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

NEUROLOGICALLY DIVERGENT – A Dramatic Recitiation

Earlier this month, the Pulitzer prizes were announced. As stated by the Pulizter organization, the music prize is awarded

For distinguished musical composition by an American that has had its first performance or recording in the United States during the year, Fifteen thousand dollars ($15,000).

The music prize was first awarded in 1943 to William Schuman for Secular Cantata No. 2 A Free Song, which had been premiered by the Boston Symphony Orchestra the previous year. The later recipients were, for the most part, well-known composers of contemporary concert music, or what some people term contemporary classical music. The winners were all men until 1983, when Ellen Taffe Zwilich won for Symphony No. I (Three Movements for Orchestra). In 1999, Melinda Wagner won for Concerto for Flute, Strings and Percussion. Since 2000, women have won more frequently, including Jennifer Hidgon (2010), Caroline Shaw (2013), Julia Wolfe (2016), and Du Yun (2017). In terms of genre, the prize had always gone to contemporary concert music, with the exceptions of 1997 when Wynton Marsalis won, and 2007, when Ornette Coleman won.

On April 16, 2018, it was announced that Kendrick Lamar had won the prize for DAMN.,

a virtuosic song collection unified by its vernacular authenticity and rhythmic dynamism that offers affecting vignettes capturing the complexity of modern African-American life.

This was the first time the prize had been awarded to a piece in the genre of Hip-Hop. Depending on whether your definition of popular music includes jazz or not, this was the first award to a piece of pop music. Several people, including straight up Internet trolls and probably some who should better, were upset enough by this award to take to social media to voice their displeasure. The Twitter account @NewMusicDrama collected the worst of these, conducted a poll, and announced the winner:

Seeing the invitation to conduct a dramatic reading, while enjoying a leisurely Saturday afternoon, I leapt at the opportunity. I now present the finished piece, NEUROLIGICALLY DIVERGENT

Early Beethoven

I’m only 21 pieces in so far, so I will need to pick up the pace to make it through all 194 pieces with opus number this year. Right now, I’m listening to the Violin Sonata No. 1 in D Major, Op. 12 number 1, and just came across my first “favorite moment” in these early pieces. It’s a cadential figure that happens in the first movement, at the end of the second theme.

Beethoven Op. 12 No. 1, first movement

From what I can recall, I first heard this piece in a performance in Von Kuster Hall at UWO, around 1987 or 1988. This moment is striking, and you can hear why it would appeal to someone who has played rock guitar. Even better than a power chord!

Was Brahms Quoting Beethoven?

I was listening to the WQXR broadcast of the Vienna Philharmonic at Carnegie Hall last night. It was an all Brahms program: The Academic Festival Overture, The Haydn Variations, and the first symphony. I grabbed my trusty old Dover score and followed along for the performance of the symphony. I was not paying too close attention, but even so, something leaped out at me at the start of the second movement. The opening theme begins with a striking similarity to the first theme of Beethoven’s sixth symphony. I grabbed my Beethoven score, and sure enough, the first seven notes of both themes are the same. The Brahms is in E Major, the Beethoven is in F major, so the Brahms is naturally written a semitone lower. Both themes are given to the first violins. The rhythms and tempo are different.

Brahms quoting Beethoven

I’m wondering if this was a purposeful quote, or just a coincidence. Also makes me wonder if there are any other Easter eggs in “Beethoven’s 10th?”

Beethoven Listening Project

Nothing like a new year to start a new project! One of mine for 2018 is to listen to all of the works of Ludwig van Beethoven, from start to finish. How would you like to join me? We’re just getting started with the Piano Trio in Eb Major, Op. 1 No. 1.

Mondestrunken

Just watched Moonlight, and, happily, it’s great. Need to watch it again on short order. Don’t know if composer Nicholas Britell truly had Vaughan Williams’ Fantasia on a Theme by Thomas Tallis in his ear as he wrote the score, but I had to listen to that piece right away on finishing the film. It fit right in.