Quartal Harmony: Tetrachords

In today’s post we will continue our series on quartal harmony with a quick look at quartal tetrachords. When we began with dyads, we saw there are three quartal dyad types: P, A, and D. With quartal trichords, there are nine types (3 * 3). With quartal tetrachords, we now have twenty-seven types (3 * 3 * 3). These are summarized in Figure 2. Of the twenty-seven, only eleven occur in our four scale harmonizations.

As before, the most common tetrachord type is based on stacked perfect fourths, PPP. This type occurs nine times in total across the harmonizations, making it once again the most harmonically ambiguous chord type. PPP occurs four times in the Major harmonization, twice in the Melodic Minor, and once in each of the other harmonizations.

After PPP, the next most common tetrachords are PPA and APP, both of which occur four times, once in each of the harmonizations. Both of these are very useful voicings. APP can easily serve as a Maj7#11 and PPA can serve as a dominant, as we shall see next.

When looking across the four harmonizations, we see an interesting detail on the both the supertonic and dominant degrees: they each share the same tetrachord across the harmonizations. In all cases, the supertonic is a PPP, and the dominant is a PPA. In other words, the II-V quartal tetrachords are the same for Major, Melodic Minor, Harmonic Minor, and Harmonic Major.

We mentioned the Viennese trichord in our last post. Any tetrachord that contains either PA or AP is a superset of the Viennese trichord. Our next most common tetrachord is PAP, which occurs three times in the harmonizations. Once can see PAP as an interlocked PA and AP, or an interlocked Viennese trichord. A Viennese tetrachord perhaps? This voicing also functions well as Maj7 11 chord.

Fig. 1: Quartal tetrachord harmonizations.

Quartal Tetrachords

Fig. 2: Statistics on tetrachord occurrences in scale harmonizations.

Quartal Tetrachords Chart

Quartal Harmony: Trichords

Today we continue the discussion on quartal harmony with trichords–yesterday’s topic on dyads was a bit of a warm-up. Things are getting more interesting now. There are three varieties of fourth: perfect, augmented, and diminished, or using our labels, P, A, and D. In order to construct a trichord, we need two intervals. That gives us a total of nine different types of quartal trichord. These are shown in the Fig. 2 chart below. When looking at the harmonizations of our four scale types, Major, Melodic Minor, Harmonic Minor, and Harmonic Major, we can make some observations about how often the trichords appear.

Two of the trichords, AA and DD, do not appear in any of the harmonizations. In AA, the outer voices form an augmented seventh, which is enharmonically equivalent to an octave. In DD, the trichord is enharmonically equivalent to an augmented triad.

We saw yesterday that P was the most common dyad, and see now that PP is the most common trichord. It occurs five times in the Major harmonization, three times in the Melodic Minor, and two times in each of the Harmonic Minor and Harmonic Major. So then, just like the P dyad, the PP trichord is the most harmonically ambiguous of the quartal trichords.

The next most common trichords are two of my favorites. They combine an outer major seventh, and inner perfect and augmented fourths, or PA and AP. I love both of these sonorities and used them often in my early composing. (Berg and Webern loved them too, so much so that they are sometimes referred to as the Viennese trichord.)

Two of the trichords occur in only one harmonization. PD occurs in only the Harmonic Major harmonization, and DP occurs in only the Harmonic Minor. These two trichords are the most harmonically specific.

I find it interesting to look at the similarities and differences among the four harmonizations. One thing that leaps out is that all four harmonizations share the same trichords on modes 1, 2, and 5, or on the tonic, supertonic, and dominant. So, for example, a riff on the tonic and supertonic trichords would be completely harmonically ambiguous; it would fit with any one of the four scales. Somewhat surprisingly, the dominant is the same for all four.

Not surprising is where the most variation occurs between the harmonizations: on the mediant and leading tone. Both of these degrees (or modes) contain both the third and the sixth, which is where all of the variation between these four scales takes place. Contrast this with the subdominant and submediant. The subdominant trichords contain the third, so the two with the lowered third are the same, and the two with the natural third are the same. For the submediant, the two with the lowered sixth are the same, and the two with the natural sixth are the same.

Fig. 1: Quartal Trichord Harmonizations

Quartal Trichords

Fig. 2: statistics about how often each trichord type occurs in the different harmonizations.

Quartal Trichords Chart

That’s it for today’s installment. In later posts, I will look at quartal tetrachords, as well as usage of inversions for the trichords we looked at today.

Quartal Harmony: Dyads

More follow-ups to my recent visit to the Aebersold Summer Jazz Workshop. I attended master classes with four great jazz guitarists: Corey Christiansen, Dave Stryker, Mike DeLiddo, and Craig Wagner. All four of these musicians gave me things to work on. One of the discussions that we got into with Corey was about quartal harmony, something I had worked with him on previous visits. I decided that it would be good to do an in-depth study on this material, so I plan on doing a series of blog posts. My ultimate goal will be to incorporate this into my jazz comping and soloing, but I am going to start out with a high-level survey of the territory. To do this, I’ll look at quartal chords of various size, dyads, trichords, tetrachords, pentachords, and so on. For each size of chord, I’ll look at all of the different possible quartal chord types, that is, all of the different possible combinations of the interval of a fourth. I’ll also look at where these different types of chord appear in harmonizations of four different scale types: Major, Melodic Minor, Harmonic Minor, and Harmonic Major; these correspond to the twenty-eight different modes I made reference to in my last post.

Let’s start with dyads. There are three different qualities of fourth that I will look at: Perfect Fourth, Augmented Fourth, and Diminished Fourth. Since we are only dealing with fourths, I’ll use a simple label and just call them P, A, and D. The first figure below shows where each of these three different dyads occur in harmonizations of our four different scale types.

Quartal Dyads

The next figure contains some statistics about how often each dyad type occurs in the different harmonizations.

Quartal Dyads Chart

There are a few things to note. First, it’s clear that the prefect fourth appears the most often: eighteen times out of twenty-eight in total. In a sense, this makes the perfect fourth the most ambiguous of the three dyads, since it occurs in so many contexts. Another interesting insight is the augmented fourth; since it only occurs once in the Major scale, it has traditionally been taken that it serves well to establish the scale type and key. However, the augmented fourth appears twice in each of the Melodic Minor, Harmonic Minor, and Harmonic Major scales, arguably making each of these scales more harmonically ambiguous than the major. Another thing that strikes me is that the Melodic Minor, Harmonic Minor, and Harmonic Major scales have the same number of occurrences of the different dyads P (four times), A (two times), and D (one time).

That’s it for today. Next time, I’ll do the same treatment for quartal trichords.

A Mode For Every Day Of The Year

I just got back from two weeks at the Jamey Aebersold Summer Jazz Workshop. It was amazing, as always, and especially important since this is the final session before Jamey retires after running the “camps” for over fifty years. I sat in on Pat Harbison’s advanced music theory class, and I got some great ideas out of that. One idea in particular got me thinking about digging deeper into the sound of specific modes. Pat mentioned something along the lines of “twenty-eight modes ought to be enough.” Since there are about twenty-eight days in a month, and twelve months, I saw how you could practice a different mode on a different root note every day of the year. Take the month as your root, and the day as your mode number. The base scales I chose were Major, Melodic Minor, Harmonic Minor, Harmonic Major. For the extra days in the month, up to three in a month with thirty-one days, I added Diminished (mode 1), Diminished (mode 2), and Whole Tone. Today is July 17, so that means my mode of the day is the second mode of Harmonic Minor, with a root note of F# or Gb. The whole scheme is summarized in the charts below.

One mode per day