It is not true in general that the higher you go on the fret board, the lower your harmonic is. Actually, if your were to play an harmonic at the 24th fret, you would hear a note sounding an octave higher than the harmonic at the 12th.
Still, however, the harmonics behave differently than fretted notes. Now, let’s get physical and explain why. On perfect strings fixed on both extremities.
Basics of perfect strings
Because it’s extremities are fixed, a perfect string of length L can only vibrate at certain frequencies. These frequencies are such that the matching wavelength are of the form:
λn = 2⨉L/n
The next image1 illustrates why: the extremities don’t move, so must be on nodes of the vibrating strings.
The matching frequencies for these wavelength are:
fn = k⨉n/(2⨉L)
for a some constant k which depends on the characteristics of the string.
In practice, whenever a string vibrates, it vibrates at a combination of these frequencies. f1 is the fundamental, which determines the note you hear, the various fn≥1 are the harmonics, which are multiples of the fundamental and create the timbre of the note.
Remember that the higher the frequency, the higher the pitch.
What happens when you fret a note
Whenever you fret a note, what happens is that, without changing the other characteristics of the string2, you change it’s length; i.e. instead of having fixed points at the bridge and the head, you have fixed points at the bridge and the fret.
The fundamental frequency of the note you’re playing is thus:
fʹ1 = k/(2⨉Lʹ)
where Lʹ is the length of the string up to the fret you’re playing. Because, obviously, Lʹ < L, fʹ > f. The fundamental frequency is higher, the note has a higher pitch.
What happens when you graze a string
When you graze a string to “play a harmonic”, what happens is very different. You don’t shorten the length of the string: the whole string is still vibrating. However, you muffle some of the frequencies it’s vibrating at by preventing mouvement at a given point.
For example, if you play a harmonic on the 12th fret, that is in the very middle of the string, you muffle every alternate frequency. If you look back to the previous illustration, you can see that the frequencies depicted on the right-hand side of the image don’t make the string move in its very middle, but that the ones on the left-hand side all do. But if your finger is right there, the middle of the string cannot move.
That means that the only frequencies you allow to vibrate are the fn where n is even, f2, f4, …
The lowest frequency at which the string is vibrating is thus f2, which is the fundamental of the note you’re grazing. The fundamental is twice as high as the open string; you’re playing an octave.
If you were to graze the string at one quarter of its length, be it the first (~5th fret) or last fourth (exactly the 24th fret) of the string, you would only allow one out of four frequencies to be vibrate. The fundamental would be f4, that is two octaves above the open string.
If you were to graze the string at a third of it’s length, once again which third does not matter (the first is around the 7th fret), only one frequency out of three would vibrate and the fundamental would be f3, i.e. an octave and a perfect fifth above the open string.
You could theoretically play any note with fn as a fundamental this way, but higher harmonics have very little power.
Why do the 12th fret harmonic and fretted note are the same note
Should be left as en exercice for the reader. I’m way too nice.
We have said previously that a 12th fret harmonic sounds one octave higher than the open string, i.e. it’s frequency is double that of the open string.
Now, when you fret a note on the same fret, the length of the string is Lʹ = L/2 (we’re at the middle of the string). Thus, when you fret this note, the fundamental is:
fʹ1 = k/(2⨉Lʹ) = k/(2⨉Lʹ) = k/L = 2⨉k/(2⨉L) = 2⨉f1 = f2
The fretted note is, once again, at the octave and share the same fundamental (and thus pitch) as the harmonic.
Please notice, however, that while the frequencies are the same, the power at which the string vibrates for each frequency is different. The pitch is the same; the timbre is different. Typically, harmonics are much softer.
- Courtesy of Wikipedia.
- Technically, you would be slightly changing the tension of the string, but on a well fit guitar, the effect should be minimal. No need to worry about that in our model.