[coliveira]

It doesn't matter how good the intonation at the bridge or nut. There's the mathematical fact that we cannot get pure thirds and even fifths in modern equal temperament system. If you have a good ear you'll feel the subtle difference between notes, and they can never get exactly where you want. It's something you have to live with in modern music.

[LegionMammal978]

The thing is, unless you're playing with other instruments, no one is forcing you to tune to equal temperament. E.g., it's very common to tune a violin's A string to an A440 reference, then tune the other strings to 3:2 perfect fifths by ear. It just gets more complicated for fretted instruments like the guitar.

[coliveira]

If you tune the strings in a guitar to a perfect 4th, which is the case except the 3rd between strings b-e, the lower e will significantly differ from the high e open string. There's no way to get around it.

[dahart]

Wanting to play in any key and not be locked into a key automatically pushes musicians toward equal temperament, even when playing solo, and even on a violin. Saying no one’s forcing you is technically true but sounds pretty naive, and (forgive the pun) tone deaf to me; there’s no realistic alternative for modern music. Some people do choose to play with other tuning systems on occasion, but there’s a reason why 12 TET is so popular and widespread.

[kazinator]

Wanting to change keys freely only pushes fixed-pitch musical instruments toward equal temperament. Since many important instruments are like that, and virtually all instruments that are capable of accurate intonation not relying on ear are like that.

If an ensemble includes instruments that are equal temperament, then the non-fixed-pitched instrumentalists adjust their pitch to sound good with those.

An ensemble consisting only of instruments that can play any interval can change keys by pure intervals.

E.g. switching from the original major key to the relative dominant key can mean changing the root by a pure fifth. In equal temperament, this modulation is done by altering only a single note: sharpening the subdominant. All other notes are from the original scale. If we change key by a pure fifth, that is obviously not so; all notes are detuned off the original scale.

If we change through all the keys along the circle of fifths, perfectly purely, we arrive at the Pythagorean comma: the gap between the destination root and the original.

Another possibility is to progress the roots through the diatonic fifths of the original scale, rather than pure fifths. Like, we start with a pure, just intonated C major, and then change keys through G,D,A,E,B,F#,C#,Ab,Eb,Bb,F back to C using the notes of that pure C major scale, or sharps/flats relative to those. Then we don't run into the Pythagorean comma; but of course all the pure scales we end up using are detuned from C major, and in a different way from following pure fifths.

[kazinator]

> It doesn't matter how good the intonation at the bridge or nut.

Yes, it does.

> There's the mathematical fact that we cannot get pure thirds and even fifths in modern equal temperament system.

Those are the pennies that don't matter, if your instrument has dollar problems.

If you don't have good intonation, then you can't even properly get the approximations provided by equal temperament.

With good intonation, compensated on both ends, you have a much better experience making tuning adjustments to get better compromises for the music you are playing.

[coliveira]

Intonation is just the beginning. It won't solve the deeper issues with equal temperament.

[kazinator]

Actually, it will solve most of it for guitarists, because the tuning problems that most guitarists blame on equal temperament are actually coming from their bad intonation. The fourths and fifths in equal temperament are not nearly as bad as they imagine. The equal temperament fifth is only 1.955 cents sharper than the pure fifth (3/2 ratio). Just under 2 cents.

Guitar intonation that is accurate to 2 cents is very good, I would say above average.

Another way to look at the pitch error in the ET perfect fifth is as a percentage of the pitch, which is about -0.169 %.

Suppose a 1200 Hz tone (quite a high note, somewhere between D6 and D#6) is played together with one that is 0.169 % flat. That flat one will have a frequency of 1198 Hz. The difference is 2 Hz, and so a 2 Hz beat will be heard: two volume swells per second.

Much lower down, at 120 Hz, that will be 0.2 Hz: two volume swells every ten seconds. Basically nothing. It makes no difference to guitar chords played in the first four fret box down by the nut.

The equal temperament error is worse for some other intervals; the ET major third is a percent sharp, or around 13.6 cents, which is a lot. It is pretty jarring, even in lower registers.

That's not what the submitted article is about; tuning in such a way as to fixing the tiny error in the fourths/fifths will not repair the major third.