The Diesis is the difference between a stack of 3 pure major thirds (386,3 x 3 = 1159 cents) and the octave (1200 cents), almost a quarter tone. I used to think that the Diesis wasn’t all that important for understanding temperaments, but recently, after reading all of the works of Neidhardt and Sorge cover to cover, I have realized that tremendous insight can be gained by seeing how the Diesis is distributed in any given temperament. Most modern explanations place far too much emphasis upon compensating the Pythagorean comma and closing the circle, but this has never really been much of a challenge; the circle will always close by itself, the only question is whether it closes smoothly or abruptly. Historically, the real purpose of temperament was to control the quality of the major thirds, and a Diesis Matrix clearly demonstrates why the various major thirds are as they are in any system, perhaps more than any other type of graphic representation. It also easily demonstrates how our hands are tied by the interlocking intervallic relationships within the octave. Since octaves must always be pure, any vertical stack of 3 major thirds must include the entire Diesis somewhere (1 Diesis ≈ 2 syntonic commas). Any temperament can be viewed as a set of four stacks of major thirds which are contiguous by fifths (for Werckmeister, Neidhardt and Sorge, these primary stacks were built on C, G, D, and A). one can begin by deciding the distribution of the Diesis within each of the four stacks, and thereafter the only thing left to do is decide how to distribute the tempering among the first four fifths (historically being the notes from C and E). The sizes of the remaining 8 fifths will then be determined automatically by the previously-fixed sizes of the major thirds. Such an approach places the emphasis on third quality rather than fifth quality, as it should be.
Neidhardt used just this method to design the temperaments in his Sectio Canonis Harmonici (1724). He stated that after determining the sizes of the thirds, one need only determine the tempering of 3 fifths, the first three from C to A, since all the others will automatically be defined by the combination of these three and the decisions which have been made about Diesis distribution. Sorge also adopted the same approach to set his “marker stones” which divide the octave into three blocks for setting either equal or a near-equal temperament, which is how he designed his near-equal temperaments. His first step was to determine the Diesis distribution in the primary stack (C – E – G#/Ab – C), which in turn largely defines the nature of the thirds within each of the three trients by placing limits on the possible variation.
For each temperament, I include two versions of Diesis analysis: on the left, a simplified version I call the Diesis Stack, which only shows the qualities of the major thirds without any offset for fifth tempering. This allows the eye to more easily compare overall third quality by showing only the amount of the Diesis which has been given to each third. On the right is the Diesis Matrix, which shows the tempering of both the thirds and the fifths (it is interesting to note that, as we can learn from reading Neidhardt/1724, for any given Diesis Stack, there is an infinite number of possible Diesis Matrices!).
In both graphics, there are 5 stacks of 3 major thirds each (the fifth stack being a transposed repetition of the first), each stack related to the next by a fifth. You read the Matrix in two dimensions: horizontal shows fifth relationships, vertical major third relationships. Third quality is indicated by ascending green columns which indicate pure major thirds. These columns are broken in the middle to indicated the fact that the overall scale is not the same as that used to indicate Diesis compensation; in other words, the size of the pure major third is not shown at the same scale as the amount by which the thirds are too wide. The portion of the Diesis which has been given to each major third is indicated graphically be the space between the top of each green column and the black trace immediately above it. The numerical value is given directly beneath the upper note of the interval, and the sum of these values for any stack is always 1. The Diesis can be accounted for in a variety of ways, either by giving all of it to single very nasty third (as in 1/4 comma meantone), or by spreading it out among the 3 thirds, either equally or favoring one or another trient. Historically, the best (narrowest) thirds are always found in the 1st trient, the widest (worst) thirds in the 2nd trient, and the 3rd trient often has a mix of good and bad thirds, worse on the left side (emerging from the 2nd trient) and getting better as one moves to the right (i.e. returning to C-E, which is always the best major third).
Fifth quality is indicated by the same system of guide vectors used for the main graphs, i.e. the red vector indicates a pure fifth and the green vector indicates a 1/4 comma meantone fifth. Flat horizontal progression indicates Equal Temperament fifths.
Here is Werckmeister’s Continuo temperament in the 55-comma version, showing both Diesis graphics as they appear for all temperaments:
The only disadvantage of the Diesis Matrix is that it shows nothing about minor thirds. One could also construct a minor third matrix using the Greater Diesis (the difference between 4 minor thirds and the octave), but since minor third quality has never been the major arbiter in the construction of temperaments, it is not really worth it. Minor third quality can easily be observed on the normal graphs.