A glockenspiel bar is struck and allowed to resonate freely, producing a tone. The frequency of this tone is dependent on the bar’s physical dimensions (length, width, thickness), as well as the bar’s composition (wood, metal) and homogeneity (i.e. no variations in the density or composition of the bar). If we restrict the variables so as to allow only variation in the length, we can see that there is a relationship between the length of the bar and the frequency it produces. This relationship is governed by the equation (Rayleigh) F = c/L2, where F = frequency, L = length, and c is a constant dependent on the bar’s other physical dimensions and characteristics (Since only length will vary, c will remain constant for this exploration).
Glockenspiel bars are supported at two points along their length, both points approximately 0.224L in from the ends. These two points are the nodes, thus allowing for free resonance of the bar, with minimal damping from the supports. The inverse-square relationship c/L2 may not be fully evident to the student, although an obvious inverse proportion should be evident. A large portion of this module will have the student explore the mathematics behind determining the inverse-square relationship, as well as exploring some of the physics behind music and acoustics.
Suggested Time: Two class periods.
Advanced Preparation:
Procure some tuning forks and steel metal bars, along with the other materials. If you are planning on cutting the bars yourself, cold rolled steel works very well. Other metals, such as aluminum, do not produce a quality tone. Stainless Steel is a possibility, though its expense may be too great. Each group of students should have between 13 and 25 metal bars, enough to form at least one full octave of tones (13 bars will give one full octave between the smallest bar and the longest bar, 25 bars will form two full octaves). Also, each group should be given a length of nylon cord for supporting the bars and a wooden mallet for striking the bars. A TI-83 calculator with CBL and Microphone probe are also required. This activity can also be performed on a Macintosh computer with a ULI Microphone probe and the program Sound4.5.
Engagement:
An excellent introduction to this module is to allow the students to play with tuning forks. Lightly tap the tuning fork on a convenient surface, and listen to the resulting tones. The students should hear an initial ping, which will dampen and die quickly, and a longer, deeper, truer tone, which will resonate for some time. A CBL or ULI Microphone probe can be used to pick up the frequencies: the ping will be represented by very dense waves initially, while the true tone will be represented by a longer lasting sinusoidal curve. At this point, it is an option to discuss the notion of overtones and undertones, and how they are represented on a frequency curve. Does any musical instrument produce a truly pure tone? What is meant by a pure tone?
Allow the students to explore the tones produced by the bars. They will discover that the placement of the bar relative to the cord supports will be crucial to the bar’s quality of resonance. Through trial and error, they should discover that the best placement of the bar is to have the cord be directly below the two nodes.
Exploration/Modeling:
Once the students have figured out how to produce a decent tones from the bars, have them lay out their bars in ascending order of size, using the cord as supports, thus creating a rudimentary glockenspiel. The shorter bars produce a higher tone, while the longer bars produce lower tones. Their task will be to determine the frequency generated by the bars, and compare the frequencies to the lengths of the bars. Plotting frequency vs. length will produce the desired inverse-square relationship. However, this poses a problem of its own: how to recognize the inverse-square graph as compared to other possible inverse proportion graphs.
Brief Mathematical Aside:
Inverse proportions are functions of the form y = k/xn, where n is greater than 0, and k is the constant of proportionality. The problem arises in that the graphs of the inverse-linear proportion y = k/x and the inverse-square proportion y = k/x2 are very similar in appearance. How does one determine the actual inverse proportion in such a case? A common method, and the one the students will use in this exploration, is to convert the axes of the graph into logarithmic scale. Given a general function of the form y = k/xn, the task is to determine the value of n. By taking logarithms of both sides, we get log y = log k - n log x. Setting y1 = log y and x1 = log x, and recognizing that k1 = log k is a constant, we have a linear equation y1 = k1 - nx1. A simple linear regression on x1 and y1 will give a slope n, which will then determine the actual exponent of the equation y = k/xn.
Procedure and Discussions:
The students will lay out the bars in ascending/descending order by size, adjusting each bar to produce quality tones. When students have collected data on frequency and bar length, they will enter this data into lists L1 and L2 on the calculator (L1 should be length, L2 frequency). Have them produce a plot and discuss the nature of the plots given. To develop a plot using logarithmic rescaling, have the students convert each list by taking the logarithm of the members of each list. Let L3 = log(L1), L4 = log(L2). Plot L4 vs. L3, and a linear graph should be evident. The slope n will be the desired power of the inverse relationship. Will n be close to 2, or exactly 2? Discuss with your students why or why not. From this, develop your final model of the inverse-proportion between frequency and length.
A further exploration can be used to relate the notion of an octave with the bar lengths, as well as using the inverse-proportion developed above. Each bar is p times as long as the bar previous to it. If b0 = length of the first bar, then b1, the length of the next bar, will be p times b0, or b1 = pb0. Therefore, b2 = pb1 = p2b0, b3 = pb2 = p3b0, and so forth. One octave, or twelve bars further, will have b12 = p12b0. Two bars, one one octave higher than the other, will differ in length by a factor of 1.414 (the students will measure this). Setting b12 = 1.414b0, we have 1.414b0 = p12b0, or 1.414 = p12, therefore p = (1.414)(1/12) = approximately 1.029. One will note that the square root of 2 is very close to 1.414.
Some students will notice that bars two octaves different will differ in length by a factor of 2, i.e. b24 = 2b0. Therefore 2b0 = p24b0, so p = 2(1/24), which is also approximately equal to 1.029.
In the formula y = c/xn, if L is halved, then F is increased by a factor of 4, which is two full octaves, as expected.
Physics:
Mathematics:
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