continued…

None of these is close enough to provide a suitable definitive answer. Also, even if you decide on one over the others it is possible to see that modulation or transposition becomes hopelessly out of tune. Unless you assume that half steps are always 16:15 even though it is not consistent with other scale intervals.

Assume 1–flat 2 is 16:15

Then 1–M3 is 9/8 x 10/9 = 5:4

Then 1–4 is 9/8 x 10/9 x 16/15 = 4:3

Then flat 2–4 would also be 4:3 (a modulation from the I chord to the flat II chord)

The primary reason the just system was developed in the first place is that the resulting intervals are extremely consonant. There are no “beats” that can be heard in the intervals of an equal temperament system. All the intervals of the major scale in just intonation sound very “sweet and pleasing”. With the advent of the digital age it is possible to use the just system concept in a shifting manner to achieve chromaticism, modulation and transposition. More work and experimentation is necessary in this area to come to a conclusion as to its usefulness. My own experimentation in this particular area has me a little confused about the results. In some cases I prefer the beats that I am accustomed to when I hear a major third interval, the just intonation of a third sounds very two-dimensionally flat. However, there are times I find I want a fifth (and also a sharp fourth) to be a bit less shifty in their equal temperament beating.

As stated before, the relative ease of manipulation results in my use of just interval ratios. If approximated they are near enough to equal temperament to be of value. A further analysis of purely equal temperament intonation is something that I will tackle in a future essay.