfectly super-symmetrical, i.e. uniform, and without feature. There is no East and no West, no now and then, no up and down, no this and that, and you will permit me to add, tendentiously for the sake of our musical focus, no consonance and no dissonance.
The Chinese poet said it best.
The Tao that can be told is not the eternal Tao.
The name that can be named is not the eternal name.
The nameless is the beginning of heaven and earth.
The named is the mother of ten thousand things.
(Lao Tsu)
The Planck Epoch begins the emergence of the "ten thousand things." I hope you can appreciate the beauty of this term invented by a sharply analytical and, yet, strangely artful people. The ancient Chinese had a unique appreciation for the tricky nature of names and numbers. And it is uncanny how their conceptual language took account of the same processes which I have hitherto been describing in the language of physics, and shall presently describe in the language of sound. Thus, they see the symmetry breaking significance of the spontaneous appearance of yin (negative energy) in the midst of yang (positive energy), and vice versa.
Traditional China said nothing, in fact, about what exists prior to the yin and the yang, beyond implying that it has something to do with the Tao. Significantly, they implied that the Tao is perfectly comprehensible, and that it can be comprehended by anyone, with or without knowledge, intelligence, or education. They tell variously of the marvelous savants - and of innocents - who attained to that high degree of comprehension.
We, being mere mortals, are condemned merely to wonder what such pure comprehension must be like, for we are told it is bliss to experience it. And we yearn, so pathetically, for the supposed satisfactions of that blessed state. Is the yearning for comprehension and understanding an expression of our wish for a return to the Singularity from which we came? Keep that human yearning for comprehension alive in your consciousness, if you will, as I turn our attention to the world of sound.
Every beginner in philosophy must confront the following question: does a tree falling in the forest make a sound if there is no ear to hear it. Like all philosophical questions, the yes-or-no of the answer is less significant than the train of thoughts and further questions that it gets going in the inquiring mind. For us, it may prove productive as well, if we alter the question a little bit. Let us ask about the sound made by a falling tree when there IS an ear to hear it, in order that we may begin to speak about some of the physical and psychological properties of sound.
Through scientific investigation and observation, we know that objects like falling trees make sound by virtue of propagating their vibrations through a medium, and in such a way that they impinge upon a creature ear. The vibrating body causes a continuous fluctuation of changing pressure (in short, a wave) through the medium, be it air or water or what have you. For us, the medium of sound is usually what we otherwise like to call "thin" air. Thin as it is, it carries every sound we hear. Thus, sound is carried by waves of changing air pressure.
It is quite natural to imagine that each sound we hear is conveyed to the ear individually as a discreet entity, and that these entities have position in the inner space of our hearing as they do in the external space around us. This would be in accord with our subjective experience of sound. But the subjective experience is deceptive. Actually, the forest of sound that we hear is nothing more than the combined effect in sum of all the vibrating bodies within hearing distance, expressed by the shape of a singular wave changing over time. The vibrating bodies of falling trees, and birds, people talking, and every other sound, all of these individually create waves of pressure that interact in such a way that by the time they impinge upon the ear, they have been reduced, or re-formed, into nothing more than a single continuous wave of fluctuating air pressure.
Psychologically, we are aware of a multiplicity of sounds, yet we are in actual fact physically sensing only a singular wave. How does this happen? Apparently, the psychological awareness, as opposed to the physical awareness, arises out of the human mental capacity to analyze (or, as I would like to say, comprehend) a single sound wave in terms of a multiplicity of independent components: one that means falling trees, another that means birds, etc. This capacity for analysis is very profound and enables, in its ultimate expression, nothing less than the phenomenon of music.
We shall see how this capacity has evolved through the course of time, for there is no doubt that it evolved with the species. It is helpful, however, to dwell for a moment on the magnitude of this task of aural comprehension; and, at the same time, to acknowledge the strictly biological imperative that drives it. In other words, we may view it as an aspect of the will to live. A distinct survival mechanism is at work here, from the point of view of special selection. The advantage of acute aural analysis is clearly related to all of the other adaptations that accelerate human comprehension and understanding. The resolution of complex sounds by the brain into discreet elements carrying comprehensible information becomes a tool for resolving a chaotic stream of stimuli into the image of a rational and meaningful environment. The sound of a voice becomes the signal for love. The sound of a bird signals morning. The sound of falling trees could very well mean that one should get out of the forest.
