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Vector_Shift
09-04-2013, 01:37 AM
A little music theory, a little physics, and a little math: :D

When you play an open string, the string vibrates such that the middle of the string swings back and forth the most, while the ends of the string, which are attached to the nut and bridge, swing the least. The string wiggles like the first wave in this image:

http://www.astarmathsandphysics.com/ib_physics_notes/waves_and_oscillations/ib_physics_notes_standing_waves_on_strings_html_40 995f93.gif

When you play a harmonic at the 12th fret, you are placing your finger right in the middle of the string, and then plucking the string. This causes the middle of the string to stay put, similar to the way it does at the nut and bridge. Each half of the string wiggles back and forth the way the whole string did when you played an open note. This is like the second wave in the image. Since each part of the string is half as long as the whole string, the frequency it produces is twice the frequency that the open string produces. In other words, the note is one octave higher.

When you play a harmonic over the 7th fret, your finger is 1/3 of the way from the nut to the bridge. When you pluck the string while holding that point still, you get a wave where the points at the nut, the bridge, 1/3 of the way from the nut to the bridge, and 2/3 of the way from the nut to the bridge all stay fairly still and the three sections of the string between those points all vibrate. Since each of those sections of string is 1/3 the length of the whole string, you get a frequency that is 3 times the open note frequency. This note is between one and two octaves up from the open note.

Similarly, over the 5th fret, your finger is about 1/4 of the way from the nut to the bridge, so you get a note that is two octaves up from the open note, and over the 4th fret, your finger is about 1/5 of the way from the nut to the bridge, so you get another harmonic there that is 5 times the frequency of the open note.

So why the 12th, 7th, 5th, and 4th frets? Well, the 12th fret is more obvious than the others. The 12th fret is half way from the nut to the bridge, so you get the same note playing the 12th fret or the harmonic over the 12th fret. The next harmonic is going to be played 1/3 of the way from the nut to the bridge OR 1/3 of the way from the bridge to the nut [1]. As I said above, the 7th fret is approximately 1/3 of the way from the nut to the bridge, the 5th fret is approximately 1/4 of the way, and the 4th fret is approximately 1/5 of the way.

Why are those frets in those positions? It's because the guitar is designed to play notes in 12-tone equal temperament. That means two things. One is that there are 12 frets, and therefore 12 notes you can play, per octave (as long as you don't count both the root note and its octave). That's the "12-tone" part. The other is that the ratio of frequencies between two consecutive notes is always the same. That's the "equal temperament" part. A note that is one octave up from another note is twice the frequency of that other note, no matter how you divide the octave. Given those constraints, we can compute the ratio between consecutive notes. Let's call the ratio "X". So if an A note is 440 Hz, then the next A# in Hz will be 440 times X. The next B in Hz will be 440 times X times X, or 440*X^2. We know that the next A will be 880 Hz (twice 440 Hz). If we multiply 440 by X twelve times, then we should get 880. That allows us to write the equation,

440*X^12 = 880

We can divide both sides by 440 to get

X^12 = 2

We can then take the 12th root of both sides to get

X = 2^(1/12)

So the ratio between two consecutive notes is the twelfth root of two. The twelfth root of two is 1.0595 (rounded). So the next A# is 440 Hz * 1.0595 = 466 Hz, and the next B is 440 Hz * 1.0595^2 = 494 Hz.

The frequency of a vibrating string is inversely proportional to its length. So if an open string makes a particular note, then to make a higher note, you need to effectively shorten the string. We do this by holding the string against a fret. If a vibrating open string makes an A note, then how far does the first fret need to be from the bridge to make an A#? An A# is 1.0595 times the frequency, so the length of the string from the first fret to the bridge needs to be 1/1.0595 as long, or 0.9439 times as long. If the second fret is to make a B note, then it needs to be 0.9439 times as far from the bridge as the first fret. This means it needs to be 0.9439^2 times as far from the bridge as the nut is from the bridge.

The table below shows how far each of the first twelve frets are from the bridge and from the nut, where 1.000 is the length of the string from the nut to the bridge. The first number is how far away that fret is from the bridge. The second number is how far away it is from the nut.


