12-TET and harmonics

Question

I recently found out that 12-TET tuning does not produce perfect fifths or thirds, but that most intervals are a few cents out of key. It's actually a compromise that works very well.

I wanted to think about what this means for guitar harmonics. In a guitar with good intonation, the 12th fret harmonic will sound exactly the same pitch as fretting the note on the 12th fret.

However, if I understood this correctly, playing the 19th fret harmonic (or the 7th fret, they are the same), should not produce the same note as fretting the same string on the 19th fret? The harmonic will create a Pythagorean perfect high fifth (twelfth? - 3 times the frequency of the open string), whereas the fretted note will generate the 12-TET equivalent.

According to the chart here, there should be a difference of 1.96 cents between both. Is that correct, or am I getting my facts wrong? (of course, my ear cannot perceive such a tiny difference).

Answer 1

That's absolutely correct. We can also find it out by ourselves like this:

In 12-TET, raising a tone by fifth means multiplying its frequency by 2^7/12 (because it's 7 semitones), while in just intonation, it means multiplying by 3/2. Dividing those numbers, we get 0.9988713..., so the 12-TET fifth is slightly flat with respect to the just intonation fifth.

Now if we want to know exactly how much it is off, we can use the formula for 12-TET again: raising a tone by x semitones means multiplying its frequency by 2^x/12. So, in order to see how many semitones flat it is, we need to solve the equation 2^x/12 = 0.9988713.

However, that is easily done. First take log₂ of both sides to get x/12 = log₂ 0.9988713, and then just multiply by 12 to obtain x = 12 log₂ 0.9988713 = -0.01955... So the 12-TET fifth is truly 1.96 hundredths of a semitone flat with respect to the just intonation fifth.

If you want to get a difference that can be immediately heard, first make sure that your low E and high E are well tuned together (the harmonic at the 5th fret of the low E is the same tone as the open high E). Then play the harmonic at the 2.7th fret (I mean it) and compare with d'' at 10th fret of the high E. The harmonic is quite flat. We can also calculate the difference just as we did before: the harmonic has 7 times of the fundamental frequency, which means multiplying by 4 (two full octaves) and then by 7/4. In 12-TET, the d'' is 2 octaves and 10 semitones higher. Calculating x = 12 log₂ [(2^10/12 / (7/4)] = 10 - 12 log₂ 7 + 12 × 2 = 0.3117..., so the 12-TET note is almost 1/3 of semitone too sharp.

Answer 2

You are 100% correct in that the harmonics do NOT match the fretted notes. The fretted notes will be "closer" to (2)^(n/12)*f0 while the harmonics will be an integer times the fundamental. As for the number of cents difference I'd have to calculate it but I trust your source. The important fact is that they are NOT in tune. The just tuning of the major scale is based on harmonics and ensuring that the ratio of any note to the tonic, Do, is a rational fraction. Equal tempered tuning divides the octave, a factor of 2, into 12 even interval ratios. That is the ratio of consecutive notes in the chromatic scale should be the twelfth root of 2. This is an irrational number and cannot be evaluated exactly so we round it. This leads to subtle intonation issues that need correction. There are schools of thought that would assert that 12TET RUINED western music.

Does it matter? Maybe, maybe not. It really does NOT matter if your 5th is not 3/2*f0 since you can define the 5th any way you like. However the notes you play on any instrument are NOT the only notes created by the instrument. The natural harmonics are generates by attack and those will cause other strings to vibrate in sympathetic resonance. In just tuning I'd expect those resonances to be slightly stronger since the harmonic is "in tune" with the note being played. But in reality damping in the instrument broadens the response cure and I would expect a slightly out of tune 3rd or 5th to excite the harmonics.

These harmonics are also generated in your ear so they are always present as part of your Aural experience. The real issue is do the create dissonance where it is unwanted. Some people with perfect pitch claim that it does. Most people probably cannot tell the difference when notes are playing in a chord or in a full orchestra. If you play notes in succession in a silent room you will hear it. He can hear 1/4 tones (1/2 of a 1/2 step), and even smaller intervals. But when the perfect 5th is played next to the 12TETE tonic or other notes the slight difference may fall into the Critical Band for human pitch discrimination. At some point we cannot tell the difference. This phenomenon is pretty well understood and one can read about it in texts on physics and music or psychoacoustics.

Answer 3

of course, my ear cannot perceive such a tiny difference

It can if you're playing a chord, or two notes at the same time. Try tuning your B string to the 7th fret harmonic of the E string. Then play the open B string against the stopped string. You should hear a slow waver in the amplitude. For 2 cents, the wavering will be quite slow. For a larger discrepancy such as the fifth harmonic (the major third), it will be much faster. For this reason, many would dispute the characterization of equal temperament as working "very well."

Answer 4

I've always used the 12th and 19th fret harmonics to intonate guitars and basses. It's always worked for me! Not saying it's the correct way, but it's always been a good way - for me. And the 24th fret harmonic is probably accurate as it splits the sound evenly into octaves. But - I reckon the guitar fretboard is calculated to 12tet, so there may be a discrepancy. Help!!