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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A term coined by Christoph Wittmann, which indicates the "numerical universe" within which the frequency ratios of a tuning system may be represented mathematically.

Prime-space refers specifically to the objectification of the pitch-continuum in multi-dimensional space as a series of axes, each of which represents simultaneously one unique prime-factor and one unique dimension of space, and along each of which are equally-spaced points representing the exponents of those prime-factors. This convention is then used in graphing specific pitches onto a lattice-diagram.

The use of prime-space may be conscious or unconscious, and in Monzo's theory, plays an important role (probably the most important) in determining the finity of a tuning system.

According to Monzo's theory, a composer, performer, or listener already has in mind (consciously or unconsciously) a prime-space before encountering a piece of music. Often during the course of the piece, a harmonic or melodic event will force the subject to make a change in his conception of that particular prime-space.

The dimensions used in any prime-factor notation are determined directly by the choice of prime-space.

. . . . . . . . .
[Gene Ward Smith, Yahoo tuning-math, message 11249 (Thu Aug 5, 2004 10:16 pm)]

--- In tuning-math@yahoogroups.com, "monz" wrote:

>> i've always thought of prime-space in a very general
> sense, where the length of a unit step on each
> prime-axis can be anything, dependent on the particular
> lattice formula used by the theorist drawing the lattice.

The way to put that in mathematical language is that prime-space is a real topological vector space; this should be for some specific prime limit p and not an infinite-dimensional space, since then this definition is precise--the vector space "inherits" a topology (product topology) from the real numbers as a topolgical field.

Since a lattice can be defined in a topological vector space (as a discrete subgroup), this definition gives a precise, but rather abstract, sense in which the lattice of monzos for any given prime limit is uniquely defined.

As usual there isn't really a simple web exposition, but there do seem to be various places you can find out about topolgical vector spaces:

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