It is with a view to this basic human need to comprehend, that I should like you to take a leap with me from the pure physics of sound into a more subjective level of experience where music is born. Let us expand the definition of several musical terms that you are doubtless already familiar with, in order to make them suitable for describing some of the things we have been talking about. These musical terms are "consonance," and "dissonance," and everyone knows that, ideally, every dissonance in music must be resolved into a consonance. We will not go into the fact that composers being the recalcitrant people that they are rarely comply with this dictum in toto, however much lip-service they may give it. Rather, I would like you to entertain the idea that the terms consonance and dissonance may be made synonymous with the terms "comprehensible" and "incomprehensible." Let us, furthermore, adopt the assumption in everything that follows, that every dissonance must be resolved into a consonance - or, to put it another way, that confusion and ignorance must be replaced by knowing and understanding. We shall see how far this may take us.
Regard the world of sound as being, at first, entirely dissonant, or incomprehensible. This means that one cannot automatically subject it to interpretation, and arrive at its comprehension. One has to learn how to turn sound into a source of information, thus resolving its dissonance. One can easily imagine how the newborn infant must learn to distinguish the various components of sound through a natural process of experience and analysis. All of this is accomplished on the subconscious level, of course, and so is not directly observable. We can only surmise that something like it must be occurring.
Is it not, perhaps, as easy to imagine the ways in which humanity itself has grown from aural (and musical) infancy to maturity through a process in which sound is analyzed with more and more nicety, less and less crudity? I will refer to this process as the progressive resolution into consonance of the natural dissonance inherent in sound. It could be phrased differently, more positively, be saying that this is a process in which sound is resolutely, if only gradually, made comprehensible, i.e. made consonant.
The first level of resolution distinguishes the properties of high and low (i.e. pitch), and of loud and soft (i.e. amplitude). This is elementary, and within the capacity of most animals. The most highly developed animals (including the human animal, of course) move on to analyze a fantastic range of different kinds of sound. In the case of the human, his faculty of speech alone is a true marvel of sound recognition and comprehension.
We have to an exceptional degree a talent for using sound for our own purposes; and, with the invention of music, we consummate this talent for aural utilitarianism. For, our experience of sound is not unlike our experience of other things. We not only respond to things: we also make things. Musically, we create instruments that create sound.
First, we create instruments that have naturally recognizable timbres, such as clappers, scrapers, bells, and whistles. We go on to create instruments which are capable of profound subtlety and power of expressiveness in the hands of a sensitive musician. But, in every step of its development, the craftsman's art discloses one primary motivation: to resolve a chaos of sound into what might be termed a focused sonic isolate which the listener can grasp, remember, and understand.
Thus, we learned to separate pitch from amplitude, as a variable component of the instruments we could conceive and build. People, it turns out, are most powerfully pleased by sounds which have been resolved by the material bodies producing them into distinct pitches. A sound in which a single pitch dominates the perception has a unique fascination for us. Take the sacred bowls of Tibet as an example: these bowls project individual pitches with such power and purity as to transport their listeners, even into something like Nirvana. The art of crafting these vessels is the product of a highly honored tradition. Clearly, the value of an individual bowl resides in the recognizability, or comprehensibility, of the sound that it makes. If its maker has succeeded in molding all unwanted irregularities out of its shape, so that it creates no confusion of conflicting highs and lows, comings and goings of loud and soft, but rather a clearly distinguished tone of distinct, constant pitch and controlled amplitude, then its recognizability is likely to be very high. And, it's value very great.
The aim is to build an instrument that will produce and hold a clear and distinct pitch. To be technical, we would describe this as a process in which the craftsman learns to create an instrument that, when played, will vibrate at a selected frequency (or set of frequencies), for pitch is dependent on the rate, or frequency, at which the sounding body vibrates. For reasons that we can't go into now, it happens that fast frequencies create high pitches, and slow frequencies create low pitches. Let us try not to get lost in mechanics and specifications.
