Twelve-Tone Games & Opusmodus

Following on from my earlier post on Josef Hauer’s system of atonal composition, I’d like to share a few short pieces inspired by his zwölftonspiel approach.

The previous pieces I posted (Tropic Preludes) were based on loosely on concept of a klangreihe as a continuum, but I progressed through the component tetrachords freely, as the intuition and mood took me. The zwölftonspiel approach to composition, however, generally involves a more systematic, ordered approach to music, which can ultimately be resolved to the ordering of the initial twelve-tone row. In a zwölftonspiel movement through the continuum – or harmonic band – usually occurs at a steady rate and the rhythmic units and dynamics are usually very simple (see this – rather large – pdf for a single-page overview of zwölftonspiel principles).

Some notes on each short piece:

1. Zwölftonspiel for Leigh Wright (Winter Solstice 2013)
Using Hauer’s 14th trope, ordered into the hexachords (f#, g#, a#, d#, a, f) and (e c d c# g b). This piece is in two halves: in the second half, the hexachords both rotate one step left, to become (g#, a#, d#, a, f, f#) and (c d c# g b e), creating two distinct, but similar, harmonic bands:

The basic ‘pulse’ of this piece is quarter beats: a new tetrachord each quarter beat. In each beat, two of the voices will in fact play 1/8th notes. Which instruments these are is also determined by the harmonic band, depending on the distance between the ‘new’ (red) tone current in the tetrachord and the ‘new’ tone in the preceding tetrachord. The resulting scheme is:

0 rows difference = Voices 1 & 2 play 1/8th notes
1 row difference = Voices 3 & 4 play 1/8th notes
2 rows difference = Voices 1 & 4 play 1/8th notes
3 row difference = Voices 2 & 3 play 1/8th notes

A ‘melic design’, used to assign the tones from each tetrachord to voices, was created using chance procedures and applied to both sections. Below is the melic design for section A – repeated notes are tied, giving note lengths of up to 1/2 duration.

You can see how the melic design translates to the music below – the recorder (top staff in the notation) corresponds to the red line in the above scheme, switching to the green line after 12 beats (corresponding to the fifth bar below).

2. Zwölftonspiel for Kim Cascone (18/12/13)
This piece was fairly intuitively written. The continuum here – (b#a f g f# c) (b d# c# e g# d) is repeated four times using a static melic scheme. For each repetition a subtly different rhythm scheme cycles between the voices based on patterns of 1/4., 1/4 and 1/8 beats to produce a more complex, polyphonic texture.

3. Zwölftonspiel for Layla Legard (16/11/13)
This was written on the fly using Logic as an example for one of my students interested in Hauer’s twleve-tone ideas. The continuum was input on the piano roll as a series tetrachords, which were freely split and articulated between four voices. These four voices then became the basis of a melic design, that is: the material is repeated four times, each voice playing and alternate voice’s material until the cycle has been completed.

4. Melancholic Supercontinuum (09/01/14)
For me, this is the most successful experiment. My original intention in looking at Hauer’s music was to try and explore some new approaches for the next Pneumatic Consort album, however I also hope to extend this piece over the summer into something more significant: a companion to Angelystor, perhaps? This uses the idea of a ‘supercontinuum’, composed of two inter-modulating harmonic bands. I’ll have to keep the exact composition of these continua secret for the moment, but they are both loosely inspired by certain aspects of John Dowland’s Lachrimae Antiquae. In this study, the supercontinuum (24 tetrachords) is played through twice, with different voicings, progressing to the next tetrachord every bar.

Melancholic Supercontinuum, spectrogram.

Notes toward a “free klangreihe” composition for the Pneumatic Consort.

I’m also very happy to announce that the release of a new composition tool, called Opusmodus, is imminent. This is an environment for computer-assisted composition based around LISP scripting. I’m pleased to have been involved in a small way in the development of this software and was delighted when the developer, Janusz Podrazik, told me that Dominik Šedivý (see my earlier Hauer post) had been brought in to help develop a series of functions related to the trope system.

The Opusmodus composition environment.

The trope functions provided by Opusmodus have potential not just for facilitating composition and ‘contemplation of the twelve-tone universe’, but also for musical analysis: functions exist to look at a twelve-tone row as a series of pitch-classes and return the corresponding trope from Hauer’s system.

The core functions of interest to those exploring Hauer’s music are:

gen-trope: Generate a list comprising of two hexachords from a given series of pitch-classes, intervals or pitches.

trope-hexachord: Retrieve either or both hexachords from Hauer’s set of tropes as pitch-classes, intervals or pitches.

trope-intervals: Analyse a given twelve-tone row (pitch-class or pitches) and return two intervallic hexachords.

trope-analysis: Describes a twelve-tone row in tropic terms – I think it’s worth giving an example here:

(trope-analysis ‘(4 9 5 1 0 2 3 11 9 6 8 7 10))

=> 12-tone Row: (4 9 5 1 0 2 3 11 6 8 7 10)
Trope: (1 1 2 1 4)
Number: 29ab
Integer Chord: (((8 11 15) (7 10) 6) ((2 5 9) (1 4) 0))
Chord Structure: ((gs4b4ds5 g4as4 fs4) (d4f4a4 cs4e4 c4))
Voice Layer: (((ds4) (fs4 g4 gs4) (as4 b4)) ((e4 f4) (a4) (c4 cs4 d4)))

There are also a series of klangreihen functions for defining and generating the tetrachordal ‘harmonic bands’ that underpin Hauer’s compositional process. These functions are: klangreihen, klangreihen-rnd and klangreihen-series. I’ll be covering Opusmodus in more detail in a future post.

Opusmodus will hopefully be publicly available in the next few weeks, for Mac systems running OSX.