The question that often arise for me when learn about a new theoretical concept is, "could I improvise using that?" or perhaps "could that concept inform my practice in some way?" Coming from a university system that emphasizes practice-led research means that I often think in terms of linking my academic research with my practice.
As way of example, here's something I was practicing today.
The book is Elliott Carter Studies, edited by Marguerite Boland and John Link. In Brenda Ravenscroft's essay "Expression and Design in Carter's Songs," she points out an interesting theoretical property of the all-triad-hexachord (ATH). Before I get to that, a little background on the ATH.
The ATH is a six-note chord that contains all combinations of three notes. It's usually described using set theory, which counts the semi-tones above an indeterminate pitch: 012478. So from a bass-note of C, the notes would be C, C#, D, E, G and Ab. It's inversion, 014678 also contains all combinations of three notes.
A little background information on three-note chords. Firstly, there aren't that many providing you regard inversions as identical, and don't specify register. For example, 016 (C, Db, Gb) is theoretically identical to 056 (C, F, Gb). The simplest way to check if two chords are the same is to name all of the intervals in the chord and see if they're identical. So the two chords in the last sentence both contain a minor 2nd, a perfect 4th and a tritone. Therefore they are identical.
So, all of the three-note chords are: 012, 013, 014, 015, 016, 024, 025, 026, 027, 036, 037 and 048, and all of those are contained within the ATH, 012478.
Seeing as I'm into Carter, I often practice incorporating this chord, in various transpositions and inversion into my playing. Sometimes I try and link it back to some sort of tonal reference, but most often I use it as a way to structure my "free" improvisations.
Ravenscroft states: " . . . in this passage Carter explores the hexachord's 'complement union property,' in which the ATH always results from a combination of a member of the subset set-class (0167) with any (04) set class member, provided there is no pitch-class duplication between the tetrachord and the dyad."
This means that a major third (04), combined with any 0167 that doesn't contain any of the pitches in it, will form an ATH. Here it is:
The first two 0167s are the two that don't duplicate pitches from the major third. The last chord, in brackets, is the second chord inverted and placed to show that it's simply a semitone-up transposition of the first chord.
Ravenscroft's observation provided me with a new way to think about the ATH, and perhaps improvise with it in a different way; a new way of thinking about something often results in new musical outcomes.
I also experimented with making the major third in the lower staff the first and third of a major tonality or third and fifth of a minor tonality and then using the 0167s as structures to improvise around. Sounded pretty interesting! They don't tend to sound very linear; the intervals themselves are the most prominent sound. To me that suggested that they're better used as a basis to play "around" rather than to be used exclusively, at least to my taste.
Of course, I also wanted to find out all the major thirds that could be combined with a single 0167:
The fact that there is four major thirds that combine with a single 0167, while there's only two 0167s that combine with a single major third is due to the 0167 being a symmetrical structure itself; two of the 0167s that combine with the major thirds are inversions of the others.
While improvising I also discovered another voicing I liked. Improvising often causes new ideas to bubble up from the depths. This voicing made the ATH sound even more tonal, which might come in handy:
In this case we have a 014678 where 0 is the pitch B, but with a C-minor triad in the bass. The inversion of the ATH looks like this:
A few month ago I read Dmitri Tymoczko's A Geometry of Music (2010). A point that fascinated me was that three-note chords voice-lead most efficiently to major-third transpositions of themselves: a C-major triad leads well to major triads E and Ab, for example. It's an interesting read with some points that make me want to improvise and compose with some of the ideas.
In the bass clef are minor triads a major third apart with efficient voice-leading. The top stave contains the notes that forms an ATH with each of the triads. Interesting that each of the three-note chords voice-lead pretty well to one another, not as well as the triads, but well enough to sound connected I think (not counting the transition across the bar-line). Here is the same exercise using major triads:
So now the task is to get familiar with all these in all keys. I'll probably do this by combining methodical practice (simply playing each voicing in all keys with the metronome) with more improvisation-based practice, where I use each chord as a point-of departure for improvising and try to sometimes return, other times progress through, each voicing in all keys.
Needless to say I have a lot to work on, but I'm excited to work hard on continuing to develop my language for improvisation. If I remember I'll make a recording of my practice and post it on here.
If you have any questions that can't be answered by a quick google search post them in the comments and I'll do my best to answer.
P.S. I also just noticed that the four groups of major thirds in the second example can form two major 7th chords a tritone apart. Tymoczko's book also talks about how three-note chords voice-lead well to similar chords a major third apart (as stated above), but also that four note chords voice-lead well to the same chords transposed minor thirds away. Tritone = two minor thirds!
P.P.S. This post brings the views of my blog to over 30,000, so thank you!