Eventually, our ability to resolve the dissonance (or, rather, the relative confusion and incomprehensibility of sound) had advanced to such a degree as to be capable of abstraction, and we ended by giving particular names to individual pitches. It is worth noting that all of these developments happened apparently in the unrecorded pre-history of humanity. The earliest relics of civilized people, as for example those found at Ur in ancient Chaldea, or those found from a comparable period in Sumeria, show well developed musical instruments dating at least as far back as the 3rd-millennia b.c. Reliefs on stone show these instruments being played by orchestras of musicians with apparent pomp and circumstance. The musical age of man is very long, and reaches back before memory begins.
With the isolation of pitch as an independent variable of sound came a truly remarkable and unexpected discovery: namely, the realization that every sound we recognize as having a specific pitch is in actual fact not at all a single specific pitch, but rather a collection of pitches. The recognition of the so-called harmonic overtone series, to which I refer here, is also of very great antiquity. It was known, at least, by the 6th century b.c. when Pythagoras demonstrated that the consonant pitches used in the Greek scale were the same as those which result when a vibrating string is divided into its aliquot parts (in other words: a string and its divisions into portions equal to the inversion of the whole number series).
We would express this fact more technically today. We would say that every pitch consists of a so-called fundamental tone, plus a regular, and invariable series of higher tones which sound simultaneously with the fundamental but at higher frequencies, and at lesser degrees of amplitude. The assumed name of a pitch is merely the name of its fundamental. Above it, other pitches are vibrating and sounding in our ear.
Thus, it is clear that, by ancient times, an important result and manifestation of the harmonic overtone series had been noticed. By studying a single stretched string in motion, Pythagoras proved that the intervals of Greek music in general drew their existence out of the natural characteristics of a string in vibration.
More important still, Pythagoras discovered that the vibration of a string was governed by an exact series of specific mathematical proportions. It is hard to overestimate the effect of this discovery on every aspect of human thought. Pythagoras, as you may remember, also taught that number was in general terms connected with the essential nature of the universe. What impressed Pythagoras about the harmonic overtone series was the fact that it was constructed out of an invariant series of whole numbers (or, rather, their inversions). By showing that a thing so human as music was based in some profound way on the proportions between the aliquot parts of a vibrating string, Pythagoras gave great impetus to the more generalized idea that something perfect underlay the imperfection that was evident everywhere on the surface of things.
The idea that numbers underlie nature is practically ubiquitous among the civilized cultures of the world. But, the Greeks had a particular fondness for it. Following Pythagoras' lead, they attached the significance of numbers to astronomy, with the extrapolation of the harmonic series onto the motion of the planetary bodies. Thus they described what they called the Music of the Heavenly Spheres, which has come down to us in its Latin form as Musica Mundana. Well into the Middle Ages, this idea held sway. Boethius, one of the philosophers - and music theorists - who carried the burden of classical metaphysics into the world of the Church, established the relationship of music to astronomical numerology as a virtual dogma. In music, after him, it was simply impossible to talk about scales and modes without bringing the planets into the discussion, along with a host of other metaphysical speculations in their trail.
Something comparatively odd happened sometime around the 15th and 16th centuries. It became the fashion among music theorists to ignore the metaphysics of music. Practical considerations were the order of the day: scales, modes, dissonance treatment in voice-leading, and the like. Boethius had hardly touched on these technical issues. But, suddenly, material factors and techniques were the focus of attention in every field of human endeavor, not only in music. We may all now assess the results of this sudden shift in consciousness: people went from ignoring the Music of the Spheres, and from putting the sun at the center of the solar system, to inventing the spinning mill, through to the sewing machine, the bomber airplane, and thence to Hiroshima. We needn't rehearse that drama in this hall. My point is simply that music is many thousands of years old, and the theoretical approach to it characteristic of the last four or five hundred years may be considered as only a temporary fad. And, thus, you can see, my friends, that it may not have been so extravagant for me to have begun this lesson in music with a consideration of the creation of our universe. No classical scholar would have taken the slightest exception whatsoever to the approach I have presented here.
Let us return, then, to the momentous discovery embodied in the Pythagorean demonstration of the harmonic overtone series. As we look at the fundamental and its overtones displayed on a music staff, it is evident that only some of the pitches that are otherwise familiar to us appear (there is no C-sharp, for example), and that some pitches are upper octave duplicates of lower pitches (the pitch C having the most reflection in the upper octaves). (See Fig. 1 above.) It is worth bearing in mind that the harmonic overtone series is infinite, in theory. We will restrict ourselves, however, to those overtones that are clearly audible by the ear, remembering that the amplitude of the overtone series generally diminishes in proportion to its height above the fundamental.