1st fret: 0.944, 0.056
2nd fret: 0.891, 0.109
3rd fret: 0.841, 0.159
4th fret: 0.794, 0.206
5th fret: 0.749, 0.251
6th fret: 0.707, 0.293
7th fret: 0.667, 0.333
8th fret: 0.630, 0.370
9th fret: 0.595, 0.405
10th fret: 0.561, 0.439
11th fret: 0.530, 0.470
12th fret: 0.500, 0.500

You can see from the table above that the 7th fret is 0.333 from the nut, which means it is 1/3 of the way from the nut to the bridge. That's why you can play a harmonic there. The 5th fret is ever so slightly more than 1/4 (0.25) away from the nut, but very close. The 4th fret is a little bit more than 1/5 (0.2) away from the nut. That means the spot on the string that needs to be held still to play the "4th fret harmonic" slightly behind the 4th fret, where "behind" means closer to the nut. While it is not entirely coincidence that these frets line up with places to play harmonics, that is not the reason they were placed there. In fact, the next harmonic can be played 1/6 of the way from the nut to the bridge, which is not directly above any fret. 1/6 is approximately 0.167, which is between the 3rd and 4th frets, closer to the 3rd.

Note, however, that everything I have written above is theory. In practice, you won't find those frets or those harmonics in exactly the spots that the math suggests. The reason is that you have to bend the string down to make contact with the fret. Bending the string stretches it, which puts more tension on the string. A string with more tension on it produces a higher frequency than one with less tension. As a result, the frets are not placed exactly where the mathematics would suggest. In addition to that, we compensate a bit more by changing the length of the string slightly at the bridge. This is called setting the intonation. When set properly, the strings usually, if not always, end up being different lengths.

What that means is that, in practice, not only will the best spot to play each harmonic not be right over the nearest fret, it also means the best spot to play each harmonic is different for different strings! The A string is typically longer than the D string, so to play the "4th fret harmonic" on the A string, you actually need to play it a little bit closer to the bridge than the spot where you play the "4th fret harmonic" on the D string. This is what people mean when they say you must find the "sweet spot" for each harmonic.

I hope this helps your understanding and your playing of harmonics!


[1] You can play harmonics on either side of the 12th fret. The harmonic that you get by playing over the 7th fret can be played approximately over the 19th fret because the 19th fret is approximately 1/3 of the way from the bridge to the nut. If you play the note at the 19th fret and then play the harmonic over the 19th fret, you should hear almost exactly the same frequency. If you play the note at the 7th fret though, you will get a different frequency, one octave lower than the harmonic played above the 7th fret. That's because when you hold the string to the 7th fret and then pluck, the part of the string between the 7th fret and the bridge vibrates all together without any point along the way staying still. That length of string is approximately 2/3 of the way from the bridge to the nut, so the frequency produced will be half that of the frequency produced by playing the harmonic at the 7th or 19th fret or by playing the 19th fret. Half the frequency is, as I said, one octave lower. It is no coincidence that the 7th and 19th frets are 12 frets apart. That's how 12-tone equal temperament works. ;)

Vector_Shift
09-04-2013, 01:41 AM
It is, of course, possible to have unequal temperaments as well as equal temperaments with a different number of tones per octave. Here is a great explanation of 19-tone equal temperament:


http://www.youtube.com/watch?v=RZ7WbmhCsqs

Brick235
09-04-2013, 03:36 AM
Really good info VS. Thanks for taking the time to post it all. That guys guitar is crazy. You need some mighty skinny fingers to hit the 30+ frets! I could spend my life learning 12 tone so I'll stick to that for now. :p

Vector_Shift
09-04-2013, 07:19 AM
Really good info VS. Thanks for taking the time to post it all.

I'm glad you appreciated it!



You need some mighty skinny fingers to hit the 30+ frets!

Yep! Did you know there are 12TET guitars out there with 36 frets?!

http://www.vintagewashburn.com/Images/EC%20Electrics/EC36%20fret%20zoom.JPG

They aren't as close together as a those on a 19TET guitar, but they are still damn close!



I could spend my life learning 12 tone so I'll stick to that for now. :p

You've got that right!

