Nuno Trocado

Continuing with the exploration of three-pitch sets, it's now time for the very special (012) trichord.

Let's take a C, a C# and a D. We can put the three notes in the same octave, a cluster of pitches as compact as possible. Or we can move each of the notes to another octave, up or down. Now, how many ways are there to voice a chord consisting of just those three notes? Let's use an additional constraint: no intervals larger than an octave between consecutive notes. Starting from a middle C, we can write the six possible combinations like this:

These are all instances of the (012) trichord class.

This expresses in a very particular way the sonority of the twelve-tone equal temperament, focusing on the smallest available intervals and its inversional equivalents.

The (012) trichord is also found in a simple ornamental technique: surrounding a target note with its chromatic neighbors above and below. Some people call this an enclosure, in the context of the analysis of bebop melodies and solos. The idea then is to perceive the target note and its neighbors as a single entity, and play all three pitches simultaneously. This is a form of integrating ornaments into harmony, a transformation that - we can speculate - is prevalent in the historical development of western music, e.g. leading tones being incorporated into chords. Why not take this idea further and promote the chromatic neighbors to the same harmonic hierarchy of chord tones?

For a Cmaj7 chord with the 9th, #11th and 13th¹, this yields:

This is what I'm practicing in the video.

While I go through the harmonization of the target notes - C E G B D F# A and back - I also cycle through the all possible voicing types seen before, changing the octave of individual pitches.

That's it. In a way, for me these are all possible voicings for a C major chord, although offering a particularly striking sonic effect.

Here's what's going on:

Append four combinatorial trichords¹: such that all twelve (4x3=12) notes together constitute an aggregate (the twelve notes of the chromatic scale).

The resulting row is not used exactly in dodecaphonic fashion, because we don't have to progress ineluctably from tone to tone; rather, we define four different places (the four trichords), and we can linger on any of them for as long as we like, or go back and forth between them, or use some as consonances and the others as dissonances. The integrity of the trichord, however, should be preserved—in principle all three notes are sounded, simultaneously or in succession, before we progress to the next trichord. This is similar to the technique of triad pairs, but dealing with other three-note groupings beyond the major and minor triads, and with four groupings in total instead of only two.

This idea is thoroughly explored in John O'Gallagher's book Twelve-Tone Improvisation—A Method for Using Tone Rows in Jazz.

Of the existing trichord classes, one that I've been focusing on is (014)². We get a nice intervallic combination of thirds and sixths together with minor seconds (or ninths, etc.) and major sevenths. It also captures the augmented second sound associated, e.g., with the harmonic minor scale, while being a building block for other frequent musical structures. For what it's worth, in a 2009 study, participants classified it as "non-familiar dissonant", associated with words like "repulsive", "hard", "cold", "cruel", "excited", "unstable", "dangerous", "oppressive", "unfamiliar" and "odd". Given that, what's not to like?

Building a row from juxtaposing (014) trichords is something that has interested many composers. It offers a high degree of symmetry and combinatorial possibilities. Examples of this specific type of tone rows are common, for example, in Schoenberg (op. 29; op. 41; op. 50c "Modern Psalm"; "Die Jakobsleiter") and Webern (op. 18, no. 2; op. 24; op. 29). One important feature is that two (014) trichords can be put together to generate a hexachord of class (014589)—e.g., {C, C#, E, F, Ab, A}—otherwise known as the augmented scale, or Babbitt's third order set. This hexachord is also related to Messiaen's third mode of limited transposition.

There's more than one way of combining the trichords to form the row. I like to use a structure with the following sets:

In the video, I'm practicing the trichords as simultaneities and in open voicing. I take each one of them and go through all the rotations³, ascending and descending using the entire range of the instrument. Here's the score:

This is just for illustration. Playing off the written notation is a very inefficient way of learning this or other concepts.