# Does just intonation work for any given key, or is it impractical even for a work in a single key?

According to my understanding any non-chorded instrument, like a string instrument or horn, can use a just intonation, so the problem only arises with chorded instruments like pianos. So, for example, in a symphony that does not have a piano I suppose using just intonation is possible.

On the other hand, let's imagine a piano concerto. As long as the concerto is in a single key, then is it possible to use a just intonation tuning, or will there still be problems?

• The requirement to fix a temperament arises when the notes are fixed in pitch. It does not depend on whether an instrument can play chords or not; it's just easier to bend pitches if you're playing only one at a time. For example, a guitar player can bend individual notes in a chord to approach just intonation. Any instrument where it's not possible to bend the pitches must choose a temperament; this includes for example the hammered dulcimer, which is not primarily a chord-playing instrument. May 29, 2019 at 16:23

## 2 Answers

Impractical for any key.

```C = 1/1
D = 9/8
E = 5/4
F = 4/3
G = 3/2
A = 5/3
B =15/8
C = 2/1
```

Now check chords; the major chords:

```C-E-G    1 : 5/4 : 3/2  or  4:5:6 ratio (multiplying by the LCM)
G-B-D  3/2 :15/8 :18:8  or  4:5:6 ratio
F-A-C  4/3 : 5/3 : 6/3  or  4:5:6 ratio
```

The minor chords:

```D-F-A  9/8 : 4/3 : 5/3  or  27:32:40 ratio  <-- !
E-G-B  5/4 : 3/2 :15/8  or  10:12:15 ratio
A-C-E  5/3 : 2/1 :10/4  or  10:12:15 ratio
```

So the ii chord is already different.

• It's unclear what your point is. Consider the fact that, prior to Well-Tempered Clavier, keyboards (harpsichord in particular) were retuned to the key of each piece (just intonation). May 29, 2019 at 12:22
• Yes, but not using Just Intonation. One of the meantone or other recirculant tunings was used.
– ttw
May 29, 2019 at 12:59
• ahhh, thank you for the correction. Always glad to learn stuff. May 29, 2019 at 13:14
• So it's possible to tune a keyboard in just intervals if you avoid certain chords. For example, in the key of C you can have a just F major chord if you avoid D minor (and D major as V/V). If you tune for the just D minor (and a just V/V), you lose either the just major third between F and A. (Actually, you can also lower the D to preserve the just F-to-A third, but then you lose the just perfect fifth between D and G. Since G major is the dominant of C major, you won't want to do that.) Note that you run into the problem before you get to chromatic chords. May 29, 2019 at 16:14
• @CarlWitthoft furthermore, I doubt it would have been common to retune harpsichords for each piece. It takes some minutes to tune a harpsichord. You could shift one or two notes between pieces, and I've been to concerts where that has been done, but if it's not done at intermission someone has to talk to the audience for a bit while the instrument is retuned. Earlier in the baroque period, pieces were almost all written in closely related keys that would not have required retuning. Erratum: strike "either" from the previous comment. May 29, 2019 at 16:33

I think it comes down to taste at that point for keyboards. Equal temperament is most common though. A few folk guitar players tune there guitars to just intonnation cause it can give a more pure sound. Coming from a wind band experience and also chior we always try for just intonation as it sounds more in tune and overtones sound nicer, but most of the time the musicians just use their ear and it natrually occurs. Practically and to keep things simpler for piano and mallet players sacrificing being slighty out of tune in every key makes their life simpler and allows you to shift from key to key with out nasty sounding intervals. Hope this helps :)

• What does it mean to tune a guitar to just intonation? Do they tune the b string a just third (5:4 ratio) above the g string? If they do that, and tune all the fourths to a 4:3 ratio, then the top e string will be vibrating 3.95 times the frequency of the bottom e string instead of 4 times. Or are you talking about guitars with frets that aren't straight across the neck? Jun 2, 2019 at 5:03
• Also, there are lots of pieces that sink in pitch when you sing them in perfect just intonation. To keep from going flat you have to modify the intonation slightly. Jun 5, 2019 at 4:37