An Instrument Is an Egg

What distinguishes a space as opposed to a mere set of points is some concept that binds the points together. Thus in Euclidean space the distance between points tells how close points are to each other …. As Frechet [a pioneer in the development of topology] pointed out, the binding property need not be the Euclidean distance function. In a metric space, which can be a two-dimensional Euclidean space, one speaks of the neighborhood of a point and means all those points whose distance from the point is less than some quantity… However, it is also possible to suppose that the neighborhoods, certain subsets of a given set of points, are specified in some way, even without the introduction of a metric. Such spaces are said to have a neighborhood topology.  (Morris Kline, Mathematical Thought, 1160, quoted in De Landa, 2002, 185; De Landa’s emphasis)

Piano diagrams are the result of a collaboration between David Pocknee and I.  Methodology is detailed here.

The piano diagrams are attempts to render visual the conceptual frameworks that mediate our engagement with instruments. When we approach a piano, the cardinal ordering of keys, hammers, and strings is not the instrument we have access to. All interactions with instrumental mechanism take place via some conceptual ‘screen’ that binds points along the surface into a neighborhood, which is ingrained into the musculature in training. As such the act of composing can be understood as the construction of one of these neighborhoods – an act of reframing instrumental space by shifting points of attraction along its surface, which grant us access to instruments by distinguishing spaces of possibility out of what before was potential only.
“In the beginner’s mind there are many possibilities; in the expert’s mind there are few.” (Suzuki, 1995 p. 13)
Imagine that first time you sat at the piano, completely ignorant of any technique. The surface of the keyboard extends along this horizontal plane, and aside from the black key/white key dichotomy, any key is just as likely as the next to be played. There are no learned exercises to regurgitate, no scales, melodies or chords to grab onto. This is keyboard surface as more or less ontologically symmetrical. The piano we perceive as beginners may be closer to what Manuel De Landa terms a flat ontology, consisting of unique, singular individuals, differing in spatial distribution but not ontological status. In that first lesson the teacher establishes basic postures and regions of bodily interaction conducive to ‘proper’ technique, and isolates a particular position to be explored more in depth. We have already raised the status of certain spaces, initiated the construction of a conceptual screen or neighborhood, and broken the ontological symmetry of the beginner’s mind. Pedagogical practices enacted in training can be understood as the emergence of arborescent, hierarchical structures built from accumulations of general types (scales, chords etc.) and particular instances (individual works that make up a given repertoire), a process of progressive differentiation that qualitatively ‘forms’ the player via a given repertoire. What these diagrams seek to express is pedagogical practice as a progression of symmetry breaking transformations that actuate privileged zones of activity along the keyboard, in construction of a particular neighborhood, within polarities established by repetition (De Landa, 2002). As Henry Threadgill would say practice makes permanent.
Every practice session applies a function to the practitioner - our domain - in order to more closely align their ability with their desire (whether musical or otherwise), producing a codomain.  We could view the initial domain as the instrument/performer assemblage in its beginner stage, the instrument as pre-differentiated, ontologically symmetrical, purely potential. Via the teacher’s imposition of a pedagogical filter on the student’s activities, every lesson and practice session applies a surjective function to the instrument/performer assemblage, creating a codomain that in strengthening certain elements necessarily omits others. Through repetition, applications accumulate over time producing successive domain to codomain relations, gradually constructing the conceptual screen that informs our engagement with instruments. Pedagogical practice, in determining the nature of differential relations, plots the points of attraction that progressively differentiate a given instrumental space.
These constructs imposed by pedagogy, depicted on keyboard surfaces in the diagrams are subject to changes in position and formation but without some procedure to step outside of our cultural conditioning a flat ontology is not possible for us. One cannot perceive an instrument outside of a given conceptual/pedagogical constellation that binds points of attraction into a neighborhood. As such I will need to introduce one more term to illustrate this reality, that of an ordinal series. Unlike a cardinal series (one, two, three…), which is defined in terms of bijective functions and quantitatively defines the nature of the elements it orders, an ordinal series (first, second, third…) implies asymmetrical relations between abstract elements; it orders elements without defining them quantitatively, only requiring that a given element be in between two other elements (De Landa, 2002). Via the differential relations employed by pedagogical practice, ordinal distances establish the neighborhoods that for us define instrumental surface.
This analogy of recurrence to size is a way of illustrating the virtual contortions a conceptual screen imposes on instrumental surface via desire. The diagrams point to a danger in certain narrow pedagogical approaches that tend to emphasize a repertoire at the expense of all others. Overemphasis of a particular set of neighborhoods over time risks hardening the interface between performer and instrument, effectively turning instruments into stone. Aided by the diagrams, one can imagine what such an instrument would look like. These are the instruments that we have allowed to petrify, to calcify in their fixity. It is not the instrument that turns to stone but ourselves.

Bennett, J. (2011, September 27). Retrieved January 2014, from https://www.youtube.com/watch?v=q607Ni23QjA
De Landa, M. (2002). Intensive Science and Virtual Philosophy. London: Continuum.
Suzuki, S. (1995). Zen Mind, Beginner’s Mind. Dixon, T. (Ed.). New York, NY: Weatherhill