Notice, if you will, that the lower six overtones include the pitches making up what we customarily call the tonic triad: i.e., the tonic, the dominant and the mediant, (or, in other words, tones on the First, Fifth, and Third degree, respectively) of the scale that begins on the fundamental. In the example, the lower overtones include the tonic C, the dominant G, and the mediant E, of the C Major scale, and these are, of course, the tones of the C Major triad.
In a similar way, the seven notes tones of the C Major scale can be derived from the first six overtones of three separate harmonic overtone series: that of C, G and F. And, we should hasten to add that the twelve tones of the familiar chromatic scale may be derived from the first 13 overtones of the same three series.
Why bring F into it? you may ask. Clearly G, being the third overtone of C, and normally one of the most audible of its overtones, is highly resonant and harmonious with C. In fact, people of all musical cultures recognize the importance of the interval of a Fifth, which is formed by the simultaneous performance of these two pitches.
Understand, then, that the relationship between C and G is precisely parallel to the relationship between F and ITS third overtone, which is C. Obviously G is to C in an upward direction, what C is to F in a downward direction. The symmetrical relationship among these three individual fundamental pitches is based on the natural power of the third overtone, and results in the fact that they share overtones in common. We experience this as a natural affinity between these tones.
Can it be that the notes of the tonic triad, the major scale, and the chromatic scale, being collections of pitches so central in importance to our music, are only coincidentally related to the harmonic overtone series. Music theorists are in universal agreement that the coincidence is necessary and perfect. Evidently, paleo-men and women found that the natural resonance created by the singing of these pitches is pleasing to the ear. Beginning with one pitch, one has to move on to some different pitch or other, if one is to sing a song. The choice of the most resonant pitches seems logical enough.
But as we delve further into the implications of the coincidence, the perfection dissolves. We see instead an evolving and dynamic relationship between the human consciousness and the nature of sound. The connection between the harmonic overtone series and our scale isn't written as law. It is not a noun, it is a verb. It gives rise to a process.
Let us characterize this process. Above all, it is a process with an implicit goal, and the goal is nothing other than the resolution of the inherent dissonance in sound. This is the gist of my thought. And, be clear, I am speaking of the dissonance that is inherent in the complex structure of the harmonic overtone series as found in Nature. The theorists have spoken of its perfection, but I will speak of its imperfection.
Of what does this dissonance consist? Why, the very multiplicity of overtones within each musical tone is dissonance enough. That dissonance is expressed by the question that it begs. What is the true identity of all of these pitches that hide within the shadow of the fundamental, and what are their mutual relationships? In the beginning, this is not at all clear. Remember that I have asked you to identify the concept of dissonance with incoherence and lack of understanding. The harmonic overtone series represents a sonic reality. It can be perceived without being conceived. In other words, until this sonic reality is understood musically, there is confusion, and therefore dissonance.
Let us put this another way. It is easy enough to register the coincidence of our musical materials with the materials that come ready to hand in the harmonic overtone series. The mere agreement in material, however, misses something of the spirit of the conjunction. For, when the ancient ancestors had analyzed sound to the point of abstracting from its natural structure the notes that rise diatonically from C to C(in the manner of our common C Major scale), they had not thereby explained WHY anyone would want to go to the trouble. After all, why should we make that arduous journey higher and higher? Only to go down again?
The question is as real for us as it was for them, and it is by no means a vain question. Indeed, the import of a great deal of recent music, since the Second World War, has been that there is no good reason to go through the scale from C to C. Whole compositions have thus been based on a single tone, or a single chord. In one celebrated work by John Cage, a pianist comes to the piano, sits, and does nothing for the specified duration of the piece. There is in this instance apparently not even sufficient reason to make a single tone, let alone a chord, hence a lapse into what could be called "musical silence."
