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Encyclopedia of Microtonal Music Theory

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periodicity block

[Joe Monzo, with Graham Breed and Paul Erlich]

A periodicity-block is a geometrical model of a closed musical tuning system defined by unison-vectors. The term was used by Fokker in his lattice theory to describe one multi-dimensional region of the lattice which could substitute for another, by means of unison vectors. It is the unit-cell of a periodic tiling of the lattice.

This concept is allied with that of finity, which Monzo developed independently of Fokker. The periodicity blocks quantify the finity of the system.

Periodicity blocks enclose a certain number of discrete categorical intervals or pitch-classes, and the unison vectors are intervals which are small enough that pitches within the block can represent or imply pitches outside of it which have different prime or odd factors, which is related to Monzo's concept of bridging.

See also Fokker periodicity-block. For a detailed explanation, see Erlich, A Gentle Introduction to Fokker Periodicity-Blocks

Paul Erlich comments:

That's correct, but Fokker generally set the overall prime-limit (either 5 or 7) beforehand and all unison vectors connect pitches (or intervals) within this limit.

Erlich has pointed out that Monzo's concept of 'bridges' refers specifically to the kinds of prime-factors and not the numbers of prime-factors, which is what Fokker described with the periodicity block concept [see also xenharmonic-bridge].

This number of pitches in a periodicty-block can be calculated by a matrix determinant, using the prime-factors of the ratios at either ends of the bridges or unison vectors to fill the matrix.

Some observations on the history of the term:

. . . . . . . . .
[Gene Ward Smith]

This is Gene Ward Smith's formula for finding what he calls a "notation" for the ratios enclosed within a just intonation periodicity-block: