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Music as a Trigonometric Function

Mr.Zbynek Loebl
Mr.Zbynek Loebl (4 posts) Head of Maths View Profile

All IB students have to do an “internal assessment” (IA) in mathematics in order to get their IB Diploma - this is a 12-20 page mathematical exploration on anything that the students are interested in.

 

Some past examples of successful IA topics include:

  • Finding the optimal trajectory of a basketball

  • Exploring different strategies when playing Monopoly

  • Modelling the best “kick” in karate

  • An exploration into the number e≈2.718...

  • The mathematics behind shuffling cards

  • Exploring the birthday paradox

 

As you can see, the topics largely vary from one student to another and this is actually the point - the IB exploration needs to be individual and show considerable personal engagement with the topic that the student chooses for themselves.

 

In fact, the focus of the work is not to be able to recall or even research complex mathematics, but rather to apply the mathematics that students learn about in school in a context of their own choosing - furthermore, it also assesses how well students are able to write up their thoughts and mathematical ideas in a formal piece of writing.

 

BCB had already received some great explorations and the current Y13s are now busy improving their work before final submission.


However, one IA draft stood out and I will summarize the key components of it here. I will not publish parts of the students' work, but just summarize what that student did as I find it really quite remarkable.

 

The student started to think about music and how sound is created. They realised that sound can be modelled by a periodic wave function like the one below.

 

 

However, this function is too smooth and a sound that would not be pleasant to our ears.

 

In fact, here is a comparison between a piano and guitar sound waves and how they compare to the sine curve that we teach to our students from Y10 onwards.

 

 

Clearly, there are similarities but also major differences.

 

The student then explored why these differences happen: in fact, each time a string is plucked to create sound, a wide range of vibrations happen at the same time due to the fact that the string vibrates simultaneously in different ways. 

 

 

These eventually diminishing overtones can be modelled as trigonometric curves so that the final tone is modelled as an infinite sum of slowly receding sine functions.

 

 

If we put correct numbers into this formula, we get a very exact model of the sound produced by any instrument. For example, the following is a generated sound wave of a guitar and it is very similar to the one above that was measured empirically.

 

 

I have not included the precise mathematical calculations, they are tough to understand, but not beyond the understanding of an average Y12 IB student. However, the careful application of theoretical knowledge to this particular setting is remarkable and is truly in line with the independence and risk-taking ethos of the IB Diploma - no one step includes really complex mathematics, but taken together, it shows a great balance of independent research, thought and curiosity.