Sine Waves and Music
PreAP PreCalculus
How do you think Beethoven was able to create his music compositions?
Introduction
Day 1: Sine Waves
Sine Wave = Frequency
Cycle = One repetition of a wave's pattern
Frequency = The number of cycles per second (measured in Hz)
Period = The time duration of one cycle (the inverse of frequency, P = 1/f )
Wavelength = The length of one period of a wave
Amplitude = A measure of a wave's change over a single period
Looking at the picture above:
2. The sine wave with the longest period is _________________ , and the sine wave with the shortest period is _________________ .
3. The sine wave with the shortest wavelength is _________________ , and the sine wave with the longest wavelength is _________________ .
4. What relationship exist between frequency and wavelength?
Using an iPad to Generate Waveforms
Use your headphones, and take a couple of minutes to explore the different waveforms (sine, square, saw). Also, you can tweek the amplitude and fundamental frequency. To hear your waveform, press the play button.
What changes to the sound occur when you:
 Change waveforms?
 Raise/lower the amplitude?
 Alter the frequency setting?
Signal Analysis
On iPad:
We will be using the Auto Function Generator app again. At the top of the page you can select your waveform and change your frequency setting in Hz.
Select the waveform for "Sine" (the farthest waveform to the left).
Set your frequency at 650 Hz.
 Sketch a diagram of the wave.
 What do you notice about the graph of the sine wave as you change the volume?___________________
 What is another label that could accurately label the volume slider? _____________________
 What happens to the graph as you change the frequency slide? __________________
 Describe the sound that is created by the sine wave? _______________________
Song Analysis
 What type of song was played? _________________
 Sketch a diagram of the wave produced by the song over the full twenty seconds.
 Label the highest frequency, the lowest frequency, the largest amplitude, and the smallest amplitude.
 What affected the frequency throughout the recording? _______________
 What affected the amplitude? __________________
All Music needs Math!
Create your own wave from music
 What did you record for your sound? _________________
 Sketch a diagram of the wave produced by your sound over the full twenty seconds.
 Label the highest frequency, the lowest frequency, the largest amplitude, and the smallest amplitude.
 What affected the frequency throughout your recording? _______________
 What affected the amplitude? __________________
Day 2: Seeing Music, Hearing Waves
Sine Waves on sheet music
In this activity, you will calculate the frequencies of two octaves of a chromatic musical scale in standard pitch. Then, you will experiment with different combinations of notes and related sine waves to observe why some combinations of musical notes sound harmonious and others have a dissonance. When you check out note combinations, you will listen to those combinations played on a keyboard to associate the sounds with sine waves. You will be using your Desmos app for graphing the functions.
The Chromatic Scale — A Geometric Series
Musical pitches (notes) are determined by their frequency, which is measured in vibrations per second, or Hertz (Hz). The notes on a piano keyboard form a chromatic scale.
A chromatic scale divides the octave into its semitones. There are twelve semitones, or half steps, to an octave in the chromatic scale.
The white keys on a keyboard are A, B, C, D, E, F, and G. The black keys are named relative to their adjacent white keys. For example, the black key between the C and D keys is known as either C sharp (C#) or D flat (Db).
 The A note below middle C on a keyboard has a frequency of 220 Hz. Using this value, calculate the frequencies of the terms that generate a twooctave chromatic scale. Calculate each value using the original value of a = 220 and the formula ar^n, where r = 2^1/12 .Round each frequency to the nearest whole number.
 In the table above, compare the frequencies of notes that are one octave apart. For instance, compare A in the lower octave (left column) with A in the higher octave (right column), compare A# in the lower octave with A# in the higher octave, and so forth. How do frequencies an octave apart appear to be related?
The sine wave related to a musical pitch has the following form, where A is the amplitude of the sound (or the volume, measured in decibels) and B is the frequency of the note (measured in Hz): f (x)= Asin(Bx)
 Based on the frequencies in the above table, write the sine functions to represent both the low and high octaves for the C notes. (The value of A represents the volume of the note, so any value can be used. For the remainder of this activity sheet, let A = 2.)
 Then, graph the sine function for each note on your graphing calculator, and change the viewing window to show two cycles of the curve. (To do this, set Xmin = 0, and set Xmax to twice the value of the period; the period is equal to 2π divided by the frequency.) Graph the sine waves for notes in both octaves in the same viewing window. Draw the graph, and record the scale, frequency and period below.

Based on your observations above, describe where the graphs meet. With use of your iPad and the Auto Function Generator app, play C notes that are an octave apart. Describe in your own words how the notes compare.

When all three of the above sine waves are graphed, they intersect at the point (0, 0).

What are the coordinates of the second point where all three sine waves intersect? (The first point of intersection should be the origin.) ( ______ , ______ )

From the origin to the next point of intersection, record the number of cycles for each of the sine waves.

Why is this song stuck in my head?
 Using the sin function formula f(x) = 2sin(Bx), where B represents the frequency of the note, graph the first line of your sheet music.
 Sketch or Screenshot your graph.
 Does your song have consonance or dissonance?
 Why do you think this is a catchy song? Or if it isn't, why not?
Hope you enjoyed our journey through music and math!
Resources:
TEDEd: http://ed.ted.com/lessons/musicandmaththegeniusofbeethovennatalyastclair
Drexel University: http://drexel.edu/excite/initiatives/summerMusicTechnology/
I have also created a workbook to go along with the lesson that you can find at my TeacherPayTeacher page: https://www.teacherspayteachers.com/Product/SineWavesandMusicExploration1705387