Write-Up #12

Guitar Strings and Frets

What's the relationship?

 

First of all, what is a fret? According to the website howstuffworks.com, the frets are the metal pieces cut into the fingerboard of the guitar at specific intervals. When your fingers press down a string on the fret, the length of the string is changed. The vibration causes various tones to be produced.

 

 

The frets can be measured from the open string, which I will name fret 0, to each one thereafter, fret 1, 2, 3, etc. The length of a guitar string can be measured from a fret to the bridge of the guitar.

The question is: What is the relationship between the fret number and the length of the guitar string to that fret?

 

I will explore this question using a set of data of fret numbers and corresponding string lengths in centimeters. Click on this link to open the data in Excel.

 

Fret # String Length
0 64.5
1 60.7
2 57.4
3 54.1
4 51.1
5 48.3
6 45.5
7 43
8 40.6
9 38.4
10 36.3
11 34.2
12 32.3
13 30.4
14 28.7
15 27.1
16 25.6
17 24.2
18 22.8
19 21.6
20 20.4

 

To obtain an idea about the relationship between the fret number and string length, we can graph the data as an xy-scatterplot.

 

Just by looking at the graph, one may conjecture that there exists an exponential relationship between the two variables. The equation of an exponential function is of the form:

If an exponential relationship does exist between the two variables, b must be negative. This is due to the decreasing orientation of the set of data points. Excel can test this conjecture by graphing an exponential trendline on top of the graph of our data. If the curve fits most of the data points, there is a good chance that our conjecture is correct.

The exponential function seems to fit our data points extremely well. Excel also provides the exponential function corresponding to the curve. In this case, the function of the curve is:

Thus, the a-value of the exponential equation is 64.368, and the b-value is -0.0576. How can we use our given data to arrive at the values of a and b?

The a-value is relatively obvious. The entire length of the string from the open string, or fret 0, to the bridge of the guitar is 64.5 centimeters. This is our initial measurement.

It is a little more complicated to calculate the value of b from the given data. We will start by calculating the ratio of successive terms of the string length. In other words, calculate the value of B2/B1 in Excel, then drag this formula down to fill the remaining cells with ratios. The ratios are displayed in the chart below. Notice that there seems to exist a common ratio among successive lengths. I included the arithmetic mean of the ratios to find a good estimate of this common ratio.

Fret # String Length Ratio
0 64.5 0.941085271317829
1 60.7 0.945634266886326
2 57.4 0.942508710801394
3 54.1 0.944547134935305
4 51.1 0.945205479452055
5 48.3 0.942028985507246
6 45.5 0.945054945054945
7 43 0.944186046511628
8 40.6 0.945812807881773
9 38.4 0.9453125
10 36.3 0.942148760330579
11 34.2 0.944444444444444
12 32.3 0.941176470588235
13 30.4 0.944078947368421
14 28.7 0.944250871080139
15 27.1 0.944649446494465
16 25.6 0.9453125
17 24.2 0.942148760330579
18 22.8 0.947368421052632
19 21.6 0.944444444444444
20 20.4 0
Arithmetic Mean = 0.944069960724122

The arithmetic mean is approximately 0.94407. Notice that if we subtract 1 from this common ratio, we obtain approximately the value of b.

0.94407-1 = -0.05593

Thus, the a-value of the exponential function is equal to the initial measurement of our data at fret 0. The b-value can be computed by subtracting 1 from the common ratio of successive string lengths. Since the values are so close, I will continue to use the value of b provided by Excel.

We can enter the equation into Excel to compare the calculated values with the measured data.

Fret # String Length Exponential Function Results
0 64.5 64.368
1 60.7 60.7651610204552
2 57.4 57.3639820072373
3 54.1 54.1531755444363
4 51.1 51.1220860012219
5 48.3 48.2606541692426
6 45.5 45.5593838793583
7 43 43.009310486918
8 40.6 40.6019711209969
9 38.4 38.3293765988573
10 36.3 36.1839849124291
11 34.2 34.1586761988174
12 32.3 32.2467291117749
13 30.4 30.4417985157222
14 28.7 28.7378944282891
15 27.1 27.1293621414963
16 25.6 25.6108634556031
17 24.2 24.1773589633454
18 22.8 22.8240913257679
19 21.6 21.5465694841515
20 20.4 20.3405537556378

As you can see, the lengths are approximately equal. Thus, the function that produces the graph representing the measured data is:

We can use this function to make predictions about the lengths of the strings for further fret numbers. For example, what would the string lengths be if the guitar had fret numbers 21 and 22? We can plug in 21 and 22 for x in our equation above to find the string lengths of approximately 19.2 and 18.1 centimeters, respectively.

What would happen to the graph if the guitar has a shorter neck? For example, what if the length of the string from fret 0 is 50 centimeters instead of 64.5 centimeters?

 

We use similar reasoning to find the equation for the relationship between the frets and the string lengths. In this case, the initial string length is 50 centimeters. Thus, a = 50 in the exponential equation:

We can assume that the common ratio between successive string lengths will remain the same no matter what the initial string length is. Since b = common ratio - 1, the b-value will remain the same in all cases, i.e., b = -0.0576. Thus the equation to relate fret numbers to string lengths is:

We can enter this formula in an Excel spreadsheet to determine the string lengths corresponding to the first 20 fret numbers (or more if one so desires.) We can also create a graph to visualize the relationship.

Fret # String Length
0 50
1 47.2013741458918
2 44.5593944252092
3 42.0652929595733
4 39.7107926308273
5 37.4880796119521
6 35.3897774354945
7 33.4089225134523
8 31.5389410273714
9 29.7736271119635
10 28.1071222598411
11 26.5338958790217
12 25.0487269386768
13 23.6466866422152
14 22.3231220701972
15 21.0736407387959
16 19.8940960225602
17 18.7805733931033
18 17.7293774280449
19 16.7370195470976
20 15.8002064345931

We can graph the equation in Graphing Calculator to see a continuation of the graph beyond our calculated points. As the fret number (x-axis) increases, the length of the string (y-axis) decreases.

Finally, we can generalize the relationship between the fret number and the string length with the following equation:

, where a is the initial length of the string in centimeters. One can explore the graph of this relationship in Graphing Calculator by entering different a-values or by animating the value of a. View an animation of the relationship as a varies between 10 and 80 centimeters.

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