The connection between physics, engineering and music as an example of STEAM education

We describe the lesson scenario 'Design & play a guitar in your own tuning—that will work even if you cannot play a guitar' as an example of the iMuSciCA STEAM pedagogy.

This scenario is built around the concept map shown in figure 2 and it is aimed at students in the age-range 14–18 years old. From the point of view of physics tuning strings is a question of varying the value of the influence factors which determine the frequency of a string. In order to strengthen the understanding of these physics concepts we applied the iMuSciCA STEAM pedagogy by connecting the physics concepts to related insights in the engineering and musical fields. In order to make the student experience these connections one can think of different learning paths, starting from one discipline and connecting to the others. Whatever path one will follow, it will start in a certain discipline, turn at some point to another one and come back over and over again, until one understands the problem more deeply, thanks to the different viewpoints, and can solve it.

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We present here one possible path the students can go through when working on this STEAM task. The students will go in this example from musical questions towards engineering and scientific ones. The goal of this scenario is indeed to play a guitar, which is clearly a music-related activity. However, the idea is to make this task feasible for all students, even those who might know nothing about playing music. Therefore we propose a learning path which brings the student to engineering: i.e. design a guitar with a special tuning, so that it will be possible to play it without holding knowledge about the guitar chords. However it becomes soon evident that some scientific understanding of the underlying physics is necessary in order to proceed in this task. In order to understand what they have to do, students have to investigate the physical parameters of strings (like thickness, length, tension, nature of the material...) and how these influence the pitch of the produced tone.

2.1.1. The 3D virtual instrument design on the iMuSciCA workbench.

iMuSciCA provides students an online ICT-workbench where they can find virtual tools and activity environments to experiment with in the mentioned STEAM fields. One specific activity environment will be crucial for the presented task: the 3D Virtual Instrument Design. When students open it they get a standard tuned acoustic guitar, but they can alter the following parameters: the nature of the material the string is made off, the density, the thickness and tension of the string (figure 3). Therefore they can experiment by tuning the guitar in another way. In order to analyze the influence of the string parameters on the frequency produced, students will use the virtual monochord on the same environment, as the length of the strings can be changed there as well, while it is fixed in the case of the guitar.

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In the following we will illustrate the proposed example learning path in more details. We will also underline the connections between concepts in the different disciplines: physics, engineering and music.

Step 1—Music: Perform with a guitar that is tuned in a way that it can easily be played.

The idea is that the performance should be feasible even for a student which is not a guitar player. So the task does not consist in designing a standard classical guitar, but one that is tuned in a way that it makes it easy to be played. This could be done, for example, by tuning the guitar so that it can be played just by striking simultaneously all the strings together on their full length or by altering the length of the bunch of strings all together (thus not needing to form chords with the fingers like real guitar players do). You can achieve this when you tune the different strings in a way that the strings produce sounding intervals like an octave or a fifth (which will sound harmonic). A lot of tuning systems used in the folk music are based on this idea (like e.g. the DADAAD, a variation of the open D tuning) and students can look them up as a good starting point to tune the guitar they have in mind.

Step 1, which is clearly music related, is both the starting point and the goal of this STEAM-task. Students will work with the virtual guitar on the 3D Virtual Instrument Design of the iMuSciCA workbench. However the use in parallel of a real guitar is also advised. Indeed, while a virtual guitar can be tuned in a way that cannot be done with a real one, using both virtual and real experiments enhances student's conceptual understanding [4].

This first step brings us quite immediately to the next one: as such a guitar does not exist we will first need to design one. However the structure of the learning path will bring us back to this music step later on.

Step 2—Engineering: Design a guitar that is tuned in a way that it can easily be played.