Rutg3rO
09-04-2013, 09:05 AM
Great info! Thanks maestro :) I love the readable way you describe it. I wish more musical theorie was written this way.
I think I can theorize the answer based on what you describe here, but do pinch harmonics work the same way?

mbarsott
09-04-2013, 03:12 PM
Thanks, VS, this will be very useful. I always wondered how to calculate the fequency of certain notes, other than doubling the ocatves. Now I know... that mutliplying by 1.06 (rounded) we get a semitone above.
The practical application I will make of this is in EQ. Sometimes I want to emphasize a certain set of notes in the neck of the guitar by applying EQ to that region, But I never could tell more or less precisely the frequencies of the notes in that region... until now!

Vector_Shift
09-05-2013, 03:46 AM
Great info! Thanks maestro :)

Glad you liked it!



I think I can theorize the answer based on what you describe here, but do pinch harmonics work the same way?

Yes. For example, if you are holding the string to the 5th fret, then there will be multiple pinch harmonics you can play, one 1/2 way between the bridge and the 5th fret, one 1/3 of the way from the bridge to the 5th fret, another 1/4 of the way, 1/5 of the way, etc. Since the 5th fret is further from the bridge than, say, the 8th fret, the spots where you can play pinch harmonics will be further from the bridge if you are holding the string to the 5th fret than if you were holding it to the 8th fret. Let's say you play a 5th-fret pinch harmonic at the point 1/4 from the bridge to the 5th fret, and then you want to play an 8th-fret pinch harmonic. As you move your fretting hand closer to the bridge, you'll need to move your picking hand closer to the bridge as well, but only 1/4 as far as you are moving your fretting hand. One way to think of it is the fret you are holding the string against becomes like the new nut.

When you play a pinch harmonic, what you are doing with your picking hand is touching the string as you pick, basically doing the job with just your picking hand that both hands were doing separately when playing a natural harmonic. You are holding a spot on the string relatively still with your thumb or other finger on your picking hand while the pick sets the rest of the string in motion. Since the pick is so close to the thumb or finger holding that spot on the string still, you need more force to pluck the string. I think that's what people mean when they say you need to "dig in" when playing pinch harmonics.

At least, this is the theory. I'm terrible at playing pinch harmonics, so I'll let someone who can actually play them describe how they're played.

Vector_Shift
09-05-2013, 03:53 AM
Thanks, VS, this will be very useful. I always wondered how to calculate the fequency of certain notes, other than doubling the ocatves. Now I know... that mutliplying by 1.06 (rounded) we get a semitone above.
The practical application I will make of this is in EQ. Sometimes I want to emphasize a certain set of notes in the neck of the guitar by applying EQ to that region, But I never could tell more or less precisely the frequencies of the notes in that region... until now!

Cool!

1.06 is close enough to compute frequencies within an octave, but you'll be a few Hz off as you go further away. But since you can just double of half the frequency to compute octaves, you can get to the desired octave first, and then use 1.06 to get to the right semitone. Of course, a few Hz is not a huge deal when you're dealing with EQ because the roll-off is not instantaneous anyway.

Rutg3rO
09-05-2013, 08:50 AM
At least, this is the theory. I'm terrible at playing pinch harmonics, so I'll let someone who can actually play them describe how they're played.

haha, that sounds familiar. But you gave some better insights in the pinch harmonics too now, so I'm gonna play with them a bit tonight :D Thanks :)

civilianemail
09-05-2013, 09:32 AM
I find it fun to play harmonic arpeggios. To do so, fret all of the notes as you normally would for your desired chord then find the frets which correspond to your fingering 12 frets higher(or 5 or 7 for that matter). Next, pluck each string with the thumb of your pick hand while pressing down on the plucked string at the higher fret you've identified with your index finger . It may take a few practice plucks to get used to. The payoff is that you can now play any note on the guitar you like as a harmonic if you fret the string in the appropriate spot. Simple to do and sounds amazing.

Rutg3rO
09-05-2013, 10:50 AM
I find it fun to play harmonic arpeggios. To do so, fret all of the notes as you normally would for your desired chord then find the frets which correspond to your fingering 12 frets higher(or 5 or 7 for that matter). Next, pluck each string with the thumb of your pick hand while pressing down on the plucked string at the higher fret you've identified with your index finger . It may take a few practice plucks to get used to. The payoff is that you can now play any note on the guitar you like as a harmonic if you fret the string in the appropriate spot. Simple to do and sounds amazing.

