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and the Art of Music
by Dane Rudhyar

Chapter 7

The Harmonic Series
Part One

When considered as a series of fundamental and overtones, each of which has a strictly defined frequency, the harmonic series is an arithmetic series. The archetypal arithmetic series is the series of whole numbers, 1, 2, 3, 4, 5, and so on. Such a series is formed by the endlessly repeated addition of number one to itself. However, no instrumental or vocal fundamental tone actually produces a complete and endless series of harmonics, and the overtones it produces do not all vibrate with the same intensity.
      The harmonic series is thus an archetypal model. It is either the process according to which the cosmogenic energy of Sound operates as it radiates or emanates from a creative spiritual source and descends step-by-step into progressively denser fields of objective and material existence, or it is an ideal concept abstracted from the experience of hearing some overtones when a variety of fundamental tones are sounded out by setting material instruments (including the human vocal organs) in motion.
      In this sense the harmonic series is a myth — the myth of number translated into musical terms. It is based on the association of two factors: an arithmetic progression of whole numbers and a geometric progression which at first apparently takes the form of a series of octave intervals (see figure 1). The octave is a musical interpretation of the ratio two to one (2: 1); two notes are in octave relationship when the frequency of the higher is twice that of the lower). (1) A series of octave-sounds follows a geometric progression and the frequencies of the sounds can be expressed exponentially: 2, 22, 23, 24, 25, and so on (or 2, 4, 8, 16, 32, and so on).

      Any series of repeated intervals is a geometric series. All series of intervals can be compared with the series of other intervals. The interval of fifth (3:2) has been considered especially important during the last five millennia, and a series of twelve fifths includes slightly more than seven octaves. One can also compare twelve fourths and five octaves. The meaning of these comparisons is discussed later in this chapter. Comparisons between other series of intervals may or may not be significant.
      An arithmetic series refers to the transmission of power which, released from a creative source, becomes differentiated. For instance, in the government or a large corporation the power wielded by the top executive descends through several levels of authority before it reaches the realm of concrete, material results. In music, this level is the actual vibration of a resonant instrument or voice. On the other hand, a geometrical series refers to consciousness, because consciousness is implied in or is the product of the relationship between a self and an other. Consciousness develops through the progressive complexification of relationships. It expands by including an ever greater number of differentiated relations — in music, an increasing number of different intervals.
      Thus, in the pattern of a harmonic series of fundamental and overtones, the relation between two successive harmonics — the interval between them — diminishes in scope; the ratios 2:1, 3:2, 4:3, 5:4, and so forth, become increasingly smaller. On the other hand, if we focus on the geometric series of octaves starting with the fundamental, we see that each successive octave contains more overtones than the preceding one. Thus, while the number of overtones increases per octave, the intervals between the overtones become smaller (see figure 1). Before the eighth octave is reached the intervals between successive overtones become so small that the human ear can no longer distinguish them clearly; the harmonic series becomes a rising continuum of sonic vibrations.
      The first octave contains no intermediary harmonics; the second contains one; the third, three; the fourth, seven; the fifth (16 to 32), fifteen; the sixth (32 to 64), thirty-one; the seventh (64 to 128), sixty-three. The last interval within the seventh octave is the expression of the ratio 128:127; it is so small an interval that the ear cannot distinguish it from the following interval, 129:128. The last interval of the fifth octave (the ratio 32:31) was used in Greece as the characteristic enharmonic interval. It was slightly larger than a quarter tone in the modern Western scale. The Pythagorean comma — the difference between the musical space of seven octaves and twelve fifths — is about one-eighth of a tone.
      An octave divided into equal intervals approximately the size of a comma would contain forty-eight eighth tones. No instrument, except electronic ones, could be tuned to finer intervals. Thus for most practical purposes music spans seven octaves of vibrations, approximately the extension of a piano keyboard (that is, from about 27 to 2,456 vibrations a second plus three half tones at the top of the keyboard).
      The traditional esoteric world view divides the universe into seven levels of being. The lowest is physical matter, the foundation of all activities and of all the changes we perceive with our senses. It is the level at which the resonance of material entities (including musical instruments) to the descending current of power released by a creative will-emanating source becomes audible as a tone — that is, as the complex vibration of the material body or vocal organ.
      This tone is intended to communicate the purpose of the originating source in releasing such a creative or transforming power. But the audible tone not only contains the original creative or informative purpose implied in the descending power of Sound; it also is conditioned by and reflects the physical limitations and special characteristics of the resonant instrument. These characteristics result from the molecular nature and shape of the instrument. Because of them the ascending harmonic series produced by the audible resonant tone (the fundamental) is never a perfect arithmetic series of overtones. Only some of these overtones can be heard, and of those a few are more intense than others. The result is the specific timbre (or quality) of the tone.
      Thus we never hear a complete (theoretically endless) series of equally intense overtones in any instrumental or vocal tones, because all we hear are sounds produced by the vibration of material entities. Moreover, we do not actually hear Sound, but the resonance aroused in material instruments by the impact of inaudible currents of energy and will or psychic activity (emotions). Nevertheless, the ascending series of overtones, incomplete and uneven as it is, is symmetrical to the descending series. Overtones can only occur as component parts of an ideal arithmetic series, of which the fundamental of the instrumental tone is number one.
      This statement may seem arbitrary and illogical, but it would not seem so to a modern physicist thinking in terms of quanta (that is, of the discontinuous release of energy) and of the specific orbits in which electrons moving around a proton must revolve. The harmonic series thus appears to be an inherent structural factor both in the dynamic process of the release of descending Sound (or will power) and in ascending overtones generated by the symmetrical reflection of the material instrument's resonance to the impact of the Sound current.
      The ascending and descending series are (in principle) symmetrical if considered as series of ever-decreasing intervals (octave, fifth, fourth, major and minor thirds, and so on). But if considered as the musical notes of Western scales, the notes of the descending series are not the same as those of the ascending series (see figure 2). An ascending progression starting from the note C as fundamental will produce a G at the place of the third partial; but a descending series starting from the same note C produces an F at this place in the ascending series.

