Well temperament

A well temperament is a scale (formally an irregular temperament) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified.

Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally (equal temperament), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a regular temperament), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than meantone or pythagorean tunings, which were considered their own categories of temperament due to the objectives of tempering at the time. Additionally, note that 12edo was considered a well temperament.

In terms of modern xenharmonic theory, a well temperament can be considered a 1-to-1 detempering of an equal tuning, and this is the sense used on the wiki. As a result, they have historically attracted very little attention from xenharmonic theorists interested in abstract formalism - in fact, they emerge from the opposite set of constraints as regular temperament theory. While regular temperaments tune each generator to the same size and allow arbitrary numbers of notes, well temperaments fix the number of notes and allow the tuning of each generator to vary, and are thus more useful when tuning fixed-pitch instruments such as keyboards.

Derivation

The historical concept of 12-tone well temperament may be derived as the solution to the problem of 'wolf fifths' in the circle of fifths. In the extreme case, this leads to equal temperaments; well temperaments leave all the fifths unequal but close to just to achieve other goals.

In meantone[12], 11 tempered fifths of 697.5 cents produce a wolf fifth of 727.5 cents. This may be solved by only tuning, for instance, nine of the fifths flat, that way the meantone major third is still accessible in a wide range of keys, but the detuned fifths become only about 709.2 cents: much more acceptable

Alternatively, one might choose to vary the tunings of the flat fifths; eight 698.5c fifths, two 695.5c fifths, and two 710.5c fifths produces a large range of this tunings, many of which may be reasonably heard as 5-limit.