# Just intonation within a piece

Say I have a piece in C major. I want just intonation and select pitches for C major.

What should happen if the piece stays on an F major triad? Should I switch the intonation to F major for that chord?

Does it depend on the function, if it's a new tonal center or an actual subdominant? Isn't such a distinction quite subjective?

Key signatures change all the time, especially in jazz, should I adapt pitches to the current chord (is associated scale)?

Background: I'm trying to create wave files in just intonation for training. I'll eventually use pieces with modulations.

• Pat gives a good answer below. As a small thought, it kind of depends on what you value more? You might want to check out the composer Ben Johnston -> en.wikipedia.org/wiki/Ben_Johnston_(composer) He adjusts intonation relative to chords as he uses them. The result is that the entire frequency center ebbs and flows with the music instead of sort of "slotting" into place with contemporary equal-temperament. I would recommend starting with his string quartets - No. 5, then No. 6, then probably No. 9, then go as you please. Sep 6, 2017 at 15:52

I think there are actually two separate (though interrelated) issues that you're asking about here. On the one hand, there's a question about music with pitches tuned to the overtone series of a particular fundamental ("should I base the tuning on the same tonal center?"), and on the other hand there's tuning based on simple ratios between integers ("how should an F major chord be tuned?"). I think your question is ultimately more about the second issue, but the first is significant as well.

If I am tuning something entirely based on the overtone series above C, then the I chord (c-major triad) is pretty clear-cut: I would use the C, E and G that appear in the first five partials and will thus have the standard just major triad relationship 6:5:4. This means that the E will have a frequency that is 5/4ths of the C (5:4 is the standard just major third) and the G will have a frequency that is 6/5ths of the E (6:5 is the standard just minor third). Automatically, this also means that the G will be 6/4ths (= 3/2nds) of the C (3:2 is the standard just perfect 5th). Even something as simple as your example of moving to the subdominant chord, however, is rife with complexity. For one thing, the first pitch in the overtone series that is close to an F is the eleventh partial, but that's nearly a quarter-tone higher than the F that is a fourth above (= a fifth below) my C fundamental. Although it might be cool to play around with this kind of subdominant, it definitely isn't what is usually meant by the term. The next partial that is a candidate for F is the 21st partial. It's definitely closer to the expected pitch, but still not very close, plus these higher partials are rarely what people are talking about when they talk about trying to use just intonation for better tuning.

This is where thinking in terms of just ratios rather than a specific fundamental makes far more sense. We almost certainly want the root of our F major triad to be a just perfect fourth or fifth away from the C, which means it needs to be 4/3rds or 2/3rds the frequency of the C. If we went a pure just-intonation major triad, then we just add an A and a C that are in the 6:5:4 relationship discussed above. Happily, this gives us the same C pitch (give or take an octave) as our original fundamental; these kinds of same-note interrelationships are common with the basic standard harmonies of a major key, but all kinds of complications ensue when more complex relations are explored. Theoretically, we could say that the F major triad is being tuned according to the "key" of F major, but that would miss the point: the 6:5:4 triad on F is absolutely part of the key of C major even if its notes aren't present in the earlier partials of the overtone series of C.

From a practical performance perspective, then, the issue tends to just revolve around 6:5:4 major triads and (often) 15:12:10 minor triads. Those ratios are entirely relative of course. For the most part, this means that the thirds of major triads need to be a bit flatter than equal temperament would dictate, and the thirds of the minor triads need to be a bit sharper than equal temperament would dictate. If my tonic were 100 Hz, then my F would be about 133 Hz (roughly 4/3rds of the tonic). My A would be about 166 Hz (roughly 5/4ths of the F), and my C would be 200 Hz (roughly 6:5ths of the A, or twice the original tonic). The reality is both vastly more complex, but also vastly more intuitive, and I would strongly advise against being overly dogmatic about any of it. Context can, and often does, override such slavishly local details about idealized relationships.

Just intonation usually has frequency relations c-e-f-g-a 1:1-5:4-4:3-3:2-5:3 so the f-a-c triad is 4:3-5:3-2 which is again 4-5-6 in frequency ratios, like c-e-g is. So the F major triad is already in just intonation and there is no reason to switch anything.

Jazz and just intonation -- well, that's not really going to work very well. Jazz chords are usually formed by stacking triads in a sort-of ambiguous manner, and when stacking intervals in just intonation you lose the ambiguity as the notes have a special relation to the root of the scale that becomes more pronounced the higher you go.

An instrument tuned in just intonation, I believe, will sound best in one key - that which it's tuned to. The close harmonies will sound o.k. - as in C, an F or G chord, but take the tune into D, while tuned to C won't sound too good. Obviously something like a keyboard can't be re-tuned as soon as the piece goes into another key. This was the main reason we now use the 12 edo tuning.

If the piece is for something like a violin, then the player will most likely adapt the placing of fingers to actually put the new part exactly into pitch. Same thing tends to happen with well trained voices, too.

• You are correct, but that isn't what the question is about. I wonder about when to change the tuning within a piece. As soon as a modulation occurs, I guess, but maybe for every accidental? Or even without accidental, if the tonal center manages to briefly change anyway? Sep 6, 2017 at 9:02
• The tuning will never be out by as much as a semitone, so accidentals won't fix it. Are you asking about changing the tuning of the instrument, or somehow changing the dots themselves? I wonder if harpsichords et al had two keyboards so that certain pieces such as you suggest could be played more in tune.
– Tim
Sep 6, 2017 at 9:06
• Changing the tuning of the instrument in the middle of a piece, just as a violinist might, yes. I mentioned accidentals as a cue that such a change might be required. A modulation. Sep 6, 2017 at 9:12
• Some baroque organs had double black keys just for that purpose, imagine how hard it is to play them. Sep 6, 2017 at 9:12
• @Gauthier - changing the tuning of a violin? I don't think the open strings would be touched. It's more a question of adapting notes slightly, so they are in tune. It mostly would happen automatically in good players, and they possibly wouldn't even be aware that it was happening - just like singers.
– Tim
Dec 31, 2018 at 10:17