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Encyclopedia of Microtonal Music Theory

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"Badness" is a function of the complexity and error of a temperament, which increases if you increase either complexity or error. A common example of a badness function is C e · err, where C is the complexity and e is a fixed exponent. The idea of a badness function is to compare and rank temperaments, with a lower badness number being better.

Logflat Badness

A logflat badness measure chooses the exponent e of complexity C so that there is a badness such that below it an infinite number of temperaments can be found, but if e were to be raised, only a finite number could be found below any badness cutoff. If L is a regular temperament of rank r (or dimension r - 1), so that:

and so forth. If d is the number of primes in the prime limit p, so that d = pi(p), then the exponent e should be chosen as d / (d - r) to achieve logflatness.

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