Natalya would like to acknowledge the amazing support of her friends Wendy Cho
, Carolyn Meldgin, Antoinette Evans, Will McFaul, Aaron Williams, and her fantastic students and colleagues at Countryside School
and Math Zoom
, especially Chris Antonsen, Kim File, Harold Reiter
, Jeffrey Huang, Kashyap Joshi, and Priscilla Wang. Frequency and Music
Our abilities to recognize patterns in music using sine waves help to “see” innovative ways of problem-solving. For teachers, a great introduction to teaching frequency theory can be found here (http://illuminations.nctm.org/Lesson.aspx?id=2359
). Students might enjoy discussing the activity found in NCTM Illuminations to explore more with the mathematics of music. Sound travels through energy in the form of wavelengths, which can be described using the function of the form f(x) = A sin (B x). For a nice overview of sine waves, watch The Math of Music
. Frequency theory has many interesting and unique properties, some of which lead to harmonic analysis in upper-level math classes. Where Math Meets
has more exploration and analysis. Not all frequency theories hold true. Read this webpage: The Complication with Consonance
for more analysis and exploration. Patterns and Music
A great overview of geometry of triads, along with reflection and rotation, can be viewed here the Geometry of Consonance
. Some Fibonacci fans might be delighted to know this intricate and beautiful pattern can be found on piano keyboards
. Music theory studies how to use patterns to create music. An excellent analysis of Moonlight Sonata can be found here
. The final piece in this video plays a Bach piece. More information about the Fugue can be read about in a fascinating exposition, Gödel, Escher, Bach: An Eternal Golden Braid.
The “Baroque” (“broke”) style has introduced hundreds to the idea of composing music using patterns, and some would argue Bach’s pieces are more mathematical in nature than artistic. About the Composer
As indicated, Beethoven was going deaf when he wrote Moonlight Sonata.
Listen to Beethoven’s Moonlight Sonata
here. The piece was widely received by his audience partly because of its moving harmony combined with a gorgeous series of triads. The movie, Beethoven Lives Upstairs,
is a classic that explores the story of the musician who composed it.