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. ;)

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. ;)