In C major, if the third note is 14 cents sharp from JI, does the V/VI have a third that is 28 cents sharp?

Question

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.

Answer 1

ALL notes in a JI system are a series of ratios in relation to the tonic. So what path you choose to get to any particular note is important in determining what its tuning is going to be.

This means in JI a modulation is possible, but will bear little resemblance to a modulation in a tempered tuning system.

The consonance of a secondary dominant is probably going to be heard relative to its target, but the tuning of that note is relative to the path from the tonic that got you there. And at this point, if you're still thinking in common practice, you're going to run into trouble since the chord progression you use to resolve the V/VI back to I (maybe with a V/II -> ii -> V in the middle) might get you at a COMPLETELY different tonic note if it's not a strict reverse order of the intervals you used to get to the V/VI chord.

But the point is that yes, as you modulate in 12-TET, you get farther and farther away from JI. Obviously the mechanical limitations of the piano meant that you could never play a JI modulation on a keyboard instrument until electronics. But at the same time, I don't know of a single keyboard work in the classical repertoire written for a non-tempered tuning system (tempered being different from equal temperment). Bach's keyboard works were meant to be played on a well-tempered keyboard, wherein every key had a different sound to it due to the unequal distances between the pitches.