Well temperament: Difference between revisions

A '''well temperament''' (or '''circulating temperament''') is a [[tuning system]] which is regarded as an approximation of an [[equal tuning]], has the same [[equave]] as that equal tuning and preserves the transposability of the equal tuning, but is not exactly the equal tuning being approximated. Historical well temperaments were formed by stacking a combination of pure [[3/2]] and flattened fifths and had 12 nearly equal steps.

One of the advantages of these tunings is that because they are not quite equal, each chord (or key) has a slightly different character because the interval sizes have changed slightly.

Despite their sharing the word "temperament" in their names, well temperaments and the modern notion of [[regular temperament]]s are two different concepts. In fact, a well temperament is an ''ir''regular temperament. But sometimes one can guide construction of the other.

 

 

 

* [[Kirnberger III]] – Kirnberger temperament III

# four tempered fifths (C–G, D–A, G–D and A–E) are flat by 1/4 syntonic comma (→ [[quarter comma meantone]])

# seven [[3/2|pure]] fifths

* [[Werck3|Werckmeister III]] – Werckmeister temperament III

# four tempered fifths (C–G, D–A, G–D and B–F#) are tuned flat by 1/4 comma (''Werckmeister did not specify whether the syntonic or [[pythagorean comma|Pythagorean]] comma should be used, so either is acceptable'')

* [[Werckmeister IV]] – Werckmeister temperament IV

# five tempered fifths (C–G, D–A, E–B, F#-C# and Bb–F) are tuned flat by 1/3 comma

 

# five tempered fifths (D–A, A-E, F#-C#, C#-G# and F–C) are tuned flat by 1/4 comma

# six [[3/2|pure]] fifths

 

* [[Septenarius]] – Septenarius temperament (Werckmeister VI)

# six tempered fifths (C-G, G-D, D-A, B-F#, F#-C# and Bb-F) are tuned flat based on division of string length

# five [[3/2|pure]] fifths

* [[Young I]] – Young temperament I

=== Detempering or deregularizing ===

Well temperaments can be obtained by [[Detempering|detempering or deregularizing]] an equal tuning. This implies going from a [[rank]]-1 temperament to a multirank temperament by adding one (or more) extra generator(s) – a common choice is to add a pure [[octave]] –, which creates an imperfect generator at the end of the generator chain. Whereas historical well temperaments often make use of irregular patterns of fifth sizes around the circle of fifths, detemperaments have identical generators all along the circle except for the imperfect generator.

A similar process is to pick a mos scale with the desired number of tones and a [[step ratio]] close to 1. If the step ratio is [[superparticular]], then it is also a [[maximally even]] scale. In that particular case, the resulting well temperament is not only a detemperament, but also a subset of a finer equal tuning, where individual steps are usually [[comma]]-sized. If the superset of the particular detemperament or deregularization is a fine enough equal tuning, it can have sisters with other [[superpartient]] step ratios.

== Relation to regular temperaments ==

Through the lens of regular temperament theory, a well temperament can be viewed as a result of applying an irregular [[tuning map]] to the abstract intervals of an [[equal temperament]] (i.e. a rank-1 abstract regular temperament), though tuning maps in the technical sense are defined to be regular. However, note that when nejis are considered well temperaments in this sense, the JI ratios the intervals are said to represent and the actual JI ratios of the neji tuning must be distinguished, and the JI ratios that occur in the neji should not be assumed to be consistent with the val.