are characteristically lower than the two original tones, but they may be higher. Two tones of frequencies 1,200 and 500 can produce a differential tone of frequency 700, or a
summation tone of frequency 1,700, and still other combinations (see the entry on combination tones in the Harvard Dictionary of Music, p. 185).
Chapter 7
The Harmonic Series Part Two
The Seven Levels of Being and the Symbolism of Number
According to the great religions of the last five millennia, and to most of the
metaphysical systems dealing with the origin of the universe (or more generally with the origin of being), all there is began in unity. The religious mind usually personalizes the concept of unity as God. Metaphysicians speak of "the One." Sankaracharya, founder of India's Advaita (or nondual) system of metaphysics, speaks of "the One without a second." The great (and actually insoluble) problem of metaphysics and religious cosmogonies is how to explain or interpret the passage from unity to duality, and from duality to the quasi- infinite multiplicity of entities active in the universe. Religions speak of God's desire to create, of the primordial Eros that moves the One to produce out of its unity a second, perhaps a mirror, image. Hindu metaphysics interprets such a process of doublement or replication as the great illusion, the essential Maya, root of all existence. The harmonic series of fundamental and overtones provides a very significant and experienceable realization of the relation between pure unity (the One) and duality (the second) by referring to the unique and mysterious character of the octave in music.
The remarkable fact is that two sounds an octave apart are given the same name, even though one of their frequencies is twice that of the other. They have for our ears an identical nature. They are the same note, even though they are obviously not the same sound. Are we conditioned by our culture to feel that two sounds at an octave interval are the same note, or is the feeling of their identity innate — that is, rooted in an intuitive grasp of the nature of a metaphysical-spiritual process, which is none other than the basic process of cosmic existence and the primary manifestation of what we call life?
Although the harmonic series of fundamental and overtones is an arithmetic series, and the archetype of all such series is the series of whole numbers created by the endless addition of number one to itself, the term addition may be misleading. Philosophically the series refers to the self-reproduction, self-multiplication, or self-replication of the One. All numbers are born out of number one. The birth process begins with the characteristic act of self- duplication. Duality emerges out of unity: the One produces the Other, which is identical to itself — a mirror image (as it were) — yet itself in a new role. This duplication process can be repeated; its repetition produces a geometrical series: two, when duplicated, produces four, which when duplicated produces eight; the eight duplicates into the sixteen, and so on.
This series might be considered a repeated process of reflection, giving rise to a series of mirror images. But sonically the series produces octave-sounds. They are not merely reflections of one another, for each one is the source of series of overtones, and each new octave of the harmonic series contains more overtones than the preceding one. Each new octave symbolizes a level of being one more step removed from the original unity — sonically the fundamental, philosophically the One.
Some of the religions of India speak of the One as Shiva, and the second as Shiva's shakti — his power, which is personified as his feminine counterpart, Shakti, the beloved. In the Tantric cosmogony, once Shiva has created Shakti and consummates his union with her, he retires (as it were) and becomes a mysterious presence beyond the cosmic manifestation to which Shakti gives birth. The mother, having produced the child, becomes the manager of the generative process and of its results. She rules over the universe of concrete and multiple entities — over all subsequent generations, each of which begins with a reflection of the primordial mother. Each of these mothers (like each of the subsequent octave-sounds of number two) also manages and rules over her own progeny and her own level of being. The initial process of duplication of the One into the second has its source in a release of power from the One. This is what religions interpret as God's "desire" for creation, for self- revelation, or for the exteriorization of the immense potentialities of his infinite being — the One desiring to be many. More impersonally, this desire is the motion that operates
throughout the universe. Motion is everywhere. Matter is the incredibly rapid motion of subatomic particles, which themselves are nothing more than whirlpools of motion. All life is God's self-revelation through rhythmic, self-duplicating motion. In its primary spiritual aspect, this motion is Sound — the descending power of the One into multiplicity, the descending harmonic series.
Is this descent endless? The rational intellect can find no reason for an end to the process of self-replication. The harmonic series issued from the fundamental One can theoretically extend ad infinitum. But infinity is only an intellectual concept; it is the negation of limit. The concept of existence, however, must include limitation. Existence can only be conceived in terms of wholes, and all wholes must have limits or boundaries, metaphysical if not physical.
Thus the octave symbolizes and defines a whole of sound-the limits within which motion, as a creative factor, cyclically operates. Metaphysically, the octave is the most fundamental whole because it originates in the first act of self-duplication, of which all further acts are replications. But the momentum of the creative release of the power of the One does not stop with number two. The power release acting through number two produces number three, the symbolic child. A new relationship is established between the mother principle and the child principle. This relationship also replicates itself, generating in music a series of fifths (the ratio 3:2) — a new limiting and cyclic factor.
The geometric series of octaves and the geometric series of fifths interact; and a time comes when both series reach almost the same vibratory frequency: twelve fifths extend a
little farther than seven octaves, the difference between them being the Pythagorean
comma. What this means becomes clear once we understand the character of the interval of fifth and experience the psychic response it normally produces in human beings. To do so, however, we have first to deal in some detail with the field of cosmic and psychic activity and consciousness represented by each successive octave of the harmonic series. What we will discuss is a series of abstract numbers and proportions, the harmonic series as an archetypal pattern that can be interpreted numerologically, descending or ascending as the occasion requires.