For reasons we cannot discuss tonight, however, it is true at least up until our own uniquely revolutionary era, that all of our musical history has been engaged in the progressive resolution of the dissonance native to sound. And the nature of the musical impulse is exemplified at its highlest and in its subtlest form through the way in which it accomplishes the resolution of the dissonance inherent in the harmonic overtone series. In practice, this is easily defined: for, when the fundamental is sounded, a great many other pitches are sounded simultaneously at higher intervals. The entire purpose and meaning of musical melody is to rise into the dimension of the overtone series, draw its pitches into coherent and comprehensible relationships, and then proceed to connect them psychologically, with all their individual character, to the fundamental pitch around which they revolve.
The literal truth of this statement can be verified by any one of you the next time you have the opportunity to hear one of the old ecclesiastical melodies, either from the Gregorian Chant, or from Protestant hymnals. These melodies, performed without accompaniment, express with perfection the fundamental mission of all musical invention from a technical point of view: that is, they begin with the fundamental, rise into the realm of its overtones, discourse for a time upon their inter-relationships, and then draw them down with finality to their resting place again upon the fundamental. And, to this day, we refer to any musical piece as being in C, or G, or what have you (in short, according to its fundamental tone), because it will invariably proceed according to this venerable principle of shape and method (or alter it for a purpose).
I would not wish you to think, however, that the dissonance of the overtone series resides only in the fact that there is a simultaneous sounding of a multitude of pitches in every musical tone. We must also recognize that the pitches of the harmonic overtone series are dissonant also in their harmonic relationship to each other. To understand how this is so, we must consider yet another meaning of the word dissonance.
It is time for us to revert to a more strictly physical definition of dissonance. My edition of the Grove's Dictionary of Music reads as follows: "Dissonance is any combination of notes which on being sounded together produces beats; that is, an alternate strengthening and weakening of the sound, arising from the opposition of the vibrations of either their prime tones, or their harmonics or their combination tones, which causes a painful sensation to the ear."
This all sounds rather complicated, but there is a fairly simple analogy that may make it clear. It is the so-called water torture, wherein water falls on your head incessantly, but only solitary drop by solitary drop. It has been suggested by some scientists that the effect of musical beats on the ear is experienced as physical torture and is based on a similar physical response. I promise you that we will not go so far afield as to delve into the medical dimensions of music tonight. Every successful Glass Bead Game must observe strict limits. Suffice to say, musical beats are rather like drops of water. As the number of beats per second increases, the ear begins to register a certain discomfort. (Beyond a certain point, on the other hand, the beats become so numerous that the ear cannot distinguish them as beats, and then their unpleasantness is nullified.)
The basic point, and the practical effect of this physical response is that, in music, the combinations of tones which produce the least (or the most) beats per second are considered consonant: i.e., the intervals of the Octave, the Fifth, major and minor thirds and sixths being most consonant. Those combinations of tones which produce beats within a certain specific range of annoyance are considered dissonant: notably, major and minor seconds and sevenths.
Here is how Sir James Jeans puts the matter in quantifiable terms. "Beats are a wavy throbbing effect produced by the sounding together of certain notes, and most noticeable in unisons and consonances, when not perfectly tuned to one another. . . .Sound is conveyed to our ears by the waves into which the air, or other medium, is thrown by the vibration of what is called the sounding body. . . . According to Helmholtz they are most disagreeable when they number about 33 in a second, which is nearly the number produced by the sounding together of treble C and D-flat. From that point they become less and less harsh till with such an interval as treble C and E, which produces 128 beats in a second, there is no unpleasant sensation remaining."
Having defined dissonance in this way, let us look again at the harmonic overtone series. We might conclude from the lower parts of the series that the overall phenomenon is consonant. Indeed, it has been the cherished desire of a good deal of theoretical scholarship in music to prove, or at any rate to assert, that the overtones series is, in fact, the natural consonance to which all music refers. Indeed, the flawed notion that the overtone series is consonant has been used for generations especially by those shallow critics who declaim against the dissonant idioms of contemporary music, as if they were defending the barricades of civilized morality against the barbarians. Regrettably, their musical piousness is not always open to rational debate. To the objection, for example, that the 7th overtone in the series is manifestly dissonant by any definition, being a B-flat against the fundamental C, the traditional theorist merely says "nevermind," and he declares that the 7th overtone is too faint in amplitude to matter. But this will not do. The pitifully tiny sound of a mosquito is enough to drive a man crazy. Loudness is not the only issue. There is kind and quality, too.
