Title
Piano Frequency
Problem Statement
There are
52 white keys on a piano.
The
frequency of each key is the number of vibrations per second the key’s string
makes.
Is this
relation a function? If not, explain
why?
If so, predict
an equation that models the data.
Problem setup
Can we find the specific frequency associated with any of the white piano keys? Since each key makes a unique sound, is the frequency produced predictable for each key?
Plans to Solve/Investigate the Problem
Using the Excel program, we can enter the number of the piano keys with the corresponding frequency as given in the original problem. By showing that each key has a specific and unique frequency, we establish that the relation is a function.
Investigation/Exploration of the Problem
We created the resulting chart using the Excel program:
Piano
Key |
Frequency |
1 |
27.7 |
8 |
55 |
15 |
110 |
22 |
220 |
29 |
440 |
36 |
880 |
43 |
1760 |
We asked the question, “What would the graph of this function look like?” Still using Excel, we created a scatter plot graph of the data set. Then, in order to find the different piano keys and coordinating frequencies, we needed to predict what those points would be. For instance, what would the frequency be for piano key number 4? We know that it will fall between 1 and 7 on our data table. We therefore wanted to create a trendline on the Excel graphing program to predict the location of piano key 4. When we looked at the resulting scatter plot options for a trendline, we found that the graph of the exponential function looked most similar to our scatter plot of the original data on the piano keys. By selecting the exponential trendline, the resulting graph closely approximated the shape of our scatter plot data and the formula for finding the frequency of any key appeared on the graph – y=24.994e^{0.0989x}.
The trendline formula for R is a measure of the probability of a point occurring on the resulting line graph. R^{2} must therefore be between 0 as having no possibility of occurring, and 1 being a perfect match with all points the same. In our data, R^{2} = 1 showed that the probability of finding the corresponding frequency for any key is a perfect match.
Author & Contact
Jill Jackson and Shirley Crawford
scrawford@rockdale.k12.ga.us