Syzygies: Two-Part Takes on Wythoff's MatrixVolume IIfor piano solo[5:23]Wythoff's matrix (2-dimensional array) is generated from two columns thereupon excluded from the matrix proper: the integers from 0; the floored products of their succesors and Phi (half of one more than the square root of 5, i.e., the "golden ratio"). Generation proceeds by line, applying the Fibonacci summation rule (each next term shall sum the two immediately preceding terms) to each integer pair:
Taken to infinity, the result yields all positive integers, each exactly once. Most striking immediately is that each matrix line's first number is the lowest not previously named.0 1 | 1 2 3 5 ...
1 3 | 4 7 11 18 ...
2 4 | 6 10 16 26 ...
...
Since Fibonacci family sequences modulo-N yield finite cycles, and as most such cycles invite partitioning into complementary halves, it occurred to me to tease some Wythoff matrix lines accordingly into two-part melodic structures. For the number of lines to be so treated, and so for the number of movements, I chose 15 -- recalling Sebastien Bach's Inventions and realizing that, within that work's C2-C6 range, 15 gives (for paired non-overlapping 1-to-3-octaves-wide voices) the total of range distributions available.
To realize Fibonacci proportions on a larger time scale, I harnessed them as the number of events in a given cycle, determining movement lengths. This was done indirectly, by specifying the modulus to be applied in each matrix-line generation. Pitch per cycle member was then determined via selective range and registration constraints.
Duration for each cycle pitch was set to parallel one of: the pitch class or PC occurrence frequency; the completed serial interval or SI occurrence frequency. The result for each movement was then scored in four separate interpretations.
A1. Homophonic, dynamics parallel duration (whole=fff, 64th=ppp).
A2. Homophonic, with durations range inverted, originally longest
notes becoming shortest and vice versa (whole=ppp, 64th=fff).
B1. Polyphonic, via retrogression of one part's order of durations.
B2. Polyphonic, again with the durations range inverted, as in A2.
Finally as systematic punctuation in all movements, a rest replaced each note having the least frequent duration value. The pairing of opposites -- especially the mutually inverse duration ranges in A1 vs A2 and B1 vs B2 -- inspired the work's title. In this volume movements with wider single-voice ranges are assigned the lesser overall lengths and vice versa. This counters the relationship of Volume I, with profound consequences for both structure and content.
Notation of rhythm in this score needs special explanation. I have superimposed two ways to indicate note duration: time-proportional horizontal spacing; length-specific note glyphs. The former (with some whitespace slivers inserted to clear note/barline collisions) is intended to serve as the principal guide in performance.
The latter, nodding to tradition, is meant to enhance perspective on the grouping of details. It references a value spectrum based, not on reciprocal powers of 2 as historically, but on the consecutive counting numbers (to avoid masses of ties that a metered notation of non-metrical duration sequences would entail). Its vocabulary consists of seven note-graphics, each alternatively modified via tenuto.
In either notation, this music's few apparent "beats" are inadvertent, and simultaneity in its polyphonic textures is virtually absent. To a pianist possibly bent on performing the work, I grant that substantial rhythmic liberty will be needed to approach a controlled (repeatable) rendition.