A Circulating Temperament is one which has consciously been devised in order to make all twelve major keys more or less usable without any of them being noticeably shocking or jarring, thus allowing one to “circulate” freely. None of the thirds are very pure, and this lack of a clear incontrovertible reference means that the bad thirds do not sound quite so bad by comparison. Thus while we may often find Pythagorean “thirds” in these systems, their presence is less offensive. The historical record contains two basic types of circulating temperaments: Pragmatic (earlier) and Rational (later).
A special note about “Harmonic Waste”
Some modern authors would have us impose the limitation upon circulating temperaments that no fifth be wider than pure, which is sometimes justified as the avoidance of a condition described with the pejorative term of “harmonic waste”. This concept has little if any basis in historical theory, though one could find a justification for it in a passing comment of Sorge, should one be so inclined; Sorge’s work, however, represents the last gasps of non-equal circulating temperaments and is hardly representative of any important historical practice. “Harmonic waste” is something which only exists in the minds of certain modern theorists who delight in coming up with arcane mathematical means by which it can be “objectively demonstrated” that certain temperaments are “better” than others. Unfortunately, such numerical juggling has about as much relevance to understanding temperaments in a historical sense as phrenology does to modern neuroscience. This “wastefulness” didn’t seem to trouble ancient musicians, since many temperaments are guilty of the sin of profligacy by including the dreaded wide fifths. Admittedly, many of them are properly mollified meantones, but even from the very first appearance of circulating systems – those of Werckmeister – we find wide fifths, and the man who was perhaps the greatest master of Baroque temperament theory and practice, Neidhardt, proposed a number of rational circulating systems with wide fifths.
How to properly treat Harmonic Waste