In the key of C major, the note "E" in equal temperament is 400 cents above the root, but in just intonation it would be a ratio of 5:4, or 386 cents above the root.
If this is true, is it not also the case that shifting tonal centers as used in secondary dominants would cause this property of out-of-tune thirds to accumulate?
For instance, now we will build a note using the E as a root, a note many of us already consider to be noticeably sharp. That note will take the role as the root of a secondary dominant, whose third is G#, an augmented fifth in the key of C, which is doubly out of tune in comparison to the E.
Although the G# is quite out of tune in the key of C, in relation to its target it is only as out of tune as the original E was in the key of C, a difference that most people agree is tolerable.
So is the consonance of the secondary dominant structure heard with respect to its original key, C, where its augmented fifth is damningly far from the just interval, and thus the structure is unstable and alien within the key, or is the consonance of the structure understood by relation to its target chord, wherein it remains reasonably stable? What would the effect of this difference be in terms of hearing the resolution?
I've used a sequencer and some pitch editing to hear this chord progression for myself both in just intonation and in equal temperament. I'm embarrassed to admit that I can't really hear a very big difference, although I can generally tell the difference between just intonation, equal temperament, and microtonal music.
