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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

When a lattice-diagram of a temperament utilizes a closed geometry to represent the tempering-out of the unison-vectors which define it, an extra dimension must be invoked to represent the warping of the dimensions which represent the generators of the tuning.

For a 2-dimensional tuning, the result is either a helix for an open-ended temperament, or a torus for a completely closed temperament such as an EDO. For a tuning of 3 or more dimensions, the result, since it invokes dimensions lying outside human abilities of visual perception, resembles a crumpled napkin. Below is an example, showing one means by which 31-edo tempers 3,5,7-prime-space:

The metaphor was originally coined by Brian McLaren in conversation with Joe Monzo in 1998. The crumpled-napkin geometry option is a feature of the Lattice window of Tonalsoft's Tonescape® software.

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