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By lightly touching a guitar string at specific nodes (most commonly at the 12, 7th and 5th fret) and releasing your finger immediately upon plucking the string - a clear ringing tone is produced. These tones are called harmonics.

When fretting the strings on the guitar, the closer you move towards the body, the higher the note. So a note fretted on the 7th fret will be higher in pitch than a note fretted on the 5th fret on the same string. A note fretted on the 12 fret is higher than either.

But when playing harmonics - it's just the opposite! A 12th fret harmonic produces a lower tone than a 5th or 7th fret harmonic played on the same string.

What causes this seemingly counter-intuitive phenomena to occur?

EDIT: Also wondering why on the 12th fret harmonics the fretted note corresponds with the harmonic note, but this is not the case on any of the other harmonic nodes.

  • 3
    Your fretboard shortens the string to the length of fret->bridge. Your harmonic "shortens" the string to the length of node->[bridge or nut, whichever is closer]. – NReilingh Mar 3 '15 at 4:53
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    The 9th fret node is the same harmonic as the 4th fret node. Each successive harmonic divides the string up into a smaller division, so all but the first harmonic (12th fret) occur more than once. – NReilingh Mar 3 '15 at 6:00
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    You don't have to immediately remove your finger. It can stay there and you can even touch again in the same spot - the string isn't moving there. – Tim Mar 3 '15 at 8:41
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    You know you don't have to stop at just the 12th, 7th or 5th frets. Any position that cuts the string in an integer partition, ie there are an integer number of waves on the string, will give you an harmonic. (See Edouard's image.) – Sam T Mar 3 '15 at 21:19
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    @RockinCowboy You have a misapprehension in your question: harmonics don't always get higher when you move towards the nut (e.g., 9th fret harmonic is higher than 7th). One thing maybe not clear in Édouard's answer is that every harmonic other than the 12th fret one has more than one place you can play it. The 5th can be played at the 24th, which also sounds the same note as the harmonic. See this: upload.wikimedia.org/wikipedia/en/thumb/3/34/… – Todd Wilcox Sep 8 '15 at 15:57
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+100

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.

A string of length *L* can only vibrate at some given frequencies - courtesy of Wikimedia

The matching frequencies for these wavelength are:

fn = kn/(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:

1 = k/(2⨉Lʹ)

where 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:

1 = k/(2⨉Lʹ) = k/(2⨉) = 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.


  1. Courtesy of Wikipedia.
  2. 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.
  • It may be worth noting that plucking an undamped string causes it to vibrate at all of its harmonics, but only the lowest is generally audible. Grazing a string will nullify all harmonics which would require the string to move at that spot. If one grazes the string at the midpoint and then at the 1/3 point, that will nullify all odd harmonics, and then all that aren't multiples of three, leaving only those which are multiples of six (which may be quite faint). – supercat Mar 3 '15 at 22:32
  • @supercat. Interestingly when you pluck a string there can be no null point at the place where you pluck the string, because you are forcing it to move by plucking. I've heard it said that plucking near the 7th harmonic null point gives a purer tone because of this. Also it can be why the 3rd and 4th harmonics (7th and 5th fret) sometimes don't seem to work because the right hand is plucking in the wrong place and cancelling the harmonic. – Noel Walters Sep 10 '15 at 12:49
  • @supercat Higher frequencies are audible as well. Our brain, however, doesn’t interpret them as independent sounds, but as part of a whole. We don’t hear a bunch of sinusoids, we hear a note whose pitch is the lowest frequency of the string (well, most of the time) with a particular timbre. When the power of higher frequencies change, the timbre is different. That’s how we differentiate a guitar from, say, a trumpet (well, the unstable attack of the sound is also very important, possibly more). – Édouard Sep 10 '15 at 13:16
  • @Édouard: It's also why power chords used with distortion sound an octave lower than played. The three notes of a power chord have fundamentals with a 2:3:4 ratio, and distortion creates sum and difference tones with their harmonics, and all those sum/difference tones are multiples of the sub-harmonic. – supercat Sep 10 '15 at 13:37
  • @supercat There’s that question I’ve wanted to answer for a while: your remarks about power chords would make for a good start for an answer. – Édouard Sep 10 '15 at 13:45
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Although I definitely see what you're saying, it's not strictly true that harmonics closer to the nut will be higher. What's happening with natural harmonics is you are dividing the string into to equal parts. An open string will not only vibrate at its fundamental frequency but also at integer multiples of that frequency, each getting higher and quieter. The resulting series of notes is the harmonic series. When you place your fingers on the nodes, you are stopping certain harmonics from ringing, giving you the pure tone.

Your examples, the 12th, 7th, and 5th fret divide the string into 2, 3, and 4 parts respectively. On the twelfth fret, there is only one node in the middle of two parts. However, on the 7th fret, there are two nodes dividing three parts. The other node is on the 19th fret. For the 5th fret harmonic, there are two more nodes: the 12th fret and the 24th fret. Of course, if you pluck while your finger is over the 12th fret, you'll only get the lower harmonic. But you can pluck on the 5th fret and place your finger over the 12th fret, you will indeed find a node there and the note will not dampen.

