Music and math are inseparable. As such, the capacity for cross-curricular activities in project-based learning are many. For those students who are musically inclined, here are some ideas to incorporate the two subjects into some really interesting PBL units.

One of my childhood memories was watching a film in elementary school made by Disney, featuring Donald Duck. It was called Mathmagic Land. From this video, I learned about the perfect ratio, as well as the harmonic series of notes. Music featured prominently within the film.

To this day, as a musician and one who loves to solve puzzles, I believe wholeheartedly in connecting and bridging the two subjects in cross-curricular activities.

The following quote from Richard Feynman set to animation illustrates the correlation between science and nature. This concept can be easily translated to math and the arts, specifically music.

Richard Feynman – Ode To A Flower from Cub Studio on Vimeo.

Why Cross-Curricular Activities?

The purpose of this article is to present you with some interesting tidbits that bridge the subjects of math and music. Hopefully you’ll become inspired to create some great ideas for “performance-based” math PBL.

Just Temperament vs. Equal Temperament

Here’s Eugenia Cheng from the University of Chicago. She explains how math was used to solve a problem involving what musicians call Just Temperament vs. Equal Temperament.

Before Bach’s Well-Tempered Clavier, scale pitches were based on a pure harmonic series. A string is divided into halves, then thirds, then fourths, and so on. This creates various components of a major scale. However, it presented a problem when constructing scales in all 12 keys, since not all intervals were the same. Thus, each scale had their unique qualities, but intonation varied throughout.

Then came along the “12th root of 2” to essentially even out the half step intervals, allowing all scales to have the same tonality. Confused? Just watch:

From Math to Music

Here’a another interesting video that has a musician redesigning an ancient tried-and-true instrument, the violin, to his own thought-out mathematical specifications. Though we don’t really get to see his violin up close, or understand the mathematical equations on the board, it could make for a very interesting PBL. The idea here is trying to improve on instruments or understanding the evolution of acoustic instruments:

The World’s Ugliest Music

Solving the problem of the “ping” used in sonar systems was accomplished through the use of mathematics. The perfect ping had to have a specific “patternless-ness” in order to be effective. Too much pattern, and the signal “confuses” itself.  

This interesting puzzle sent Scott Rickard into the realm of submarines and a look into 12-tone compositions by Arnold Schoenberg. Employing an 88 x 88 grid of seemingly random pitches, a map of the “perfect ping” is notated and codified into standard music notation for the sake of “hearing” the ugliest music. In other words, if music with pattern is considered the most beautiful, it stands to reason that music with no pattern is the ugliest. Michael Lindvall of the New World Symphony performs the piece on piano:

Ok, so hopefully, you’re inspired a little bit and can see how math informs music and maybe coming up with some PBL lessons that bridge the two subjects.

This one is adopted from an art/math lesson idea from Teach21:

  1. A world-reknowned symphony orchestra is planning to commission a composer to create an original work for an upcoming performance celebrating the use of math in music. It will be premiered at their final concert of the season. You are planning to present your idea for consideration. You’ve heard through the grapevine that they will look favorably at pieces of music that utilize mathematical concepts, specifically those involving patterns, ratios, and fractals. You must create an original work and present it to the board of directors. During this presentation you must be able to explain how you used mathematical concepts to create your composition.
  2. The modern music world is in need of a new method of music notation. Using cues from standard midi, note on, note off, volume events, devise a modern system for telling session musicians exactly how music is to be played on paper, using universal symbols present in the digital world. In other words, rewrite music notation rules from the ground up. Teach the new system and have someone play it.

Here’s inspiration for this for the teacher. Withhold this video during the project run, so as not to taint your students’ ideas.

Hummingbird Music Notation Intro from Blake West on Vimeo.

Cross-curricular activities in a math class have their challenges. In a sense, they incorporate the best of design thinking with a performance aspect that musicians are well familiar with. Because math underlies so much of music theory, you and our students can come up with numerous ways to incorporate math, music and PBL.




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