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Essential Harmonies

1. The Modes

Knowing the modes is essential, basic information for any modern musician or composer, whether your interest is jazz, rock, classical, early music, minimalism, or some hybrid of all of them. We will be looking closely at the connection between "modality" and "tonality" in all of our music.

Here they are, both in their original settings at different levels of the basic diatonic scale, and also transposed into the key of C. The diatonic modes are as they were understood in Renaissance times. The second grouping is of the modes as we think of them in modern compositional techniques. Lydian is the most "overtonal" of the modes, with all the pitches dervived from the overtonal fifths. As the modes progress, we introduce more and more reciprocal energy, until we reach the phrygian modes, which is pure reciprocal tones. Recall (since you are playing this at the piano) that the phrygian and ionian modes are polar opposites of each other.

2. Polarity Theory in Practice (Beethoven Waldstein)

We can see from the previous discussion of Harmonic Polarity that there would appear to be a symmetrical and balanced relationship between the essential harmonies of tonic, dominant and subdominant. This theory seems to me to be in keeping with how we experience music, and helps to link classical harmony with popular harmonies, in that we can see various musical styles as having more 'overtonal' or more 'reciprocal' energy, or as having a balance between the two. Beethoven's Waldstein seems evenly balanced between these energies, as evidenced by the statements of 1) tonic, 2) dominant of the dominant and 3) dominant in the first phrase, followed by tonic, subdominant of the subdominant and subdominant in the second phrase. This could be put in shorthand form as (refer to the full score for reference):

These symbols are those developed by Hugo Riemann in the late 19th century, and are still used in Europe, although not so much in the U.S. I find them useful, as they help to really teach and demonstrate harmonic function. It will be good for all our future work if you understand this symbolism and what it implies for harmonic relationships:

The most interesting and important piece here is the subdominant of the subdominant. In non-dualistic analysis, which is what is taught at most American colleges, this doesn't exist. There is no concept of a "IV/IV" but to me, this is one of the drawbacks to the usual, non-polar or non-dualistic system. Monistic approaches don't have a way to explain this Bb harmony, except to call it unusual or unexpected, when it seems so clear that Beethoven is swinging to the subdominant by way of its subdominant, in exact polarity to the swinging up to the Dominant through its dominant:

This approach makes the entire subdominant realm not just "something to proceed the Dominant" but an expressive realm all to itself, in equal balance with the dominant. This approach helps to embrace non-Classical, popular music forms as well. If we consider the blues, with its strong movement towards the Subdominant and its tendency to cadence V-IV-I rather than IV-V-I, we can see that this approach is not "wrong" or "uneducated," as much academic scholarship can tend to imply, but simply that blues and rock tends towards expressing the reciprocal realm, whereas more tonal music, even more tonal pop music like standard Broadway tunes and the like, swing more towards the overtonal. Think of a tune like "Let it Be," with its strong reciprocal chord progression:

Verse: C G Am F C G F C
Chorus: Am G F C C G F C

and compare it to a show tune like, oh pick anything!, Edelweiss from the Sound of Music:

A Section: C F C F G // C F C G C
B Section: G C F F#dim G7
A prime: C C7 F Fm C G7 C

with its strong emphasis on tonal, perfect cadences.

Now, if we interpret to the above graph to mean:

we can transform the picture into a graph of the relative harmonies.

Here, can interpret this graph to mean, still thinking in the key of C:

  1. A minor is the Relative Tonic
  2. D minor is the Relative Subdominant
  3. E minor is the Relative Dominant
  4. G minor could be the subdominant of the relative subdominant, but also the minor Dominant in C.
  5. B minor could, at a stretch, be considered the minor dominant of the relative dominant, but it is also the ii of the relative tonic.

What is worth noting here is that this is far as we can stretch this system and still stay within a diatonic realm. Once we go all the way down to the G minor, we loop back up to G major, because the next fifth below the G is of course our starting pitch of C.

Similarly, B minor would reach up to F#, which takes us to the farthest pole in diatonic harmony. Typically, the B reaches up a tritone to F, as in the cycle of fifths progression, thus keeping the system within the diatonic realm of the original tonic key. This is explained more completely in the next section.

Finally, when the roots of these primary and relative harmonies are paired together, we can see that, through modal shift, some seemingly distant harmonies, such as E major or A major, can be seen to be a part of an extended C major tonality. This is of course exactly what Beethoven seems to present in the first movement of the Waldstein, with the second theme in the key of E major and some time spent in the parallel minors of C, E and B.

2. The Major-Minor Diatonic System around the Cycle of Fifths

Looking at harmony more traditionally now, but with Polarity in the back of our minds, we can review the basic system in the keys of C and A minor, which you can then transpose to the other 12 keys. Fluency in 12 keys is an essential part of music making, whether classical or jazz. I will assign some keyboard exercises to help you gain such fluency.

The best summary I know is to look at the chord progression around the cycle of diatonic tones in any key. One motion, down by fifths, with a cross over at the tritone between B and F, creates the overtonal cycle, and the other, the reciprocal.

