Nuno Trocado

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.