Instead of plucking the string with your thumb, you can also pluck them with middle and ring finger. "Walk" the strings so to say.
Secondly, you can also tap the strings instead of plucking them. Fret the chords and tap 12 frets higher. (After reading this piece of Vector, I'm also gonna try other positions myself ;) )

sh1kamaru-
09-07-2013, 11:16 PM
great post VS ! I would just add several things :)

actually there as many nodes of a given harmonic as its fraction. What that means is that not only can you play a given harmonic on either side of the 12th fret, you can also play them at different places on the fretboard. For instance, the 4th fret harmonic in Red Barchetta can also be played on the 9th fret and 16th fret.

Also, this means that the 4th overtone (5th fret, 24th fret) is also present on the 12th fret. For this reason, when you play a harmonic on 12th fret itís much louder because there are more overtones stacked onto each other, even if you may not recognize them.

As a consequence, getting some overtones with a clean tone can be quite a challenge, and one workaround for this is to play with a lot of gain. The gain will tend to compress the tone, so weaker overtones will have a higher volume than they normally have.

For extreme use of harmonics, see Citizen Erased by Muse, Matt Bellamy plays the 7th and 8th overtones of the A string, which are placed around one third of the distance between the 3rd and 4th fret, and three quarters of the distance between 2nd and 3rd fret. I spent quite some time trying to find the sweet spot for those :D

Vector_Shift
09-09-2013, 03:57 AM
actually there as many nodes of a given harmonic as its fraction. What that means is that not only can you play a given harmonic on either side of the 12th fret, you can also play them at different places on the fretboard. For instance, the 4th fret harmonic in Red Barchetta can also be played on the 9th fret and 16th fret.

Yeah, I wasn't sure how well those spots would work since I hadn't really tried them, but they seem to work fine.

Vector_Shift
09-09-2013, 04:00 AM
One thing I realized while playing harmonics on my bass earlier is that when you try to play a harmonic, you have to be careful where you pluck the string, not just where your fretting finger is. For example, if you're trying to play a 4th fret harmonic, you're trying to divide the string into fifths, where each fifth of the string vibrates and the four points between those five pieces (the nodes) stay fairly still. If you try to pluck the string 1/5 or 2/5 of the way from the bridge to the nut, then you're plucking where you want a node. You're working against yourself. If you pluck the string somewhere between the nodes, you'll get a much louder, clearer harmonic.

To see what I mean, first find a sweet spot for a harmonic, then try plucking repeatedly as you move your plucking hand up and down the string. As you approach a node, the sound will deaden until it's practically a thud and then get louder and clearer as you move away from the node on the other side.

Plucking near the bridge (but not too close) is a safe bet for playing the 12th-, 7th-, 5th-, and 4th-fret harmonics.

Happy harmonizing! :cool:

KilaRoach
09-09-2013, 06:47 AM
I think I'll just stick to picking the yellow string at the 5th fret and bending it to match the squigily tail . . . :confused:

sh1kamaru-
05-10-2014, 12:34 PM
my mind just got blown:

https://www.youtube.com/watch?v=MYK_PF9WTRE

that is so awesome !

RichTheMighty
05-10-2014, 03:00 PM
One thing I realized while playing harmonics on my bass earlier is that when you try to play a harmonic, you have to be careful where you pluck the string, not just where your fretting finger is. For example, if you're trying to play a 4th fret harmonic, you're trying to divide the string into fifths, where each fifth of the string vibrates and the four points between those five pieces (the nodes) stay fairly still. If you try to pluck the string 1/5 or 2/5 of the way from the bridge to the nut, then you're plucking where you want a node. You're working against yourself. If you pluck the string somewhere between the nodes, you'll get a much louder, clearer harmonic.

To see what I mean, first find a sweet spot for a harmonic, then try plucking repeatedly as you move your plucking hand up and down the string. As you approach a node, the sound will deaden until it's practically a thud and then get louder and clearer as you move away from the node on the other side.

Plucking near the bridge (but not too close) is a safe bet for playing the 12th-, 7th-, 5th-, and 4th-fret harmonics.

Happy harmonizing! :cool:

BOOM - I have a miserable time with harmonics on the 5th fret but can nail them anywhere else. I am gonna play around with my position and hopefully that is it.