The musical staff notates Pitch Value Vectors whereas tablature, using fret numbers on string lines, denotes Position Value Vectors, forming a commuting algebra of Hilbert Spaces. In 2001 I demonstrated that music is semi-algebraic (Allen and Goudessenue).
Pitch Value Space is undefined without a connection to pitch, and when connected to pitch by a barycenter, becomes defined and complete.
A defined musical system must have at least 2 functions, the chromatic f(x) and the harmonic function g(x) that form a composite function with at most 1 common center (Music Multicentricity Theorem). Thus tonality is defined by the line of tonal projection that marries pitch to position to make a musical tone.
Since musical systems must have a tone generator (instrument or device) the music topos must be the triple composite function f⋅g⋅h where f(x) is a + b + c = 0 and g(x) > 0 is a scale center and h(x) > 0 an instrument center. A music cipher as defined here as an affine projection that marries R:Z pitch to position to compose a note [tone point as an orthonormal pair (position value, pitch value)]. The harmonic message is embedded in a musical system by the cipher which defines tonality, so that (harmony, tonality) is another orthonormal pair.
A cipher can also make a new note from one already known in a system.
The only algebraic operation in a musical topos is vector additions to a single barycenter according to a difference function defined by the complete lattice of the musical system, and according to the Boolean Arithmetic Operator of the Music Cipher which forms the geometry of tone value spaces by its prime ideals. The cipher model is therefore simple and natural compared to current music topology requiring two centers and several algebraic operators.
Music is composed by the finite union of notes and open intervals defined by the composite functions of the fundamental, the key, and the intonation algorithm.
Tonality, the sum total of every function, relation, and element in a musical system, is the same as the algebraic-logic interface (numeric key) of the pitch-position intonation algorithm that is precisely the triangle of cipher vectors formed between one logic and at least two algebraic sub lattices. The cipher vector defined by a complete musical lattice is also the same as the arithmetic tone values closure operator that defines tonal geometry. Specifically, the cipher is precisely the projection between the logic sub lattice and at least two algebraic sub lattices in the musical system, where the sub lattices all share the fundamental as 1 common center.
Therefore the cipher is equivalent to a point, a line, a triangle, and a sphere, reflections resulting from line-point duality in geometry. Without a common center for the R: Z cipher the musical clock is undefined: Euler's donut is dead.
The new model is a clock: the fundamental is the hour hand, the instrument position is the minute hand, and scale position is the third hand. Tonality, like time on the clock, is a vector as a composite of three functions with 1 fundamental in common. Therefore, tonality has at least two functions but at most one center.
Authors: Tuomas Korppi
Comments: 39 Pages.
A homology theory based on both near-standard and non-near-standard
microsimplices is constructed. Its basic properties, including
Eilenberg-Steenrod axioms for homology and continuity with respect to resolutions
of spaces, are proved.