The Physics Of Guitar

            Guitar is one of the most popular musical instruments around the world. Personally I like to play guitar as well. Many people say that guitar is the best musical instrument because there are many techniques, which produce unique sound compared to any other musical instruments. Unfortunately, there are many people who like to play guitar but do not know about the physics of a guitar.

Luthier, or the person who make guitars, doesn’t make guitar without physics. As you know, the important parts of guitar (both acoustic and electric) are body (which contains the bridge), neck (which contains frets) and head (which has the tuning machine). Making guitar needs a lot of calculations, especially when we calculate the frets distance.

One of the most important things in making guitar is the fret. If the distances between frets are incorrect, the guitar will sound horrible, out of tunes. Before knowing about the fret distances, it’s better to know about the scale length first. Scale length is the distance between the nuts and the saddle. Different types of guitars have different scale length. String acoustic guitar has different scale length with classical guitar. Electric guitars also have different scale length compared to both acoustic guitars. Even different brands have different scale length. Each of them have different feel. For acoustic guitar the most popular is from Martin. They usually have scale length of 25.340”. For electric guitar two of the most important scale length are fender (25.5”) and Gibson (24.75”). For Gibson guitar they have different scale length from year to year, the detail is in the Stewart-MacDonald website. The following table of scale length is also from Stewart-MacDonald website.

GUITAR

SCALE LENGTH

Classical short

650 mm

Classical long

660 mm

Fender

25.5"

Fender Jaguar

24"

Fender Duosonic & Mustang 

22.500"

Fender Bajo Sexto Baritone Telecaster

30.1562"

Gibson 24-3/4"

(see the timeline on the website)

Gibson 'Byrdland'

23.5"

Gibson long scale (used on acoustics)

25.5"

Guild acoustics

25.625"

Guild electrics

24.750"

Martin standard

25.340"

Martin short

24.840"

National

25"

Paul Reed Smith

25"

Then, after we know about the scale length, we can start to measure the distance of frets. In physics, we know that if a frequency is multiplied by 2, the tone will be the same but in the higher octave. In music there are twelve semitones in one octave. In the website of Francisco Estrada Gómez (a luthier from Argentine), he explained that one of the earliest research was done by Frederiek Chlandi (some people say chladni) in 1809. He placed a fret in the middle of the string where the second harmonic is produced. The sound of open string should have half vibration produced by second harmonic or pressed at fret 12.

We know that lambda; the wavelength is the inverse of frequency. To get the next semitones, we have the formula:

            f2 = 2^(1 / 12) * f1

The formula is made based on the condition that the twelfth frequency is the double of the first frequency; which means an octave, also, because there are twelve tones in one octave. Because lambda is the inverse of frequency, we can also make the formula:

                        1 / l1 =  2^(1 / 12) * l2

With this formula we can calculate the distance of each frets. For example, using Fender scale length, 25.5”, the distance between the first fret and the bridge will be (25.5”/(2^(1 / 12))) =  ~ 24.0688”. So the distance between the nut and the first fret is simply the scale length (25.5) subtracted by the distance between the first fret and the bridge (24.0688) that is around 1.4312. The same calculation is used for the next frets. In one of my reference pages, they even give a C++ format program to calculate fret distances. For classical guitar there is a table from Francisco Estrada Gómez that use the same formula:

Fret #

Succesive Length of a String

Succesives Spaces

Addition of spaces

0

650

 

 

1

613.714

36.285

36.285

2

579.455

34.26

70.545

3

547.107

32.348

102.893

4

516.565

30.542

133.435

5

487.729

28.836

162.271

6

460.502

27.227

189.498

7

434.795

25.707

215.205

8

410.523

24.272

239.477

9

387.606

22.917

262.394

10

365.969

21.637

284.031

11

345.539

20.43

304.461

12

326.25

19.289

323.75

13

308.037

18.213

341.963

14

290.841

17.196

359.159

15

274.605

16.236

375.395

16

259.276

15.329

390.724

17

244.802

14.474

405.198

18

231.137

13.665

418.863

19

218.234

12.903

431.766

20

206.051

12.183

443.949

 

For standard guitar, it’s easy to measure until around 24 frets. But there is also a “mysterious” guitar that I still don’t understand how to measure the frets. The guitar is called Sky Guitar, a custom guitar made for one of my favorite guitarists Uli Jon Roth. This guitar has like 30 or 40 frets, which will be really narrow. Here is the picture of the guitar taken from a fan website.

 

The picture is taken from http://www.edenwaith.com/uliroth/music/skyguitar.html

The next important thing about guitar is the string. As you know a guitar usually has 6 strings and the standard tuning is EBGDAE (from bottom to top string). Each string has its own mass, tension, and material. For classical guitar usually the top 3 strings are made from metal while the bottom strings are from nylon. There are some materials that are used for strings, including the stainless one. Some guitars also have seven strings, just like normal 6-string guitar plus one string on the top that is B string. The sound is very low pitched. The example is also the Sky Guitar, shown above.