So, back to the whole 'lower is higher' predicament. It's true that you can find progressively higher notes in the way you described, bridge to nut direction. But you can also do the same thing with nodes in the opposite direction (ie 12th, 19th, and 24th frets).

  • Interesting information. What about why the only harmonics that match the fretted notes seem to occur on the 12th fret. This will sink in eventually but I'm not quite getting it yet. – Rockin Cowboy Mar 3 '15 at 4:49
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    Your harmonics would also "match" the fretted notes for any other harmonics found on the bridge-side of the 12th fret. That is, if you had a theoretical fretboard that extended all the way to the bridge (with an infinite number of infinitely small frets in between) the harmonic nodes you found on that part of the fretboard would match the notes being played. – NReilingh Mar 3 '15 at 4:56
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    If you follow the harmonics in nut to bridge direction, they'll match up. The 19th and 24th fret match their harmonics. All the harmonics beyond this are off the fret board. If your intonation is off, though, it will be very apparent playing the harmonic and fretted note one after another. They will be out of tune. The 7th and 5th fret nodes are better understood as reflections of the 19th and 24th frets nodes using the 12th fret (half the string) as a point of symmetry. – Dan D Mar 3 '15 at 5:30
  • @NReilingh: Plucking with a finger lightly on the 12th fret will produce both the 2x (octave) and 4x (two-octave) harmonics; because lower harmonics are generally louder, the 4x harmonic will not be noticeable, but that won't mean it's not present. Indeed, plucking an open string will produce many harmonics, but they won't be as noticeable as the fundamental; one may demonstrate their presence, however, by lightly brushing the harmonic after the string is plucked. If a harmonic is audible after one touches a fractional subdivision of a string, that means it was present beforehand. – supercat Mar 3 '15 at 19:43
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After all the technical answers, try this. Play , say, the 7th fret harmonic, then press down on the EIGHTH fret. Pluck the string BEHIND - as in closer to the nut. You'll find that the note is the same. If there were more, smaller fretwires, you could do this for all the harmonics. You have been fooled into thinking the harmonic nodes only work going DOWN the neck, while notes only get higher by going UP. Think of it like a backwards guitar!

The other answers are brilliant.

4

My friend, you have just stumbled onto the Harmonic Series. This was something Pythagoras tinkered around with using the monochord, and is primarily responsible for much of how Western music sounds, is written, is analyzed, and is perceived.

Very basically, all sound travels through vibration. Since vibrations are made up of waves, each wave has a crest, trough, amplitude, and frequency. When you access a natural harmonic on your guitar, you are dividing the string at a length that neatly divides into the sum-total length of the string. In other words, when you touch at the 12th fret, you're effectively dividing the string in half. Since a string half the length produces a sound twice as high (as the frequency is twice as high), the resultant note is twice as high as the fundamental. All of the other notes that you access divide the string in similar, but different ways.

Nodes are symmetrical. If you experiment with your guitar strings in the opposite direction (past the 12th fret), you'll find that you can create the same natural harmonics by playing above very high frets and even pickups.

The reason why the harmonics go higher as you move down (or to the opposite end of the string) is that you are accessing different nodes that further divide the string into equal but smaller divisions. Just like the 12th fret divides the guitar in half, the 7th fret, 5th fret, and others divide the same string into several equal divisions. Just like the string divided in half giving you a pitch twice as high, a string divided equally into 3 or 5 or 7 parts (if amenable to the base frequency) will give you proportionally higher pitches.

This is why I said that nodes were symmetrical - it doesn't matter which direction you go from the 12th fret. You'll be able a to access the same harmonics in either direction.

Check out wikipedia for more information, but, for what it's worth, it might be a little heady for a first-time read:

http://en.wikipedia.org/wiki/Harmonic_series_%28music%29

  • The wiki link is a long read with your thinking cap on. It does not tell me why the harmonic notes on my guitar go in opposite direction of fretted. I know about 12th fret being half way point and that makes sense. But since the entire string is vibrating on a 5th fret harmonic, why is it so much higher than the 12th fret? And 12 fret harmonic is same as 12 fret fretted note. Not so on other harmonics. – Rockin Cowboy Mar 3 '15 at 4:43
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The harmonic nodes are uniformly distributed across the length of the string.

  • at the 12th fret (2x open string frequency)
  • at the 7th and 17th fret (3x open string frequency)
  • at at the (approx) 5th and 24th fret (4x)
  • at the (approx) 4th, 9th, 16th and 28th (5x)
  • and so on

for some of nodes closer to the bridge, like the "28th fret" you have to imagine where such a fret would be.

It's just that, given that we fret with our hand out over the fret board we don't usually induce natural harmonics by touching the string with our fretting hand above, say, the 12th fret. Note that you induce pinch harmonics by using your picking hand to lightly damp the string at the nodal points closer to the bridge.

