User:Zhenlige/Logical tempering

Logical tempering^{[idiosyncratic term]} means using a regular temperament (called the logical temperament^{[idiosyncratic term]}) to get “abstract” notes and intervals, but not directly the actual played notes. Different from “real” tempering, a logical temperament with torsion is allowed. The commas tempered out in the logical temperament are sometimes called “chromas”, which is a generalization of the concept of MOS chromas.

Multiple logical temperaments can be used at the same time. For example, one may first categorize intervals as alterations of 3-limit intervals, and then logically temper out 2187/2048 to get diatonic degrees.

When the actual notes are already tempered, to get unambiguous results, the logical temperament should be based on the tempered notes i.e. it should support the actual temperament.

Examples

Octave equivalence can be thought as logically tempering out 2/1.

The diatonic degrees (seconds, thirds etc.) of 3-limit intervals are the result of logical tempering into 7et, tempering out the diatonic chroma 2187/2048. Similarly, every MOS-based degree system can be thought as a logical temperament based on the equalized scale.

The conventional categorization of 7-limit intervals treats 5/4 as a major third and 7/4 as a minor seventh, which implies dominant as the logical temperament. A piece using adaptive JI with that framework can be thought as a dynamic detempering of a piece in dominant. Alternatively, 7/4 can be considered as an augmented sixth, giving septimal meantone logic.

The classical triads are made by stacking 5/4 and 6/5 intervals, meaning their difference 25/24 is a chroma, giving a dicot framework. Merging it with an actual 5-limit rank-2 temperament gives a unique MOS structure. For example, meantone gives the famous 5L 2s diatonic scale, and diaschismic gives 2L 8s.