We can measure the tension of the string. Some of important elements of guitar string are mass, length, and tension. The frequency of a string when it’s plucked can be calculated by a formula:

             

L is the length of the string from bridge to nut, which is .5 l as shown in the figure below.

While v (wave velocity) can be written as:

So, we can also write f in formula shown below:

                       

            If we assume that the string mpl (mass per length) is constant then the longer the string the higher the tension that we need to get the same pitch. In the other way, if string length is constant, the bigger the mpl the higher the tension needed to get the same pitch. Of course guitar strings have their own tension limit, so we can’t tune the string much higher than the capacity mentioned by the producer. The result will be the string will cut off itself. But, if the string is tuned lower than the capacity, it will be fine because it will have less tension. It will also be easier to play. Some guitarists change their guitar tuning for some reason. One of the reasons is to get the harmonic notes (which I will explain next) they want. My favorite guitarists who use alternate tunings are Phil Keaggy and Michael Hedges.

One of my favorite things in playing guitar is the harmonic note. The two guitarists I just mentioned use the harmonic notes uniquely and beautifully. Here is the graph of harmonic:

                When an open string is plucked, the string makes a half of a wavelength. If we touch (not press) a string at twelfth fret and pluck it, we will make the second harmonic. By doing so, the length is still the same while the string makes a full wave (velocity is doubled). It also means the frequency is doubled, an octave higher. On the third harmonic, the velocity is 3 times higher. So, the frequency will be 3 times greater than the initial frequency. For third harmonic, we can do it when we touch the string around 1/3 or 2/3 of the length of the string, which is around 7th fret and 19th fret. Starting on the fourth harmonic, there is one thing that we need to remember. If you see at the graph, at the fourth harmonic there is an intersection at twelfth fret; which is half of the string length. We know that at the same point it’s also the second harmonic. In this case, we can produce both harmonic, but with different way. To produce the second harmonic, we just simply need to touch at twelfth fret and pluck the string. But if we want the fourth harmonic, we have to touch both ¼ length of the string (which is the fifth fret) and ½ length of the string (the twelfth fret). While we touch fifth fret and twelfth and pluck the string, we make two full waves (fourth harmonic) instead of one full wave (second harmonic). In summary, nth harmonic has n times the initial frequency. Basically to produce the harmonic notes we just have to touch the string on a spot that is at nodes (zero amplitude) of the wave. On standard tuning, the harmonic notes can be used to tune our guitar. I personally like this method the best. Here is the list table of the some of the harmonic note on guitar:

Harmonic #

Fraction of String Length

Interval Above

2

1/2

octave

3

1/3

twelfth

4

1/4

double octave

5

1/5

seventeenth

6

1/6

nineteenth

7

2/7

halfsharp 20th

8

3/8

triple octave

9

1/9

twenty third

Acoustic guitar works differently compared to electric guitar. Electric guitars use pickup and amplifier to amplify the sound. In an acoustic guitar the sound is amplified without amplifier or pickup. Well there are some kinds of acoustic pickups and amplifiers to make the guitar sounds really loud (like for concert), but in this case I mean that acoustic guitar can sound loud enough just by itself. One of the most important parts of an acoustic guitar is the sound hole. The air inside the sound hole is used to amplify the sound. Just try to cover the sound hole with something tightly; the sound won’t be too loud. Or another example will be playing electric guitar without amplifier. The reason is because of the Helmholtz resonance.

To understand Helmholtz, the picture below can be helpful.  

Helmholtz oscillator consists of container of gas (it can be air) and an open hole. The example of Helmholtz oscillator will be an empty bottle. Try to blow across the top. The volume of air inside the bottle and around the open hole vibrates because the air inside is like a spring (going up and down). So, the sound can be amplified.

For acoustic guitar, the body is like the empty bottle. And when we pluck a string it also vibrates the air inside the body and amplifies the sound. This method works for all the notes on guitar. Low frequency sound (bass sound) and high frequency sound (treble sound) are amplified in the same way.

In conclusion, guitar is a stringed musical instrument that follows the rules of physics. It’s really interesting how guitar works. Its process to produce and amplify sound in my opinion is both simple and complicated. Beside all of applied physics rules mentioned here, I’m sure there must be more things to be learn from a guitar.


WORKS CITED

Billington, Ian. The Physics of the Acoustic Guitar. 28 November 1999. 12 October 2005< http://ffden-2.phys.uaf.edu/211.web.stuff/billington/main.htm>

“How does a guitar work?” Ed. Joe Wolfe. The University of New South Wales. 12 October 2005. <http://www.phys.unsw.edu.au/~jw/guitar/intro_engl.html>

Gómez, Francisco Estrada. 12 October 2005. <http://guitarristas.com/estrada/method.htm>

<http://www.cs.jhu.edu/~ihsahn/FretScale.html>

 Understanding some well known scales . Stewart-MacDonald. 12 October 2005. <http://www.stewmac.com/FretCalculator/>

Hyperphysics. 12 October 2005. <http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html>

Noyce, Ian. Noyce Guitar. 12 October 2005. <http://www.noyceguitars.com/Technotes/Articles/T3.html>

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