Beste und leichteste Temperatur des Monochordi (Jena)
The best and easiest temperament for the monochord
Sectio Canonis Harmonici, zur völligen Richtigkeit der Generum Modulandi (Königsberg)
Sectio Canonis Harmonici, about the veritable exactitude of the Generum Modulandi
Gäntzlich erschöpfte mathematische Abteilungen des diatonischchromatischen, temperirten Canonis Monochordi (Königsberg)
Complete and exhaustive mathematical division of a Canonis Monochordi tempered diatonically and chromatically
Neidhardt’s first publication is a complete discourse on how the conflicting harmonic functions of pure tuning can be resolved by the use of Equal Temperament. However, his contributions to the history of unequal temperaments are to be found in his later publications of 1724 and 1732. He was the first to propose such temperaments which employed a rationalized manner of closing of the circle by explicitly dividing the Pythagorean comma as opposed to the empirical solutions of mollified meantones and the pragmatic approximations of Werckmeister and Bendeler based on the near-equivalence of the Syntonic and Pythagorean commas. Neidhardt adopted a standard tempering unit of 1/12 of the Pythagorean comma, and each impure fifth in his various temperaments is tempered by 1, 2, or 3 of these units, i.e. 1/12, 1/6, or 1/4 of the Pythagorean coma. The total number of units must always equal 12. He also stated that no major third can be as large as the Pythagorean ditone, and therefore his worst major third is only a little more than 3/4 Syntonic coma too wide.
Neidhardt experimented with many possible combinations of both narrow and wide fifths, but in both publications, he concludes by recommending only 4 different temperaments, each one assigned to a certain social milieu: village (Dorf), small city (Kleine Stadt), large city (Grosse Stadt) and finally the court (Hof). Presumably, this represents the degree of harmonic sophistication of the music which would normally be played in each setting, the village being the simplest and the court the most complex. Thus the temperaments for a village are the most unequal, with the largest difference between good and bad thirds, and the more “sophisticated” the social setting, the closer to Equal. The list of temperaments is slightly different in the two publications, and it is very interesting to note that the later publication indicates a general movement backwards away from equality. If Neidhardt’s temperaments are ordered according to their degree of equality, with temperament 1 being the most unequal (purer natural tonalities and more acrid accidental tonalities) and 5 being Equal, the evolution of his recommendations can be represented thusly:
This small step “backwards” was probably an acknowledgement of a growing disenchantment within the larger musical community with the bad thirds of Equal Temperament after an intial period of enthusiastic interest, based upon the false (bad pun!) hope that the adoption of ET would resolve the problems of combining organs pitched at Chorthon (≈ 465) with orchestral winds pitched at Cammerthon (≈ 415), a situation which usually required the organist to transpose down by a major second. While in theory using ET would have been the perfect solution, there are many original texts which describe how musicians found the sharp thirds of ET to be too large (bad pun again!) a price to pay for the practical advantage.
The temperaments are presented here in the same order, i.e. starting with the most unequal.