Detempering

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In regular temperament theory, detempering is the process of taking a tempered pitch system and replacing each of its pitches with one or more pitches from its preimage, that is, the (typically JI) pitches that the pitch represents. Specifically, a detempered scale or a detempering has each pitch of a tempered scale (according to a fixed regular temperament) replaced with some set of JI interpretations of the pitch under the temperament mapping. If exactly one JI interpretation is used for each pitch, then the detempering is called a one-to-one detempering, or a transversal scale. Ideally the resultant JI pitch system will have a compact lattice.

A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.

The term deregularization can be used as a JI-agnostic alternative to detempering. In this sense, diasem (LMLSLMLSL) is a deregularization of semiquartal (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.

It is a distinct concept from a quasi-equal rational tuning, or neji, the main difference being that a neji replaces an equal temperament's pitches with pitches from a single harmonic series.

Examples