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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?

Tyler Durden
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  • 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. – phoog May 29 '19 at 16:23

2 Answers2

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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.

Your Uncle Bob
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ttw
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  • 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). – Carl Witthoft May 29 '19 at 12:22
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    Yes, but not using Just Intonation. One of the meantone or other recirculant tunings was used. – ttw May 29 '19 at 12:59
  • ahhh, thank you for the correction. Always glad to learn stuff. – Carl Witthoft May 29 '19 at 13:14
  • @CarlWitthoft any keyboard must be tempered; that is why we speak of just intonation but equal temperament. Pure just intonation requires moving notes to different frequencies to fit the context. I recently found a video of Pachelbel's famous canon in three temperaments, one claiming to be just intonation. A commenter noticed that the tuning was rather poor toward the end; this is where the violins have some ii 6/5 chords (including E) over the bass G (most of the piece has IV chords there). The fourth between the B and the E is indeed horribly out of tune, though it is fleeting. – phoog May 29 '19 at 16:11
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    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. – phoog May 29 '19 at 16:14
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    @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. – phoog May 29 '19 at 16:33
  • So you're saying that in the early days, even a harpsichord playing on its own tuned to one key will still sound out of tune in that one key - with no modulations? I was under the (probably wrong) impression that 12tet came about so an instrument so tuned could be played in any key, to overcome the restrictions brought about by an instrument tuned 'justly'. – Tim May 30 '19 at 06:42
  • I thought the whole point of the "Well-Tempered Clavier" was that it was a celebration of being able to write for a keyboard in any key? – Duston May 30 '19 at 13:58
  • Many of the mean-tone or recirculant temperaments can be played in different keys; there keys do sound slightly different. However, a V-I still sounds like a V-I so temperament isn't as big an effect as it might seem (else equal-temperament wouldn't work.) – ttw May 30 '19 at 14:38
  • @Tim no, I'm saying that early harpsichords and other keyboard instruments used tempered tunings. These, being unequal temperaments, sounded better in some keys than others. Equal temperament is an improvement over other temperaments (in some respects; in others it is worse, as with any compromise). Nobody ever tuned keyboards using only just thirds and fifths because it is mathematically impossible. – phoog May 30 '19 at 17:55
  • @Duston it certainly is. The myth here is that non-well-tempered keyboards were somehow tuned in "just intonation." That is impossible. You can do it using only just fifths, but then the major thirds are even wider than in equal temperament. – phoog May 30 '19 at 18:03
  • @Duston also, the so-called "well" temperaments allow you to play in every key, but, being unequal, the different keys sound different, and in some keys there may still be unusable "wolf" intervals that Bach will have avoided in certain compositions. So it was probably not possible to transpose every WTC movement into each of the 11 other possible keys. By contrast, with a modern keyboard in equal temperament, it is possible. – phoog May 30 '19 at 18:54
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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 :)

Adam Peters
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  • 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? – phoog Jun 02 '19 at 05:03
  • 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. – phoog Jun 05 '19 at 04:37