Well Temperament
Kirnberger III's Well:
- the fifths are pure, except for
- the C–G, G–D, D–A, and A–E fifths are quarter comma meantone
- the F#–Db is the "wolf" fifth (from a meantone perspective), a schisma flat
(Other variations available: Kirnberger II with pure C–G and G–D as well as half-comma meantone of D–A and A–E, and technically it is possible to do pure C–G and third-comma in G–D, D–A, and A–E.
Vallotti's Well:
- the fifths are pure, except for
- the F–C, C–G, G–D, D–A, A–E, and E–B fifths are 1/6 Pythagorean comma flat.
Well Temperaments are temperaments made by a generator that varies in size. They do temper out some commas despite variable size of primes, though usually with the same amount of steps.
Most well temperaments are 12 tone temperaments, like 12edo with pure octave and the same 5-limit mapping but varying sizes of fifths, in order to achieve closer to just thirds similarly to meantone systems in the 'near keys' (along the white keys) but closer to Pythagorean tuning in the 'far keys'; to achieve this it has to distribute the flattening by Pythagorean comma to the 12 fifths by varying amounts (which may be zero or negative). Other approaches are possible.
The Well Temperaments can be approximated by certain Equal-step Tunings. For instance, the third comma variation is approximated by 171edo (tempering out the schisma), and Vallotti is approximated by 306edo.