# The Guitar String Activity

Overview:

## Guitar String Activity: Frequency, tone and the length of a string

In this activity, we're going to explore the relation between a guitar string's length, frequency, and the tones that they create. Before we start, let's discuss the unit Hertz.

Hertz (Hz): This is a measure of frequency. Frequency is used to quantify the number of events in a given amount of time. This could be how many times you buy ice cream in a week or the amount of waves that break on the shore every minute. One Hz is defined as 1 cycle per second, so the Hertz may not be the best tool for quantifying the frequency at which you buy ice cream, but it is useful in the instance of a guitar: when you pluck a string, how many times does it cycle from that initial bent state, to the other side and back in a second? Turns out this value is what mostly determines the tone that you hear.

## Materials

-Pencil

-Sewing tape measure (ideally with metric units)

-Guitar (we included the box of strings in our picture)

-Guitar tuner (we tuned with an iphone app, but you'll find one a little further down this page)

Geared up:

## Collecting Data:

Step 1: First we want to know how long the string is. Using your tape measure, record the length of the guitar string from saddle to nut (in centimeters!)

The nut:

Step 2: Make a similar measurement, except this time from the saddle to the 5th fret. You can begin to record your data in a table like the one below. If you're feeling ambitious, you can make a table of this data for each string.

 String _____ Fret String Length (cm) Frequency (Hz) 0 (open string) 5 7 9 12

Continue making measurements from the saddle to frets 7, 9 and 12 for each string. To find which number fret you're on, count frets up from the nut. The following picture will help:

Step 3: Now we want to know the frequency of sound for each of the string lengths. For starts, let's make sure your guitar is in tune.

There are a number of different tunings that players have used over the years. The most common used in music today is (from biggest string to smallest): E, A, D, G, B, E. If your computer has a microphone, you can use the tuner below. The tuner will tell you the frequency of each note, so you can fill in the frequency columns on your table.

Remember to tell your browser to allow use of the microphone! If you don't see a prompt asking permission, refresh the page and click 'allow.' If you're using Safari, try this link: http://an-online-guitar-tuner.com/

It sounds like you're playing...

frequency (Hz):

Courtesy of Jonathan Bergknoff

Step 4: You can now record the frequencies for each of the frets in your table.

-Notice anything interesting about the ratio between the frequency of the 12th fret and that of the open string? We call this ratio an octave

-What about between the frequency of the 12th fret and that of the 5th fret? We call this a 5th.

-What about between the frequency of the 9th fret and the 5th fret? This is called a perfect third.

The cool thing about these ratios is that they apply to every note. For each note you play, you can use these ratios to find the octave, fifth and perfect third. What's the point of finding these quantities? Locate three notes that have this ratio of frequencies (4:5:6) and play them all at once to find out.

The following image should be useful in explaining how the frequencies are combined in the chord you just played. Although your chord was not likely to consist of the frequencies below, the ratio is the same, which creates the same harmonizing effect. Triad is a fancy way to say three-note chord.

There are many other ratios of frequency that make unique sounds, hence the variety of chords at a musician's disposal. For example, the major 7th chord is defined by a ratio of frequencies 15:8. The major 7th is widely considered smooth and comforting. Why do you think this is?

## Exploration:

1) What sorts of string qualities do you think effect the tone that you hear? Well you had to tighten and loosen the strings to tune your guitar, so we know that the string tension effects the tone. Let's call this T. The tension is a force, like the tension a string would have if it were holding up a 10-pound weight.  Thus we measure it in units of force or weight, either pounds or (in metric) Newtons.

It's also pretty clear that the strings differ in size: the low E string is considerably thicker than the high E string. We know this effects tone because the string tensions are roughly equal, yet the thicker strings produce lower frequencies. Let's measure this difference with the term Mass per unit length, or µ. Finally, and probably most obvious, we know the length of the string effects the tone because you're changing tone everytime you change fret. Let's call this length L.

As it turns out, we can combine all of these quantities into a convenient formula that gives the frequency, f:

A quick look at the formula shows that it matches our intuition. As we increase string tension, the freuqency goes up. As we increase the mass per unit length of the string, frequency goes down (think of the larger strings on a piano that are responsible for the lower frequencies).

-So what would happen to the frequency if you halved the length L of the string? We can try this one together:

For thist frist step we multiply the length of the string by 1/2.

Two multiplied by 1/2 equals one, and we're left with L in the denominator.

The new frequency is twice that of the original.

To check your answer, take a look at that table you constructed. When the length halved, is there an appropriate change of frequency?

-Now what about if you quadrupled the tension T? I wouldn't suggest actually trying that on your guitar, but could you lower the frequency by an octave (half of the original frequency) by reducing the tension? By how much would the tension have to be reduced?

-Suppose you wanted to play a note on the high E string of frequency 20,000 Hz, beyond human perception. Keeping mass per unit length and tension constant, how much shorter would you have to make the string? Could you play such a note?

Bonus question: In the formula:   ,  why do we multiply the length of the string by two in the denominator?