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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

Although the lattice structure, representing rational musical pitch relationships thru the implemenatation of prime- or odd-factorization, is theoretically unbounded and infinite, there are finite boundaries of the lattice which designate the limits of our perceptual abilities to differentiate between prime- or odd-affect, sonance, and "octave"-equivalence. All ratios within these boundaries can be clearly, if not precisely, differentiated.

Because my lattices are based on the premise of monophony, with rational complexity directly proportional to the size of the prime factors as well as the size of the exponents of those primes (see sonance), which translates on a lattice-diagram into distance from the central n0 along the prime-axes, all ratios lying outside the finite boundaries cannot be clearly differentiated from the simpler ones within.

These boundaries also designate the limits of interval size of categorical interval perception. They have yet to be conclusively determined, in part because of the effects of bridging. (Paul Erlich has done important work in this area - see his writings on harmonic entropy)

The important point about finity is that, while frequency vibrations may affect us in specific ways, if these affects cannot be perceived audibly, then they do not play a part in musical analysis, and therefore we recognize limits to enharmonicity.

Finity has been recognized mostly by theorists and composers who have grappled with:

Among them are Riemann, Schoenberg, Partch, and indeed any advocate of some particular tuning.

In the 1960s Fokker characterized the finite systems, which could be emulated at other parts of the lattice by different sets of exponents within the same set of prime-factors, as periodicity blocks.

My concept of bridging characterizes periodicity blocks as the finite systems which can be emulated at other parts of the lattice by different prime-factors, which distinguishes it from Fokker's conception.

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