# Well temperament

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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 temperaments are two different concepts, though sometimes one can guide construction of the other.

## Historical well temperaments

1. one tempered fifth (D–A) is flat by 1 syntonic comma
2. one tempered fifth (C#-Ab) is flat by 1 schisma
3. ten pure fifths
1. two tempered fifths (D–A and A–E) are flat by 1/2 syntonic comma (→ 1/2-comma meantone)
2. one tempered fifth (C#-Ab) is flat by 1 schisma
3. nine pure fifths
1. four tempered fifths (C–G, D–A, G–D and A–E) are flat by 1/4 syntonic comma (→ quarter comma meantone)
2. one tempered fifth (F#–Db) is flat by a schisma
3. seven pure fifths
• Werck3 – Werckmeister temperament III
1. 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 should be used, so either is acceptable)
2. eight pure fifths
1. five tempered fifths (C–G, D–A, E–B, F#-C# and Bb–F) are tuned flat by 1/3 comma
2. two tempered fifths (G#–D# and Eb–Bb) are tuned sharp by 1/3 comma
3. five pure fifths
1. five tempered fifths (D–A, A-E, F#-C#, C#-G# and F–C) are tuned flat by 1/4 comma
2. one tempered fifth (G#–D#) is tuned sharp by 1/4 comma
3. six pure fifths
• Septenarius – Septenarius temperament (Werckmeister VI)
1. 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
2. one tempered fifth (G#–D#) is tuned sharp based on division of string length
3. five pure fifths
1. four tempered fifths (C–G, D–A, G–D and A–E) are tuned flat by 3/16 syntonic comma
2. four tempered fifths (E-B, B–F#, Bb–F and F–C) are tuned flat by 1/4 Pythagorean comma less 3/16 syntonic comma
3. four pure fifths (F#–C#, C#–G#, G#–Eb and Eb–Bb)
• Vallotti – Vallotti/Young temperament II
1. six tempered fifths (C–G, D–A, E–B, F–C, G–D and A–E) are flat by 1/6 Pythagorean comma
2. six pure fifths

## Classification by approaches

There are several approaches to well temperaments. These are not strictly mutually exclusive, but they provide different frameworks that cater to various goals.

### Circle of fifths

Well temperaments can be structured around the usual uneven distribution of differently-sized fifths, but with a wider palette of fifths, such as superpyth fifths (approx. 702 ¢-720 ¢) and flattone fifths (approx. 691 ¢-695 ¢). Consequently, major thirds also come in various sizes, sometimes approximating other intervals than the usual 5/4, such as 9/7 and 14/11. For example: Carl Lumma's Cauldron.

The same idea could also be applied to other equal temperaments, using circles of other intervals, possibly with other equaves. For example: George Secor's 29-tone high tolerance temperament.

### Detempering or deregularizing

Well temperaments can be obtained by 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.

If the main generator is a fifth, then there is only one wolf fifth that closes the circle of fifths, a feature which is often associated to tunings such as quarter-comma meantone. However, these tunings are not always considered as well temperaments because they may not preserve transposability due to their higher mistunings.

If the main generator is different from a fifth, then there are multiple wolf fifths which are evenly distributed along the circle of fifths. Each wolf fifth is typically more in tune than the single wolf fifths of the fifth-generated cases, since the total mistuning is spread out over multiple intervals, but that also means that wolf fifths are more likely to be used frequently in such well temperaments.

Well temperaments based on rank-2 temperaments can be designed to follow the structure of a moment of symmetry (mos) scale. In that case, each generic interval comes in two sizes, which ensures that there will be exactly two kinds of fifths even if the generator is not a tempered perfect fifth.

For examples: Duowell, a well-tuning of Duodene

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.

Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves.

### Neji

Nejis are primodal scales that more or less roughly approximate the equal tuning with the corresponding number of tones per equave. These scales achieve consonance by ensuring that all intervals share a relatively small common denominator, instead of focusing on a few very simple intervals such as the perfect fifth (3/2) or the classical major third (5/4).

## 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.