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“ We hear from mathematicians that bees have... solved a recondite problem, and have made their cells of the proper shape to hold the greatest possible amount of honey, with the least possible consumption of... wax in their construction.
[A] score of individuals work even at the commencement of the first cell. I was able ... to show this fact, by covering the edges of ... the extreme margin of the circumferential rim of a growing comb, with an extremely thin layer of melted vermilion wax;
[T]he colour was most delicately diffused by the bees as delicately as a painter could have done with his brush by atoms of the coloured wax ... worked into the growing edges of the cells all round. The work of construction ... a ... balance struck between many bees,... all trying to sweep equal spheres, and then building up, or leaving ungnawed, the planes of intersection between these spheres.
The bees, of course, no more knowing that they swept their spheres at one particular distance from each other, than they know what are the several angles of the hexagonal prisms and of the basal rhombic plates. The motive power of the process of natural selection having been economy of wax... "
From The Origin of Species by Charles Darwin
The musical materials and structure of Economy of Wax are substantially derived from the set text’s description of, and allusion to honeycomb construction (see above). To begin with, the honeycomb itself:
The geometry of the standard two-layer lattice shown here has some distinct properties. They include:
- A lining-up of points vertically between cells, due to the nature of interlocking hexagonal prisms arranged in two displaced layers (1).
- A base for each cell with three interlocking “rhombic plates”, each of which in turn has four sides (2). This comes out of (1), using the point of intersection of the three prisms in the opposing layer.
There is also a narrative inherent to the text, aspects of which are worth pointing out.
- A honeycomb being built up from scratch (3).
- The actions of bees within the process of construction, most notably their incredible number and freneticism, and their spherical movements in distributing wax (4).
The piece as a whole has a particular discrete, harmonic event of two seconds duration every twelve seconds. The outlaying of this gesture is designed to reflect aurally the visual qualities of the honeycomb, especially its internal repetitions and geometrical consistency. In effect, these events are a series of harmonies whose character is based on (1) and (3); that is, chords that expand and contract simultaneously up and down through register vis à vis the honeycomb shape, and also slowly build up through time in the same manner, in keeping with the text’s temporality. This bumpy harmonic ‘crescendo’ happens four times, in direct relationship to the four divisions of the text, and also once in an overlaid, overarching fashion through the whole work. The four sub-‘crescendi’ are themselves arranged as a harmonic ‘crescendo’. Here is a comprehensive visual representation of all that:
The sub-‘crescendi’ also act as skeletons for the four continuous, middleground sections which contiguously make up the piece. These sections correspond directly to the divisions in the text, are pervasively (4), and are predicated on a sort of gross word-painting, where textual meaning is articulated in a general manner through textural character.
A – General excitement over the mathematics of the honeycomb.
~ Hocketed ‘moto perpetuo’ reflects the frenetic and constant nature of this attachment.
~ ‘Event montage’ canon represents simultaneous zoomings-in on “construction” of particular cells.
~ Liberamente textures indicate formidable (if not excessive) “consumption”.
B – A celebration of experimental procedure
~ Waxing and waning tremolo and trills mimic “a score of individuals work”
~ Mix between measured and liberamente cycles represents “covering the edges”
~ ‘Dance’ with extreme registers, frenzied punctuations and circular melodic lines reflects “covering…the extreme margin…with…vermillion wax”
C – An up-close, voyeuristic look at construction itself.
~ Sporadic, registrally expanding chordal enunciation, very similar to overall work’s structure and consequently to (1) and (3). Two ‘crescendi’, one reflecting the delicate diffusion of wax, the other the “building up”of cells. Registrally contained lines “sweep equal spheres” and periodically bump into each other on “planes of intersection”.
~ Short episodes of liberamente, allusive to B, imply refocusing.
D – General excitement over the mathematics of the honeycomb.
~ Hocketed ‘moto perpetuo’, allusive to A, reflects the frenetic and constant nature of this attachment.
~ Mix between measured and liberamente cycles, allusive to B and C, represents experimental procedure
~ Final ecstatic heraldry for the textual and scientific conclusion, “The motive power of the process of natural selection having been economy of wax…”
Other aspects worth noting:
~ All pitch material comes from the serial manipulation of a melodic line whose shape mirrors the harmonic ‘cresecendo’ discussed above; that is, embodies (1) and (3).
- The constant metric interchange between x/4 and x/6 is a direct result of (2), a geometry with significant interplay between 2, 3 and their immediate multiples.