Since fifths in just intonation (JI) are 2 cents wider than ET fifths, should I expect an electronic tuner to tell me that my D, G, and C strings are flatter (and getting progressively flatter) if I use harmonics to tune my instrument?

If so, this will apply to any instrument tuned using harmonics, since you can't get ET tuning using harmonics, right?


It does apply to any instrument, but for cello it's perhaps most notable because we so often play with instruments that have E-strings: violins, or else guitars. If you tune a cello in Pythagorean fifths down from an A-reference, and then a violinist tunes her e-string up a fifth from that, what you end up is a Pythagorean major third (plus three octaves) between the low C and high e, and Pythagorean thirds sound really jarring.

Of course, one solution is to just avoid using those open strings as-is. Playing all third-notes fingered allows intonating them however you like.

Nevertheless, it is common to address the problem also already in tuning, by chosing some meantone tuning instead of Pythagorean. In particular, if you tune the C so it actually makes a nice Ptolemaic third under the violin's E and then divide up the discrepancy from 3:2 fifth between the strings in between, what you end up with is ¼-comma meantone. 12-edo is also a meantone tuning, where the fifths are just a tiny bit flat but the thirds still quite notably sharp, so it is still a good idea to intonate major thirds down manually but at least not as necessary as when the instrument is completely in Pythagorean tuning.

On guitar, tuning to exclusively to harmonics theoretically gives you also a Pythagorean third, namely between g = E×(43)3 and b = E×31. In practice however, I never found this to happen; maybe one reason is inharmonicity: especially on the thick low strings, the higher harmonics start to go ever more sharp, such that you automatically tune the A-string a bit higher than Pythagorean etc., which ends you up with something more similar to 12-edo than Pythagorean. Anyway it's usually better to tune to fingered notes or to a tuner.

I personally tune my cello basically to fifths, but make each of them slightly narrow mostly just by feeling. To this end, I don't tune the strings completely open but press my finger on the nut so the higher of the pair I'm tuning goes slightly sharp. This has the advantage that it's also much quicker to hear whether the fifth is currently too narrow or too wide, because I can vary the pressure-sharpening without turning the actual tuners. Basically, it's right when there's a very slight beat in the open strings that goes away when I press my finger on the higher one. If the beat gets worse when I put on the pressure, it means I'm too wide; if I need too much pressure to get the beat to vanish, it means I'm too narrow.
Because I have a fivestring cello, I actually have that critical third interval in my instrument itself, so I can easily check in the end what kind of meantone tuning I've achieved. I don't bother getting the F to A as an exact Ptolemaic ratio, but merely narrow enough so it doesn't sound worse that 12-edo. (It's not like I can actually bow the outermost strings simultaneously anyway. In pizz I do but the mellow cello pizz sound is quite forgiving for this.)

A guitar I tune thus, starting from A: 1. d and low-E strings as 4:3 fourths around A; 2. b and e-strings as 3:1 and 4:1 over E; 3. g string so that its fourth fret matches the low E, and second fret the A. This is where the compromise needs to be made; it can make sense to change this depending on the piece you're playing. Particularly, in the open E-major chord, the G♯ often sticks out as notably too sharp, so there it can be a good idea to make the g-string extra low; meanwhile, in G major you'll want that g-b interval to be smooth so a slightly sharper g-string makes sense.

On cello, because it's fifths instead of fourths, the effect would be the opposite: the over-wideness between top and bottom strings would be exacerbated! – but actually this isn't relevant because the bow's phase-locking effect prevents inharmonicity.

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    "Pythagorean thirds sound really jarring": some people prefer them for functionally dissonant chords such as the leading tone that is the third of the dominant chord of a V-I cadence. – phoog May 24 at 16:58
  • I also meant to mention that period instrument ensembles usually tune each string to the keyboard instrument they're playing with, since the keyboard is usually tuned in a temperament where the fifths are rather farther from pure than they are in equal temperament. – phoog May 24 at 18:39
  • Thanks for the super detailed answer! I was thinking about it more yesterday and realised that this is probably also why sometimes a cellist might choose to use 2nd position A (on the D string) instead of open string A; glad then to see you write: "Playing all third-notes fingered allows intonating them however you like." – abelian May 25 at 0:17
  • @abelian well, the preference for fingered notes over open strings has multiple reasons – timbre and vibrato are in general probably more important for it than intonation. In fact, cellist of all skill grades like to use open strings as intonation-anchors, at least when practising difficult stuff. – leftaroundabout May 25 at 7:20

It can be worse on the guitar. I had a very good guitar player in one of my bands who asked me (I was also a math professor) why tuning his guitar exactly by ear for each string would fail; he had very good pitch discrimination and could hear the problems.

What he was doing was tuning fourths perfectly with a 4/3 ratio (by ear); then tuning the third to 5/4 (then the last fourth.) Good but the problem illustrates the necessity of tempering to play even in a single key.

Calling the lowest string's frequency 1 (I did say that I taught math) which allows the argument to be independent of exact frequencies, one get the following rations using exact just intervals.

E=1 A=1*4/3=4/3 D=4/3*4/3=16/9 G=16/9*4/3= 64/27 B=64/27*5/4=320/108=80/27 E=80/27*4/3=320/81

The highest E should be 4/1 or 320/80 in just intonation. So some tempering is necessary.

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  • Yeah, but tuning the b-string as ⁵⁄₄ over the g-string doesn't really make sense anyway. It's much easier to tune it to the 3rd harmonic of the low e-string. Even if you wanted the exact JI ratios, going from string to string in a row would accumulate errors (due to inharmonicity, imperfect hearing, string sag after tuning) and thus not give a good match between the outermost strings. – leftaroundabout May 24 at 11:38

IAAC (I am a cellist :-) ). While we often check our tuning via this harmonic - to - harmonic method, the preferred final test is to play open double-stops and verify 'true-temperament' fifths by the absence of undertone buzz. This ensures we don't interfere with ourselves when open strings are used.

Otherwise, I agree with leftroundabout that avoiding the open string option is best whenever possible, especially when playing with other instruments. If the passage is forte or louder, it doesn't really matter.

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  • By "undertone buzz" do you mean beats? What do you mean by 'true-temperament'? – abelian May 27 at 23:14

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