Issue No. 11: How To Combine Triads Into Twelve-Tone Structures
Introduction
Ever since I began composing over twenty years ago, I have been interested in music that explores the relationships between complex, dissonant pitch materials and those that have fewer pitch classes, are based on thirds, and are historically more familiar to audiences. Unsurprisingly, then, I am fascinated with how we may combine triad types (major, minor, diminished, and augmented) with non-overlapping pitch class content into total chromatic collections for aggregate harmonies or twelve-tone rows.
In this issue of The Theisen Journal, I will be outlining ways we can combine triad types that exhaust the equal-tempered twelve-note aggregate.
Yes, I will be colloquially referring to these units as “triads” not “trichords.” I rather enjoy the historical connotations of that term.
No, I will not be exploring set classes, interval vectors, and such. I totally could, but I’m trying to keep this issue as pithy, accessible, and impactful as possible. If you want to do this set theory stuff knock yourself out.
And I’m absolutely aware that none of this is original, groundbreaking research. The point is to provide a handy guide that’s easy to read and fun to look at as a resource for composers, arrangers, and improvisors who wish to play around with these materials. Let’s dig in!
One Triad Type (The Most Basic)
Combining triads of a single type; in this case, augmented triads. [click to expand]
An obvious, elementary approach to combining four triads into a twelve-tone structure is simply to fuse four augmented triads whose roots are separated by semitone. I must admit that I have never really explored this possibility in my own music since it creates a sound world that in my mind too strongly evokes those late-in-life Liszt piano pieces where he is attempting to be eXtReMeLy SpOoPy™. (Although I must admit that Nuages Gris is a bizarre, enchanting, and effective little piece.)
It is also worth pointing out that if one combines two augmented triads whose roots are a whole step apart, the result is a whole tone collection (meaning that this approach to creating twelve-tone structures through triadic fusion produces outcomes similar to that of simply alternating between the two available whole tone collections).
Your options are frankly limited if you wish to combine four triads of the same type into a twelve-note aggregate. Sorry, but I don’t make the combinatorial rules. I just report ‘em.
Two Triad Types (Now It Is Getting Interesting)
Combining triads of two types. [click to expand]
If we are seeking to combine triads of two different types, our possibilities expand. Here we can connect major and minor triads of varied transposition distances as well as diminished triads with an augmented triad.
Example 1 shows two major triads separated by a whole step and two minor triads separated by the same distance. We could regard the two major triads as forming a lydian hexachord and the two minor triads as being the first six notes of a dorian scale based on the lowered sixth scale degree of the lydian. Another way of perceiving this example is a major chord with its doubly chromatic submediant (flat minor vi) plus a pair of triads with the same tonal relationship transposed a whole step away. Yet another hearing is a major triad plus a minor triad a whole step under with a major triad plus a minor triad a tritone away.
Example 2 also shows two major triads and two minor triads but the distance within each of those pairs is a tritone. Alternately, we could conceive the first six notes of a major scale and the first six notes of a separate major scale a tritone away. Another way of perceiving this example is a major chord with its doubly chromatic submediant plus a pair with the same relationship transposed a tritone away.
Example 3 shows three diminished triads with a complementary augmented triad. This partition is particularly useful if one desires the ability to weave between triadic, octatonic, and twelve-tone compositional spaces. The three diminished triads are separated by roots a major third (four semitones) apart; the pitch classes in the augmented triad are each a half step above the roots of the diminished triads.
Three Triad Types (Major and Minor plus Augmented or Diminished)
Combining triads of three types. [click to expand]
There are two options if we seek to combine three different triad types.
Example X shows a major triad and a minor triad a whole step below with two diminished triads a perfect fourth apart. This option seems particularly Stravinskian; consider the octatonic possibilities of merging any of the two chords of the four. Also notice that the diminished triads could be respelled/voiced in a way to create strong leading-tone chord tension seeking resolution toward either the major or minor triad (in Example X, A in one diminished triad is the leading tone of the B-flat minor triad and B in the other diminished triad is the leading tone of the C major triad).
Example Y contains our now familiar “uncanny” major-to-lowered-minor-submediant relationship plus two augmented triads from opposite whole tone collections.
Notice the potential crossovers between Example Y and Examples 1 and 2 above.
All Four Triad Types (Maximum Diversity for This Game)
Combining triads of four types. [click to expand]
There does exist a possibility for all four triad types: a major triad and a minor triad separated by three semitones plus a diminished triad and an augmented triad.
Next Issue…
In the next issue, I’ll explore combining traditional seventh chords (and related harmonies) into twelve-tone structures. Let me know what you think about this issue by leaving a comment below, commenting on social media posts, or shooting me a DM!
I would love to hear sonorities, twelve-tone rows, or improvisational patterns you create using the information in this issue.