Well temperament

A well temperament or a circulating temperament is a scale which is regarded as an approximation of an equal-step tuning, preserving the transpositional invariance of an equal-step tuning, but is not exactly the equal-step tuning being approximated. Historical well-temperaments were formed by stacking a combination of pure 3/2 and meantone fifths and resulted in 12 nearly equal notes.

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

Other quasi-equal scales

• 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)

• 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