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A string can support multiple independent vibrational modes simultaneously at multiples of its fundamental frequency; generally, only the brain will focus on the lowest one it hears and regard the others as subsidiary to it. Brushing one's finger against part of a string will absorb energy from all vibrational modes which would require that part of the string to move, while leaving unaffected those modes which where that part of the string would naturally be stationary.

If one is careful and touches the string at just the right spot, it's possible to pluck a string, graze it lightly so as to nullify all vibration modes which would require motion at the finger's position, and then remove one's finger, leaving the string vibrating in only the reduced set of vibrational modes. If one touches the string briefly at the midpoint, and then at either the 1/3 or 2/3 point, the first touch will eliminate all modes that aren't multiples of 2x the base frequency, and the second will eliminate any that aren't multiples of 3x, thus leaving only multiples of 6x the base frequency. Such harmonics may naturally be very faint, but one can maximize their starting amplitude by plucking the string at a spot near 1/12 or 1/4 of its length (plucking the string a spot near k/N of its length for some integer k will produce very little of the Nth harmonic; plucking at at a spot near (k+(1/2))N of its length will maximize the Nth harmonic.

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So basically, the 12th fret is the mid-point on the string and the lowest pitched harmonic. Anything from 12 to the nut gets higher in pitch and the same thing goes for the opposite direction, from 12 to 24. Makes sense now when you think of dividing the string equally. The nut and the bridge are the fixed points and the 12th fret is exactly midway between them. So dividing the string in half, anything higher or lower than the 12th will have a higher pitch as it gets closer to either fixed point.

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Here are some basic ideas that are of interest to musicians. Music is based on a simple mathematical idea called 'Harmonic Series'.

https://en.wikipedia.org/wiki/Harmonic_series_(music)

The terms 'pitch' and 'frequency' can be used almost interchangeably in this discussion. If we are talking about the physics of music, these terms are closely related.

The Harmonics Series is simple idea. If we multiply a base frequency (pitch) by integer multipliers, we create frequencies (pitches) that vibrate sympathetically (harmoniously) with the fundamental and the other pitches in the series. We really don't need to go far up the series to create the intervals that create the 12 divisions in the octave that Western music and even bird calls use. Eastern music divides the octave further, we also use smaller intervals from time to time in Western music, but we have no system of notation for these smaller intervals. At least I am not aware of such a system. I would not be the least bit surprised to learn that there is such a system, or systems.

If you look at the first picture in the Wiki article linked above, we see that if we place our finger on the twelfth fret of the the guitar, we cause the string to vibrate at double the frequency, creating the interval of an octave. If we place our finger on the 7th or the 19th fret, we triple the frequency. This creates a pitch one octave plus a fifth above the fundamental. If our fundamental pitch is E the resulting pitch is a B. If we place our finger on the 5th fret or the 17 fret, we get a pitch 2 octaves above the fundamental, or a frequency 4 times the fundamental. If we place our finger on the string at the 4th,9th,16th or the 21st fret we create a frequency 5 times faster than the fundamental, this produces a note that is 2 octaves plus a major 3d above the fundamental, creating a G# on the E string of a guitar.

We can continue up the series, but we have already produced all of the intervals necessary for the 12 tone scale that Western music uses. If our fundamental is E, we have a B and a G#. E to B is a fifth. B up to E is a fourth. The difference between a fifth and a fourth is a major 2nd. The interval between an E and a G# is a major 3rd. The interval between a G# and a B is a minor 3rd. The difference between a minor 3rd major 3rd is a minor 2nd, or half step.

There is a problem with this system, however. If you use a scale based on these pitches, there are tuning problems, if you try to play in a different key, you will be out of tune.

Bach solved this problem. He divided the octave into 12 equal intervals, based on the mathematic algorithm of multiplying the fundamental frequency by the 12th root of 2 to create a half step. If you do this 12 times, you end up with a frequency that is 2 times the fundamental i.e. an octave above, and a 'symmetric' scale that is identical in every key. The frequencies are not exactly the same, but they are very close.

For example, lets look at the Perfect 5th. Using the harmonic over tone series, a fifth is created by multiplying the fundamental by 3/2, or 1.5 .

Using Bach's method (the one used in modern music), to create a fifth, we multiply the fundamental by the 12th root of 2, raised to the 7th power (because a Perfect 5th is 7 half steps). The result here is the we multiply the fundamental by 1.49831. This difference is detectable by ear. As you can see, these numbers are pretty close.

Interestingly, if we create a overtone at the 5th fret (I like to use the A string on the guitar for this. It helps to use some overdrive which compresses the signal and makes the overtones louder.), then on the 4th fret, then the 3rd fret, slightly past the 3rd fret about 1/4 of the way towards the 2nd, then over the 2nd fret, we can create the notes in a dominant 9th chord!!! A, C#, E, G, B. !!!

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