Harmonic Polarity: Introduction
Harmonic Polarity / Harmonic Dualism
Harmonic Polarity, or Harmonic Dualism, is a theory of harmony that was widely taught in the 19th Century, fell out of favor in the 20th, but is experiencing a renewal in the 21st. The fundamental concept of Harmonic Dualism or, as I call it, Harmonic Polarity, informs my work in composition and in music theory, and thus it forms the basis for my approach to harmony. It is not the only way to organize harmonic thought, but it is to me the most organic and musical. In the end, all systems of organizing music regard a choice and are not "absolute."
The theory of Harmonic Dualism or Harmonic Polarity is my choice. Let us make a start.
The Polarity of Time
Dividing the world into dualistic poles, often with a central point around which the poles balance, is a fundamental aspect of human experience. It is natural for the human mind to create dualisms which contribute to our organization of information. It is no different with music.
The concept of dualist poles centered around a balancing central point is perhaps best understood in reference to time. We know there to be a past, and we posit there to be a future, and the central pole between this dualistic structure is the present moment. When we are active in the world, the present moment would seem to be but the briefest of instances, as we continually base our actions on past experience and shape our future according to our will. Yet at the same time, in contemplative traditions, the present moment is known to be all that there is—the present moment is in fact an eternity, while past and future are but the illusions of the active mind.
This fundamental polarity of time exists in the theory of music to profound degree. Let us look at some of these fundamental polarities.
The Polarity of the Overtone and Reciprocal Series
A central polarity in music is that of the overtone and "undertone" series, which I will refer to as the reciprocal series, after my "theory guru" Allaudin Mathieu. The term "undertone" has been used in the past to describe this series, but it is an unfortunate term in that it implies that the series is a naturally occuring phenomenon, like the overtone series. This is not the case.
There is no way that a vibrating string can produce "undertones" in the same way that it produces overtones. Earlier proponents of Harmonic Dualism sought to assert this in an effort to be materialistically scientific, but this only created confusion. The reciprocal or "undertone" series is a mathematical and musical concept and serves music theory in the manner in which it creates a symmetrical system for contemplating and understanding tonal harmony.
As we saw in our study of Pythagorean thought, the overtones correspond to notes that are derived from mathematical principles using low integer relationships along the divisions of a string. The reciprocal series is simply the opposite conceptual pole to the overtone series. It can be thought of as being created by lengthening a string, rather than dividing it, by integers. But it is also, and most importantly, a concept. It is the spiritual, inwardly existing cousin of the overtone series.
First, let's recall that the overtone series produces an interesting phenomenon, which is the major triad that arises from the ratios of 4-5-6:
Here is how the reciprocal series looks, starting on the high C rather than the low C. It is created by taking the reciprocal, or inversion, of each interval of the harmonic series and going down in pitch rather than up. Thus we go down in turn by an octave, a fifth, a fourth, a major third, a minor third, and then smaller steps in the higher ratios. Thus:
What is important about this series is that it helps to explain the existence of the minor triad as a central building block of harmony. It awards equal status to the minor triad by viewing the minor triad as the result of a process that exists in polarity with the overtone series. With the reciprocal series, the ratios of 4-5-6 form the triad of F-Ab-C, which is the subdominant minor triad in the key of C, built from the top down.
Here is how the two series look when combined one with the other:
A beautiful symmetry is formed, with the major triad ascending and the minor triad descending.
There is a deeper, philosophical aspect to the relationship between the overtonal and reciprocal series. I will cover this on a separate page.
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