One of the very appealing properties of mathematical music models is that they often offer natural extensions to different tuning systems. Here I describe an application in 19-EDO - i.e. one octave is divided into 19 equal steps (EDO stands for "equal division of the octave"). All the ideas and concepts described in the previous pages can directly be applied to any equal temperament without any change! We just have to replace the mathematical structure of Z12 by Z19.

A division of the octave into 19 equal parts is a popular one among the possible alternative tunings. Like 12-EDO, it offers good approximations for the classical pure intervals - a strong point of this tuning is that the approximation for major and minor thirds (6 steps/5 steps in 19-EDO) is distinctly better than in 12-EDO. On the other hand, the perfect fifth (11 steps) is slightly worse.

The triad set I am investigating here is a 19-EDO version of the whole-tone triad set in the previous chapter. The point that interested me specifically was that 19 is prime and, as a consequence, 19-EDO cannot have a thing like the fully symmetric whole-tone scale in 12-EDO. And, like in the previous chapter, I wondered whether whole-tone music in this tuning would sound less "amorphous" or "rootless".

Here is the triad set (in units of 19EDO):

{0, 6, 12}, {0, 3, 6}, {3, 6, 9}, {3, 6, 12}, {6, 12, 15}, {0, 12, 18}, {16, 2, 5}

(Here is the 12-EDO version to compare.)

The carrier of this triad set is a scale that also derives from a series of 9 whole-tones (3 units in 19-EDO). Ths gives the following scale:

As can be seen (and heard), this scale is not regular at all - it contains semitones (2 units in 19EDO) and even third-tones (1 unit). Besides, it also has the property that it contains every interval of 19EDO - *except* pure fourths and fifths. (This, however, just as a side-remark...)

The minimal cadence-sets are a little different from the 12-EDO version - the main difference is the augmented triad which has no translation symmetry in 19-EDO and, therefore, forms a minimal cadence-set by its own - while in 12-EDO, there are always at least two triads necessary.

The all-over number of minimal cadence-sets of this triad set is 16, slightly less than the 18 of the 12-EDO version, but still clearly over average, indicating rich modulation possibilities. An interesting property is the following: If we drop the augmented triad, we get a 6-element chord set with the same carrier - and the property that *every* arbitrary chord pair forms a minimal cadence-set! This is a property that the harmonic scale triad set in 12-EDO has, as well.

Here is a musical composition in this scale and triad set. It is called "Gon Dance", in german "Gon-Tanz", which is a pun, or, more precise, a spoonerism of "Ganzton" (whole-tone), which refers, of course, to the scale the composition is based on. As for the question whether it sounds less "amorphous" than in 12-EDO, I cannot tell yet. Possibly I will be able to when there is more music available - the piece, in its current state, is not much more than 3 minutes long and contains exactly one modulation. Also, for this reason, the harmonic potential indicated by the high number of minimal cadence-sets is just starting to be explored...