This is clearly an engineering task as it is about planning, modelling and concretely realizing such a guitar. The goal is to simulate such a virtual guitar using an existing model which is actually based on scientific principles. At this stage it is not necessary to have a deep knowledge of those principles: being able to handle with a series of parameters influencing each other is enough. One option could be just to make students aware that there are correlations between pitch and certain parameters like length, tension, linear mass density of a string (without however knowing the precise relations). The teacher could give the students in that case an excel file as a 'black box' in which the correct relations are embedded: the students just insert the desired values of frequency and typical values of linear mass density and length (of the guitar strings) and they get the needed value of the string tension back. They can then use these values to design their guitar on the 3D Virtual Instrument Design of the iMuSciCA workbench.

However, if we want the students to deeply understand the relation between pitch and the above parameters, we need to go further to the next step, which is related to the physics of tone.

Step 3—Physics: Which influence factors determine the pitch of the string?

In this step students inquire the precise physics relations between the parameters which are used in step 2 and the frequency produced on a string. The students can verify or even 'discover' Mersenne's law [5, 6]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f=\frac{1}{2L}\sqrt{\frac{F}{\mu}}\nonumber \end{align}$$

with: f   =  frequency (Hz); L  =  length (m); F  =  tension (N); µ  =  linear mass density (kg/m).

The inquiry can be performed at two levels: first qualitatively and then quantitatively. Four influence factors can be inquired:

• 1.  
  influence of length on the frequency
• 2.  
  influence of tension on the frequency
• 3.  
  influence of thickness on the frequency
• 4.  
  influence of material type (and mass density) on the frequency.

The last two inquiries are chosen instead of the linear mass density, which is actually included in Mersenne's law, as those two parameters (thickness and material) are easier to be controlled. In doing this we take into account that:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mu =\frac{m}{L}=\frac{\rho V}{L}=\frac{\rho L\pi {{r}^{2}}}{L}\nonumber \end{align}$$

with m  =  mass (kg); V  =  volume (m³); 𝜌  =  mass density (kg/m³); r  =  radius (m). In section 2.3 we explain this inquiry in more detail.

Step 4—Engineering: apply these insights on the influence factors to virtually tune a guitar.

At this point students can go back to the engineering field in order to apply the acquired knowledge to concretely design their virtual guitar. As announced in step 2, they could use e.g. an excel file embedding the physics relations of step 3, insert the values of the desired frequencies for the strings of their special guitar and the typical values of linear mass density (or radius and material or mass density) and length and get the needed value of the string tension. Students could experiment by inserting values of linear mass density and length on the virtual guitar which differ from typical values used for real guitars or which are simply not available for real strings. This might also lead to values of the tension which might not be feasible with real strings. That's the interesting side of using a virtual instrument, i.e. exploring the 'extremes' of certain parameters, what cannot be done with a real guitar (e.g. if one increases the tension of a string there's a certain limit upon which the string will break).

Step 5—Music: perform by playing the tuned guitar.

Finally students can go back to the field of music they came from and use the designed and tuned guitar to perform by playing it.

For this part of the activity students use the virtual monochord on the iMuSciCA 3D Virtual Instrument Design, which allows better control of all parameters compared to the guitar. First students are asked to qualitatively investigate the influence of length, tension, thickness and material type on the frequency: they can just freely try to alter these parameters one at a time and measure how the frequency of the fundamental harmonic changes using e.g. the 2D Sound Visualisation (figure 4). At this point students can start the quantitative analysis.

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2.3.1. Length.

2.3.2. Thickness.

2.3.3. Tension and mass density.

Similarly students will start changing the tension and separately the density respectively from a value F₁ to 2F₁, 3F₁, etc and from ρ₁ to 2ρ₁, 3ρ₁ in order to recognize the relations:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f\sim \sqrt{F}\nonumber \end{align}$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f\sim \frac{1}{\sqrt{\rho}}\nonumber \end{align}$$

this time they will have to focus on the frequencies f _n measured for tension values like 4F₁, 9F₁ etc and for mass densities like 4ρ₁, 9ρ₁ etc, as such values easily allow us to identify a square root dependence.