Learning this pattern in 12 keys on the keyboard is one of the best drills you can do for yourself if you want to understand tonal music. As you play, listen to the different emotional and expressive quality of the overtonal and reciprocal motion (sometimes referred to as progression and regression, but that sounds too judgemental in my view).

Overtonal ("Progressive") Cycle:

Reciprocal ("Regressive") Cycle:

Ideally, you should then play these progressions, beginning on different voicings for the first chord, moving smoothly to the next chord. Here's how you would start the overtonal series on different voicings:

For the overtonal cycle, any minor chord can be turned into a dominant seventh chord, lending more forward motion to the progression through the use of the secondary dominant. Try it!

For the reciprocal cycle, ii or vi can be voiced as a major triad, which strangely brightens up the motion. But it is not as convincing as the change into the secondary dominant.

The Secondary Dominant Cycle

Secondary Dominants are rarely used in complete succession around the cycle of fifths, but to learn them at the keyboard in this sequence is helpful. If this is clear to you, you will understand secondary dominants whenever you encounter them, whether in classical music, popular music or jazz.

The best way to learn this is to play the cycle of fifths sequence again, but insert the tone that creates the secondary dominant before the resolution down a fifth. Note that his means the I to IV chord requires the insertion of the seventh (the natural seventh of the harmonic series, perhap?), but that the other motions only require that the minor triad be transformed into a major triad, thus creating the leading tone which resolves up a step to the subsequent root.

Then you could also insert the seventh of the secondary dominant, completing the "V7 of ..." :

 

The Cycle in Minor

The minor mode is more complex, because there are really two modes plus two tonal variants of minor that come into play when we harmonize in minor. These are the aeolian and phrygian modes, plus the so-called harmonic and melodic minor scales, respectively:

While this theoretically creates many different possible harmonies, based on these various scales, in actual practice the Phrygian mode gives us primarily the Neopolitan harmonies, and the harmonic minor gives us the major dominant chord, which usually resolves to the tonic, unless it resolves as a deceptive cadence to VI. The melodic minor, as the name suggests, largely reflects the tendency of composers to raise the sixth and seventh steps of the scale on the ascending, while maintaining the natual (aeolian) scale on the descending.

Turning now to the cycle of fifths based on the minor mode:

The overtonal cycle, when starting on the relative minor tonic, resembles the major cycle, and in fact, for a few bars it sounds just like you are modulating back to C major. But then, by keeping the raised leading tone on the relative dominant, the pull back to major is heard:

The reciprocal cycle in minor is odd sounding, but useful to understand when you intend to move along this path, if even for one motion. Note that, since there is not the perfect V-I cadence, the progression remains completely within the natural minor or aeolian mode:

Minor reciprocal motion is commonly found in musical passages where the minor tonic note moves down a whole step to the lowered (natural) 7th, with the dominant minor chord expressed in the harmony. For example, in Bach's Jesu Meine Freude (here in C minor):

Notice here an essential polarity. The harmony first swings to the minor dominant, descends to the minor subdominant, then finishes the phrase back on the overtonal dominant seventh chord.

Chromatic Reciprocal (pre-Dominant) chords
including Neopolitan sixth and Augmented Sixths

In standard, common practice tonal harmony, what is fascinating is that on the overtonal side, the primary progression is that of the perfect and deceptive cadences. Once the V chord has been stated, the motion is towards tonic or, deceptively, towards the relative tonic. Thus, in a major key the movement is down a fifth, V7 to I or up a whole step, V7 to vi. In a minor key, the movement is down a fifth (also), or up a half step (V7 to VI). This latter cadence usually is then followed by a ii dminished seventh to V7 to i conclusion:

 

On the reciprocal side, presaged by that last progression, there are a great number of harmonies which, in tonal music, are seen to precede the dominant harmony. If you can see how all these are deeply related to each, and all partake of "reciprocal energy", you will have a much easier time recognizing these when used by composers in their compositions. Play these through on the piano!

As to the Augmented sixths, far too much complication has been made of them. They are really quite straighforward when you see them as being born from the subdominant triad in first inversion, with a melodic alteration before swinging up to the dominant. The basic motion began in a minor key, which only required the addition of the raised soprano to create the augment interval. In time, as in the second example, this motion was used in major, requiring two chromatic notes:

The above augmented sixth is the so-called Italian sixth, because the augmented sixth chord is created by the simple melodic movement. Notice that it only has three discrete voices in the chord.

The German sixth simply adds the fifth above the bass note, solidifying the sense of harmony by reinforcing creating a more solid (Germanic?) four voice harmony:

It is probably easiest to understand the French sixth as being born from the same subdominant triad, but with the added sixth. This chord always partakes of a dual nature. It is both a ii7 chord in first inversion, but also subdominant triad with an added sixth degree. This is true whether we are in the major or the minor mode

From this chord we can easily derive the French sixth by the same process, placing this chord in the next inversion, then expanding out the sixth into the minor sixth: