Pieter Bakker Musica mathematica
Het ORGEL 98 (2002), nr. 2, 11-14 [summary]

Most of Andreas Werckmeister's sources can be identified easily. This is not the case with regard to his irregular circular tunings. It seems, however, that he was inspired by two sources. The first one is Musica mathematica of Abraham Bartolus, published at Leipzig in 1614. Bartolus takes over a tuning designed by Andreas Reinhard, published in his Monochordum (Leipzig, 1604): the phrygian scale is divided into 48 parts and related to the proportions of the universe. Bartolus praises this tuning: it would be possible to make music in all keys. Werckmeister sees that the tuning needs to be tempered first. The second source is a disc made by Theophil Staden, studied by Werckmeister in Harsdörffer’s Mathematische und Philosophische Erquickstunden. It visualises the idea of a circulair tuning in a primitive way.

Werckmeister's tunings known as Werckmeister III and Werckmeister IV are irrigular circular. He recommended these tunings in particular, and not equal temperament, which is regular circular, because he preferred the thirds to be purer, because the popular meantone temperament was more related to Werckmeister III en IV, and possibly because equal temperament was hard to realize accurately.