Blog #7: Math and Music Cross-Curricular lesson

Hey everyone!

Welcome back to my blog!

This week I am going to make some cross-curricular connections between math and art, specifically, music.

I worked at the Virtual High School out of Bayfield Ontario, creating math lessons, and I did a heavy focus on cross-curricular lessons between math and music concepts.

I was able to come up with a fairly simple lesson for grade 9/10 level students, as well as a more complex lesson for grade 12.

If you haven't taken music class or are unfamiliar with music theory, this lesson may be a bit challenging, but there is a great resource that can be used to help familiarize yourself with how Pythagorean made such great impacts in the math world.

Music: A Mathematical Offering - Dave Benson 

In music, the pythagorean scale (C major scale) was developed through ratios to establish harmonizing sounds. The scale is constructed through an octave and"perfect fifths". An octave is when the top note is double the frequency of the bottom note. A perfect fifth means one note is one and a half, or 3/2 of the frequency of the note 8 semitones below. So, the note G is a perfect fifth away from C since they are 8 semitones apart.

C - C# - D - D# - E - F - F# - G

From the knowledge that the note G has a frequency that is 3/2 the frequency of C, we could determine the note's frequency when given the frequency of C.

The note C has a frequency of 262 hz, so if we take,

262 x (3/2) ... we would get 393. This means the note G would have a frequency of approximately 393 hz.

This is how the pythagorean scale was developed. The next perfect fifth from G would be D.

(G - G# - A - A# - B - C - C# - D)

We can take our ratio of G and multiply it by 3/2 to get the frequency of D. 

This process continues on, which can help anyone create the music scales and find the note frequencies.

The high C would be found simply by multiplying the frequency by 2, since this is an octave.

Now rather than multiplying each frequency by 3/2, we can create individual frequency ratios for each note. For example, if we want to find the frequency of D, but didn't have to find the frequency of G, we could take 3/2 x 3/2 and get 9/4. Now, the frequency ratio does have to be between 1 and 2, since the notes in the scale have to be between the base and octave notes. So, 9/4 is greateer than 2, so we could have to divide by 2. 9/4 divided by 2 is 9/8. So multiply 9/8 by the frequency of C to find the frequency of D.

This can be a confusing process, so check out this video to help understand the relation:

The task for this lesson can be differentiated to different levels, or different grades. In this lesson, you could ask students to come up with the notes and their frequencies when given a base note. For example,

Question: The song "Shallow" from the movie "A Star is Born" is written in the G major scale. Given that the scale is produced with perfect fifths, beginning on the note G, which has a frequency of 393 hz, construct the rest of the scale (8 notes from low G to high G) and find their frequencies.

In the new grade 9 de-streamed math curriculum, there is a focus on numeracy, which includes ratios and proportions. The following expectations could be used:

B3.3 apply an understanding of integers to explain the effects that positive and negative signs have on the values of ratios, rates, fractions, and decimals, in various contexts

B3.5 pose and solve problems involving rates, percentages, and proportions in various contexts, including contexts connected to real-life applications of data, measurement, geometry, linear relations, and financial literacy

These both can be covered if we let students explore the relationship between notes through the ratio of 3/2, the perfect fifth.

In the grade 9 music course, there is a portion about Foundations of music, which covers theory and the development of music. An expectation that can be connected to this lesson would be,

C2.1 demonstrate an understanding of the origins and development of some musical forms.

There are so many great concepts within the music world that were established through mathematics. 

I hope this is a good start to a math and music lesson that you could use with your students!

See you next week!

- Jenna :)