[1][2][3] The tuning systems are numbered in two different ways: The first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord.
[a] The last "Septenarius" tuning was not conceived in terms of fractions of a comma, despite some modern authors' attempts to approximate it by some such method.
Instead, Werckmeister gave the string lengths on the monochord directly, and from that calculated how each fifth ought to be tempered.
Werckmeister designed this tuning for playing mainly diatonic music (i.e. rarely using the "black notes").
Werckmeister also gave a table of monochord lengths for this tuning, setting C=120 units, a practical approximation to the exact theoretical values[citation needed].
The resulting scale has rational frequency relationships, so it is mathematically distinct from the irrational tempered values above; however in practice, both involve pure and impure sounding fifths.
Werckmeister also gave a version where the total length is divided into 147 parts, which is simply a transposition of the intervals of the 196-tuning.
One apparent problem with these tunings is the value given to D (or A in the transposed version): Werckmeister writes it as "176", but the value is suspect: It produces a musically bad effect because the fifth G–D would then be very flat (more than half a comma); the third B♭–D would be pure, but D–F♯ would be more than a comma too sharp – all of which contradict the rest of Werckmeister's writings on temperament.