measuring and lining up the pan pipes

students playing their instruments

us demonstrating an octave

Despite the recent appearance of some good books on the connections between music and mathematics, there are very few teaching activities available that bring this exciting material into the high-school math curriculum. What activities we found online were also of low quality, and some were not even scientifically correct. This motivated us to develop a fresh set of activities that teach the connection between graphs of functions, period, frequency, musical pitches and intervals, the speed of sound, and the harmonic series. We developed and field-tested these with 7th and 8th-graders who have had Algebra I, over three weeks near the end of their school year in 2013, and they gave very positive feedback (they loved it).

In our module, students learn how to record the pressure-vs-time graphs using the free software audacity, measure the period and hence frequency of sustained sounds, related this to musical pitches, and build and measure their own instruments - sets of 5-note pan pipes - using lengths of straws that they have computed using pitches from various musical scales (blues, pentatonic, diatonic, etc). Students are guided by worksheets, able to test their understanding with homework questions, and their work culminates in a mini-concert playing tunes to the class on their pipes. The module takes around 4 1-hour class periods in total (not including homework).

This strengthens students' basic algebra skills, opens their eyes (and ears) to real-world applied mathematics and waves, allows them to form hypotheses and test them in a hands-on setting, and builds confidence playing music in front of the class. The module content and style builds upon Barnett's experience developing the college course Math 5: The Mathematics of Music and Sound, and the experience graduate student Martinez brought as a year-long visiting instructor at the Newton School in Strafford, VT, through the GK-12 project (and her experience as a flautist!)

Here is the module we developed (write-up courtesy of Martinez): Periods, Pitches, and Pipes (PDF, 11MB)

This PDF file includes lesson plans, lists of class materials, instructions with photographs, worksheets and homework, and grading rubrics for the final project. During the final mini-concert we recommend hooking a mic into a computer and projecting the free software baudline to the whole class to visualize the frequencies produced in real time.

Please contact us if you have any questions. Enjoy!

Note added Jan 2016: Gareth Roberts' wonderful book covering some of this (especially the tuning and harmony aspects) has just come out; consider getting it.

This work is enabled in part by NSF grant DMS-1216656, and by the GK-12 project at Dartmouth College. We also thank the generosity of teacher Ilene Kanoff, and the wonderful students at the Newton School.