If we do not acclaim the "consonant theory of the overtones" for its attention to the apparent facts, we can no more easily approve it for its recognition of what has been accomplished by people through thousands of years of musical development. For the resolution of the natural dissonance of sound by means of music is a difficult and lengthy task. It is not accomplished only by the individual listener, nor by the musician, nor the sovereign composer. And it is not accomplished in any individual piece of music. This is a job too large for individuals. It calls for the labors of generations of people through thousands of years. It is an accomplishment of human culture as a whole.
The resolution of the intervalic dissonance within the overtone series (for example, the kind of dissonance that sounds when a C and a B-flat are played simultaneously) has proven to be the most difficult part of the process, and it remains incomplete to the present day. Perhaps, it shall never be complete. The position of music is not so different from that of science and philosophy. It is not commonly expected that these other disciplines should arrive at a final conclusion. It may be that each stage in the development of music presents in itself a more convincing image of completion than do any of it counterparts in the sciences, but the appearance of perfection (even in Bach or Mozart) is deceptive.
To give you an idea how complex the problem has been, let me take you back one last time to our overtone series to investigate a level of dissonance that we have not yet discussed, but which is related to the other two. Here, I will ask you to think back upon what I was saying at the beginning of the evening about the broken symmetry that gave birth to the universe. I want now to show you the broken symmetry that lies hidden within the heart of music.
First, we shall consider the second overtone, which is an Octave above the fundamental. I will remind you of the mathematical proportions that were revealed by Pythagoras on a sounding string. He demonstrated that the second overtone of C, namely the C of the Octave above, is the pitch produced when the string is divided in half, and each half vibrates independently. And, we know today that the frequency of the higher C is precisely twice that of the lower C. These are just two ways of saying the same thing.
Now, look at what I shall call the Line of Octaves. You will wonder why I ask you to look at the Line of Octaves, whilst every other theorist asks you to look at the Circle of Fifths. Fear not: I shall come to the Circle, but without first considering the Line, I believe that you are in no position to appreciate the Circle.
The Line of Octaves presents an image of a perfect symmetry. No matter where you move you find a C. Each C vibrates at precisely twice the rate of the C before it, and half the rate of the C after it. When you play any of these Cs together, say the first and the seventh, no unpleasant beats are created. Like all perfect symmetries, the Line of Octaves is fully consonant.
Now, then, consider the third overtone of C, which is an Octave and a Fifth above the fundamental (which we will reduce to the interval of a Fifth by subtracting the redundant Octave). As you know, Pythagoras demonstrated that a string vibrating in three aliquot parts produces the pitch which is an Octave and a Fifth above the fundamental of the open string. This is a very precise relationship. Like the relationship of the Fundamental with the Octave, the relationship of the Fundamental with the Fifth renders a mathematical formula which will enable you to determine the exact frequency of the second harmonic whenever you know the frequency of the first harmonic, through simple multiplication. In short, what me mean by the interval of a Fifth is not something approximate. It is in fact, one might say, painfully exact, because it is in the nature of this interval to break the perfect symmetry which we have encountered in the Line of Octaves.
Without going into the actual numbers, you only need to picture what happens when you juxtapose 12 intervals of a Fifth, one above the other in consecutive order. It has been customary in music theory texts to depict this arrangement in the form of the so-called Circle of Fifths. But this practice doesn't do graphic justice to the natural facts. You remember that, on the Line of Octaves the first C and the C seven octaves higher were perfectly consonant, or as we say perfectly in tune.
The twelfth Fifth above C is also called a C in our music. But, low and behold, it is not perfectly consonant with the corresponding C on the Line of Octaves. The mathematics of the interval of a Fifth mean that the vibrational rates of the C on the Line of Octaves and the C on the Circle of Fifths differ by a small but irreducible amount. That tiny quantity is known as the Comma of Phythagoras in honor of its discoverer.
I am not the first to propose it, but I can surely lend my support to the idea that we should cease portraying the Circle of Fifths as a circle. Can we not see that it must be a spiral? Can we not call the topmost pitch by its true name, which is not C but B#.
But we should have known without counting vibrations, for there are no perfect circles in nature. Yet, there are many, many spirals. The spiral may be said to break the perfect symmetry of the perfect circle. As we saw with the fate of the Cosmic Singularity, the break with symmetry is so small as to be hardly noticeable. Nevertheless, what wondrous results it has produced. Can you picture anything so lovely as a rose and show me that it is made from perfect circles?
