Harmonic Polarity, expanded
The previous chapter looked in a simplistic way at how chords can be built above the notes of the major and minor scales. This page looks more deeply at the principles involved in the creation of triads, based on principles of polarity.
Harmonic polarity, the spreading outwards in two directions around a central tone, was seen on the first page of this chapter as resulting from the pairing of the overtonal series with the reciprocal series. This, again, was the result:
Harmonic polarity, based on this principle, can be further explored by taking the fundamental polarity of the 3-limit system and viewing it from above and below. This gives us the central pole of C, with the F below and the G above (expressed as the ratio of 2:3 on the staff).
Let's explore this polarity in great detail. This page is going to summarize a great deal about harmony in a few straighforward concepts. You will want to spend some time here to clarify your understanding of these concepts.
The Central Polarity
To review, here is our central polarity, with C in the center (tonic) and G the fifth (dominant) above and F, the subdominant, below:
F — C — G
In tonal music, there is a fundamental movement between these three tones, and the chords that we will soon build on top of them. C has a tendency to move downwards to the F, toward the subdominant, but the subdominant, as the outer pole, has a tendency to swing upwards to the G, to the dominant, which then pulls backwards to the tonic. We can think of this as the primary motion of tonal music:
This fundamental motion can also go around the other way, although this is less common in classical music and more common in popular music, especially music influenced by Blues and Gospel harmonies. We can think of this as the reciprocal motion of tonal music.
Expanded Polarity
We can now expand our system of polarity by creating another fifth above and below this central spine:
Bb — F — C — G — D
Thus in this system:
- C is the Tonic (central tone)
- G is the Dominant
- D is the Dominant of the Dominant
- F is the Subdominant
- Bb, a fifth below F, is the Subdominant of the Subdominant
The same fundamental circular motion can apply. Tonic can tend to swing downwards to the subdominant region, sometimes (especially in popular music) by way of its subdominant. Simlarly, tonic can swing outwards to the Dominant by way it its dominant. We will hear some musical examples soon that present these motions.
The following symbols can be used to indicate these harmonies. We will use them interchangeably. One is called Roman Numeral notation, and the other is called Function notation. Roman numeral notation helps to indicate the step of the scale on which a chord is based. Function notation better explains how one chord relates to another chord within a system of harmony.
We can therefore use these symbols as shorthand for the following tones. The Roman numeral symbols are above, and the Function symbols are below. Notice how they directly correspond to the labels in the graphic above.
When we build triads above these bass tones, we use the same labels as for the individual notes. Thus:
Building chords above tonic, dominant and subdominant
Using the natural notes from the two staves above, we can create the natural chords above the tonic, dominant and subdominant. We saw this on the first page of this chapter, and labeled these the Primary Triads.
Then we can also create major chords above the outer notes the central spine, the D above and the Bb below:
We call the outer chord on the overtonal side the "Dominant of the Dominant," because it is another fifth above the G. Similarly, we call the leftmost chord the "Subdominant of the Subdominant" because it is another fifth below the subdominant. The outermost chords will tend to move inwards towards the Dominant and Subdominant, respectively.
The V/V or the "Dominant of the Dominant" makes use of the musica ficta of F# because a dominant triad, by definition, is a major triad that contains a leading tone as its active tone. Thus the D triad is sounded with an F# in its center to create a leading tone to G, making it the dominant of the dominant triad.
Explaining the "Secondary" Dominant and the "Secondary" Subdominant
Notice that the outward chords partake of tones from either side of C on the cycle of fifths. Thus, with C as the center, the outward chords are called the "secondary" dominant and the "secondary" subdominant. The D chord, as a major triad, used the F# from the key of G major, and the Bb chord draws its Bb from the Key of F. This is why these chords are called the "the dominant of the dominant" and the "subdominant of the subdominant." To further explain this:
- Since D is a fifth above G, and is the next fifth over on the cycle
of fifths, the D major triad is the dominant of the G chord.
- In other words, if G were thought of as tonic, D would be the dominant of G. It's a matter of simply dialing over one direction to the right on the cycle of fifths.
- Also, C then becomes the subdominant of G.
- Similarly, Bb is a fifth below F on the cycle of fifths. Thus Bb is
the subdominant triad of the F chord.
- In other words, if F were thought to be the tonic, Bb would be the subdominant of F. It's a matter of simply dialing over one direction to the left on the cycle of fifths.
- Also, C now becomes the dominant of F.
Notice how the C major triad shifts its relationship to the outer harmonies:
- C functions as tonic, as the tonal center in the key of C.
- C can also function as the dominant of F, and thus leans towards the F chord, since motion of Dominant to Tonic is an essential motion in tonal music.
- C can also function as the subdominant of G.
Bringing all these harmonies back into the key of C, with C as tonic, these chords then take on the labels of the central spine that we already described above. Thus in the key of C major:
- C major is the tonic triad—the I ("one") chord
- G major is the dominant triad— the V ("five") chord
- F major is the subdominant triad—the IV ("four") chord
- Bb major is the subdominant of the subdominant—the IV of IV (IV/IV) chord.
- D major is the dominant of the dominant—the V of V (V/V) chord.
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