12-tone puzzle \\ triad sequences

Consideration: Complementary Hexachords

Rather than thinking single-mindedly about the division of the 12-tone chromatic scale as four sets of triads (which may or may not form recognisable triad pairs), we can also imagine the chromatic scale as being divided into two sets of complementary hexachords – six-tone scales which are indeed represented by or constructed from two adjacent triads. Using hexachord patterns instead of triads will lead to more flexibility in melodic design and a smoother stepwise motion, which may encourage explorative lyricism and the incorporation of already-familiar improvisational ‘language’ with a serial twist.

For example, we may start with the tritone-separated triad pairs solution of C Dm Gb Abm and combine each pair into a hexachord:

Example 1. Strong-strong hexachord complement

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Gb – AbmGbAbBbBDbEb

Note that as both hexachords are both diatonic and identical in construction, no one hexachord takes precedence. Additionally, in this context, since both hexachords are related to familiar (read: diatonic) melodic structures (construction being TTSTT, same as the first six pitches of the major scale) the musical result of combining these is likely to be relatively pleasant, and therefore it could be to our advantage to practice this hexachord pair as an extension of this serial improvisational approach.

These features of balance and pleasant musical effect will not always be the case, particularly in cases where one or both hexachords features adjacent semitones as part of the construction. The example below was derived from starting with a single triad pair of Cm Db (i-bII, relating to the minor Phrygian modes). I have suggested that the hexachord resembling the familiar Phrygian mode should be the ‘Primary’ (that is, that this hexachord pair would be used as an extension of contexts where the Cm Db triad pair would already be acceptable to use), and the remaining 6 pitches form the ‘Complement’, which cannot not produce a particularly strong or familiar triad pair sound.

Example 2. Strong-weak hexachord complement

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Cm – Db
Problem: two adjacent semitones in the Complement hexachord

N.B. my best solution for the complement triad pairs is D+ Esus4 – however it’s quite a complex sound with limited ‘standard’ applications!

Until now we have only considered hexachord complements as a by-product of starting with a familiar triad pair, around which to orbit our melodic designs. However, what if we take the opposite approach and start with a familiar hexachord? How about the blues scale? There’s a six-note scale that improvisers have been using continuously for over a century and to great musical result. So how does that hexachord complement look?

Example 3. Blues scale hexachord complement

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C Minor Blues

Without trying this, it’s possible to imagine a rather happy result. Two relatively familiar structures – the C minor blues scale as primary, and something from A major as a complement. However, we mustn’t forget to consider the implications of a Locrian-based hexachord in the secondary position. Take some time to sit with this idea and consider whether there is a meaningful setting in which it may be applied.