Well temperament

A well temperament or a circulating temperament is a scale with the property that for at least one interval class (the interval class is viewed as forming a closed circle within the tuning, hence the name), all of the intervals in the "circle" can be regarded as approximations of some targeted interval, but which is not an equal temperament. The targeted (circulating) interval can be a fixed just interval (like a just perfect fifth), or an n-edo interval when the goal is to produce an unequal coloring of n-edo. In the best known examples, the interval approximated is a fifth and the scale has twelve notes to an octave.

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.

Types

A well temperament may be classified by method as follows:

• Maximally even set in a large edo or another equal-step tuning
• A quasi-equal detemperament (though not all detemperaments of an edo are well temperaments)
• Neji

Historical well temperaments

• Kirnberger – Kirnberger temperament

1. the fifths are pure, except for
2. the C–G, D–A, G–D and A–E fifths are quarter comma meantone
3. the F#–Db is the wolf fifth, a schisma flat

• Vallotti – Vallotti/Young temperament

1. the fifths are pure, except for
2. the C–G, D–A, E–B, F–C, G–D and A–E fifths are 1/6 Pythagorean comma flat.

• Young2 – Young temperament II
• Werck3 – Werckmeister temperament III
• Cauldron
• Bifrost
• Teleic scales – unit step generator, patent tuning alternating *ed(16/9) and *ed(9/8)
• Kartvelian scales – unit step generator, alternating edf and ed(4/3)

Other scales called "well temperaments"

• Well tempered nonet

Articles

• An Introduction to Historical Tunings by Kyle Gann Permalink
• Circulating Temperaments by Gene Ward Smith Permalink
• Well v.s. Equal Temperament by Michael Rubinstein Permalink
• Six Degrees Of Tonality: The Well Tempered Piano by Edward Foote Permalink
• [http://www.rollingball.com/images/HT5.htm Five