I gave up on the fiddle years ago, but still wrestle with this problem on the guitar. The really obtrusive problem interval is the major 3rd interval (for example C-E in the key of C, or E-G# in the key of E). The "equally tempered" major 3rd is 17% sharp of the "natural" major 3rd , and even relatively insensitive ears (like mine) can hear the difference – which is why we often keep on retuning our guitars between songs, often to little effect.

Most singers and fiddlers will automatically sweeten these awkward notes by adjusting their strings or vocal cords. Players of fretted or keyboard instruments (which are almost always tuned to the equally tempered scale nowadays) don't have the option.   Clashes between the two scales can be a real problem even for classically trained musicians – for example in pieces written for a piano quintet (2 fiddles, viola & cello plus piano).

For struggling fiddlers, I have no solutions. But for guitarists, here are some hints based on years of painful – and occasionally humiliating – experience.

1) Have the intonation on your instrument checked (and if necessary adjusted) by a competent luthier. If it's beyond correction start saving for a better box.

2) Use good quality strings, and change them as soon as they show signs of wear.

3) Tune up using a good electronic tuner, and don't try to improve on the result by tweaking the odd string here and there – that way madness lies.

4) When playing chords, try to use fingerings that hide the 3rd in the middle – its awkwardness is most intrusive when it's on the top. (For example, when playing the first-position C chord, try using the little finger to hold down a G on the first string at 3rd fret, rather than leaving the 1st string open, as many chord charts advise.)

5) When playing a melody line which requires you to dwell on the 3rd note in the scale, add a little vibrato. This blurs the edge, and takes away the harshness of the ET Major 3rd.

6) When playing a melody line where the 3rd note in the scale is part of a continuous upward run, try hitting the note one fret behind and then immediately sliding up to the next fret. (Caution: this works fine in Jazz and blues, but may sound inappropriate in other genres of music.)

Wassail!