      The harmonic series considered as a series of intervals is like a ladder whose rungs occur at always diminishing distances from one another. If you place the ladder against a blank wall and mark on the wall the places at which the rungs touch the wall, you will have one series of marks if the ladder is placed with the smaller distances between the rungs toward the top of the wall, and another series if the ladder is reversed.
      This relationship can be demonstrated audibly using a monochord, the didactic instrument of Pythagoras's teachings. If one plucks successively the entire string of the monochord, then one half of it, one third, one fourth, one fifth, and so on, one hears an ascending series of harmonics; these are explained by the physical fact that when the string as a whole is plucked it not only vibrates as a whole, but its aliquot parts also vibrate — thus the vibrations of half of the string, a third, a quarter, and a fifth are also perceptible, at least in theory.
      If, on the other hand, the hand plucks one inch of the monochord's string, then two inches, three inches, four, five, and six inches, and so on, a descending progression of sounds is produced, which gives the hearer a symbolic experience of the path followed by the descent of creative and willful Sound. It is only a symbolic experience, because no part of a descending series of harmonics is audible. What seem to be "undertones" are combination tones (or resulting tones). These are complex auditory phenomena that acousticians consider to be subjective, in the sense that they are apparently produced in the inner ear because of the way the 25,000 extremely delicate hair cells of the cochlea vibrate. Combination tones, however, are produced only when two or more loud tones are heard. In complex and non-harmonic tones, like those of Japanese gongs or church bells, such low combination tones are often very strong. They also can be heard in a piano under certain conditions. (2)

1. Hereafter, octave refers to the interval, octave-sound to an overtone.  Return

2. The frequency of a combination tone is the difference of the sum of the frequencies of the two loud tones giving rise to them, or of multiples of these frequencies. Combination tones 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).  Return

By permission of Leyla Rudhyar Hill
Copyright © 1982; by Dane Rudhyar
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