“Mollified meantone” is the term I have finally settled upon for those temperaments which attempt to smooth out the worst inconveniences of regular meantone temperaments, thereby “mollifying” them to a greater or lesser degree.
Mollified meantones work their magic by employing several easily comprehended tricks, as can readily be observed in the graphics:
1. The number of tempered fifths is reduced from 11 to 9 or less so as not to descend too far below the highest end of the chain.
2. The steepness of the initial downward slope is often reduced, either from the outset or after a small number of 1/4 comma fifths (D’Alembert). Both of these variations reduce the total amount of descent in the sharps which must be regained in order to return to C or F, while simultaneously making some of the bad thirds less bad.
3. The other fifths are usually larger than pure in order to climb back up over a series of several steps rather than the usual huge leap upwards concentrated in a single fifth. This tends to smooth out the change from good thirds to bad thirds and back again.
Many modern sources labor under the illusion that this type of temperament is “French”, since the most commonly-known examples are the 18th century versions of Rameau and D’Alembert. In Germany, however, the idea seems to have always been present from the beginning of the adoption of meantone as the “normal” keyboard temperament. Arnaut Schlick’s tuning instruction, which is viewed by many as a definite move toward near-perfect thirds while yet clinging to wispy memories of Pythagorean, already discusses this problem of the sharp/flat dichotomy for the central accidentals. Preatorius also mentioned that attempting to tune accidentals for double function was done by “some”, but he preferred to “leave the Wolf howling in the wilderness where he belongs in order not to perturb our harmony.” Werckmeister’s 1698 conintuo temperament is perhaps the most sophisticated mollified meantone of the entire historical record. Depending upon the degree of basic tempering and the possible variation of the last few fifths, can easily approach the functional circularity of the rational Pythagorean-comma-fraction circulating temperaments of Neidhardt and Sorge.