Ostomachion: Wind Chimes from Archimedes' Boxfor 12 MIDI instruments[10:00]This work exists in two parts: Score (the piece to be heard), and Catalog (an analytical grouping of the work's source materials by geometric type). The latter, meant to elucidate the former, is presented here first.
Catalog
Ostomachion ("Bone Fight") is an ancient fit-the-pieces-back-into-their-box puzzle and the subject of Archimedes' monograph of this name, history's earliest treatise on combinatorics. The puzzle consists of fourteen 3-to-5-sided flat pieces, which can without gaps or overlaps fill their square box in, all-told, 17152 ways, reducing -- if we discount rotations, reflections and a few duplicating anomalies -- to 268. These, taken as the fixed content of four "core triangles" subdividing the square, group finally into 24 "tessellation families", whose lead items comprise the puzzle's "core squares". (See references below.)All the tessellations are plotted on a 12-by-12 grid of subsquares via points forming its inner squares' corners. Thus, horizontally and vertically, points are disbursed along 13 parallel equidistant levels.
As 13 levels apply as well to inclusive chromatic octave subdivision, these tessellations seemed to invite musical realization of some sort. Taking the "cores" as my subject matter, I first converted their X/Y specs to LilyPond score files, separately for each of: the individual pieces; their intermediate triangular groupings; the full (triangles combined) squares. Each file of the first two types details all forms (rotations/reflections) and positions that its object assumes throughout the 24 square tessellations.
With these files I assembled -- to aid in the composition process, whatever that might consist of -- a Catalog, here offered for anyone interested in the analysis.
A word about "Odd Man Out!" on page 43. This tessellation, Cutler series #164 and my favorite, is not among "the 24". It is, however, unique, in containing none of the core triangles AND not honoring the line that bisects nearly all of the original 268 squares. It is also, finally, not to be found in my composition (see Score) -- being, as I said, "out".
For anything I have gleaned of Ostomachion history and theory as well as its specifications, I am indebted principally to three sources: Netz & Noel, The Archimedes Codex, Da Capo Press 2007; Chung & Graham, A Tour of Archimedes' Stomachion; and the 4UMI Play, Stomachion.
Score
I have reordered the 24 tessellation scores so as to pair them as variously as possible: the odd numbers ascending as originally, with the evens interlaced and descending. Interlaced in turn, externally separating the pairs, are uniquely positioned isolate puzzle pieces.The puzzle's intermediate "triangle" groupings are realized (not separately, but) as instrumental and rhythmic subsets of the full-square tessellations. I.e., each triangle's several voices share a single instrument (so, 4 per square) and sync rhythmically per their original specification.
But the timing between triangles I have made variable, shifting these apart: increasingly, through odd-numbered tessellations; decreasingly, through the even-numbered. Altogether, only two full-square instances actually do fit together: the first and last. Their pair-partners are the most out-of-whack, with intervening pairs variously so and the center ones approximately matched.
Random choices (of order, possibly also selection) have played a part, as I wanted an impression of "wind chimes" (unscripted, one presumes) overall: within range constraints, instruments and volume levels are randomly assigned. Some choices made "instead" by design have the intent to appear random: a tempo scheme, and the progression of subgroup-shifting silences.
All of the above I embodied in a program that (calculating via J, template editing via sed, scoring via LilyPond and page assembling via pdfjam) generated this listening Score, and has others, complete as is. (Note: its accidentals apply once only.) The output audios -- 71 MIDI files -- I combined via Rosegarden, then converted via TiMidity to WAV and via pacpl to MP3.
There are 3 untitled movements, each of 12 phrases lasting between 1 and 24 seconds.
My grandmother's wind chimes enjoyed a breeze far less often than not. Accordingly, this piece's main character is its silence. I recommend to the listener low volume and a wandering mind.