The complaint will be raised that today we do not tune our musical instruments exactly according to the proportions of the harmonic overtone series. This is quite so. Were you to play Fifth after Fifth at the contemporary piano keyboard, starting with a C and ending with a C, you should find the lower and the upper perfectly in tune, just as on the Line of Octaves. To produce this result we have contrived to alter every other interval in the chromatic scale, to a slight if not imperceptible degree. Evidently, we have in this way made an attempt to recapture something of the perfect symmetry of the idealized Circle of Fifths. But we have done this to a purpose. And, although I think that now I need not go into the technical advantages of it, I may state for the record that we have done it in order to resolve the dissonance inherent in the harmonic overtone series.
The so-called equal-tempered tuning to which I am referring became necessary when we began to add multiple parts to our music, so that a greater and greater variety of notes were played simultaneously. It only seemed that modal music had accomplished the task of resolving natural dissonance with the employment of so-called "Just Intonation," which tunes musical pitch in accordance with the precise properties of the natural overtone series. It only seemed that nothing remained to be done. Indeed, the Roman Church, entrusted with the tablets of the modal tradition, did everything in its power to prevent its fundamental aesthetic principles and procedures from being changed. But the desire for more complex forms won out in the end. How scandalized would be Pope Gregory, or any sensitive person of his day, to hear the crudeness of our so-called even-tempered piano tuning. How lacking in brilliance and subtlety it would seem. How unresonant with the echoes of sound in nature.
We should have to explain to him that the change could not be helped. If there were no other reason, it would be simply that people can never ever wait their turn to sing. Wherever there is music, they all want to jump in and have some fun themselves. Too often, of course, they can't seem to sing the right notes, or they sing the right notes but transposed into their own vocal range. You cannot imagine what a can of worms is brought into the picture when more than two people are singing at once. For one thing, even if they sing nothing but two pitches, say C and G, the conflicting and incompatible relationships between the overtones of each note make a dissonance that simply cannot be put to rest. It must be dealt with. And, therefore, people have done whatever they could in this cause.In a real sense, for reasons we cannot fully plumb, people in the Western world took on the challenge of resolving music in which more than one note plays at a time. Thus, they brought harmony together with counterpoint, and the issue was polyphony. For nearly one thousand years, we have been wrestling with this Holy Angel.
Why do we do it? Does the resolution we find in our music function like the wish-fulfillment of a Freudian dream? Can we hope for any kind of final resolution to the dissonance of life on Earth, or dream of a time when our efforts may be brought to a conclusion?
There are no answers to these questions. Ultimately, we do what we do because it is human to do it: we are obeying our nature. We cannot live in the world of perfect symmetry embodied by the Cosmic Singularity. We should be vaporized by the high energies, just like everything else. We live after the coming of the yin and the yang, and the birth of the ten thousand things. I have tonight equated all of our musical life with the progressive resolution of the fundamental dissonance of Nature into the image of a harmonious Universe. This effort is connected with the fate of humanity in general. We must each one of us be the judge of how successful it has been. Those of us with a cheerful nature will portray the experience as a comedy. Those with a darker sense of things will agree with the philosopher who wrote of "the birth of music from the spirit of tragedy." What is sure is that, when confronted by the multiplicity of phenomena in the world, and the complexity of the forces that move all things, the struggle of our mind and spirit to understand is quickly and decisively defeated. We should consider the prospect dubious; yet, we rebel against defeat, and the battle goes on. This is what some people call the human predicament, and others call the human moral impulse.
Herein lies the origin of the hoary notion which marries Music to Good. Frequently, the Good is shortened to God, so that Music becomes the Handmaid of the Divine Will. It has been claimed that the Godhead itself can be heard: for instance, in the internal vibration created by the silent chanting of the word OM. In any case, the religious sense is everywhere allied with the musical. For my part, I would not blame the Deity for what may be only musical transgressions.
I have perhaps given you more than you bargained for when you asked your questions about the natural origins of music. And I hope you will forgive me for having, in my enthusiasm, expanded the normal span of our little talks to such an imprudent length. Quite frankly, my voice is tired; and your. . .
Well, you too are tired. It is only fair to us all to turn out the lights and go on home.
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