Polarity: Overtonal and Reciprocal Energies
Defining essential polarity
When we studied ratios, we learned that the attempt to divide the octave using the methods of the arithmetic and harmonic mean gives us the relationship of the fifth and the fourth. This also created a fundamental polarity of inversion: a fifth up is a fourth down, and a fifth down is a fourth up.
Thus we tend to view this fundamental polarity as that of a central tone, with a fifth above and a fifth below, as we have seen:
From the standpoint of Polarity Theory, also known as Harmonic Dualism, we can easily see that the C stands as a central note, above which we find the overtonal fifth, 3:2 above the C. Thus, we could think of C as being the generating tone of the G. But what about the F? Well, in addition to it being in a reciprocal relationship to the C (2:3), we could also think of F as being the generating tone of the C. Thus this central spine creates a fundamental polarity: C as generating tone, the Creator, but also itself the child of its own generating tone.
Based on this principle, let's explore how we might define all 12-notes of the chromatic scale based on the simple principle of tuning notes according to a combination of perfect fifths (3:2) and pure major thirds (5:4). The following is borrowed from the book Harmonic Experience by W.A. Mathieu, from whom I have learned a great deal about the subtle relationships between harmonic events. His work can be reached at Cold Mountain Music.
First, let's add another tone above the G, and create a central spine of four perfectly tuned fifths:
We can then posit major thirds, tuned 5:4, above this central spine. We will call these the overtonal thirds, because they derive from going up from a fundamental tone, as in the overtone series.
Similarly, and somewhat more radically, we can also find the major thirds below the central spine. These are the reciprocal thirds, found as the generating tone of the note of the central spine. They are also reciprocal in the sense that they stand as 4:5 ratio below the tone.
This basic polarity allows us to view notes as having either overtonal or reciprocal energies above and below a central spine of perfectly tuned fifths. As you will notice, this also gives us all 12 notes of the chromatic scale.
Before closing this page, it is useful to look at this information, just as we did in the related concept of Just Intonation, in relationship to the harmonic series:
We can discover the higher fifth, the D two fifths above the central tone, easily within the harmonic series, and thus the C - G and D form the basic spine of fifths that we consider "overtonal." Yet this is not the case when we go to search for the reciprocal fifth, the F below the C.
The F, the lower fifth, is readily found by simple mathematical ratios and is logically connected to the central C by virtue of its being the reciprocal fifth, and in that sense the generating tone of the central tonic. In musical practice, the fourth of the scale is common in a great many styles of music. And yet the perfect fourth is nowhere to be found in the harmonic series. You can go up an infinite number of partials above the central tone and you will never find the perfect fourth, the F. The pure A exists as the next step in the scale, and the well-tuned B follows before the next iteration of the octave C. But the F? You'll never find it.
This points to an important point, which is the ultimate limitation of the harmonic series in determining music theory. While I have stressed its importance up til now, it is also important to stress that music theory and musical organization transcends the mere physics of the harmonic series, and moves into a more deeply philosophical choice of reciprocal as well as overtonal notes. There will be more to be said about this in chapter 5.
We should now look at how modes are used in contemporary practice.
NEXT: 4E The Modes in modern practice