Trigonometric Graphs

Lesson

- To explore trigonometric functions in real world settings.
- To practice graphing sine curves.
- To understand the different elements of a sine curve.
- To explore what makes notes sound dissonant or consonant.

- Graphing calculator ( or graphing application)
- Paper
- Different colored pens or markers
- Piano (can use an online piano)

Did you know that sound can actually be modeled by sine waves? When you’re listening to music on the radio or on your phone your brain is interpreting sound waves that are traveling through the air. These sound waves are in the form of sine waves and their appearance changes depending on the note played and its volume. For instance, when the sound being produced is louder, the amplitude of the sine curve will be larger. Furthermore, when the noise produced has a higher pitch the wavelength is shorter and thus the frequency is greater.

Frequency is measured in the number of vibrations per second and denoted by a unit called hertz (Hz). The average adult human can usually hear sounds within the frequency range of $20$20 vibrations per second and $20000$20000 vibrations per second ($20$20 Hz to $20000$20000 Hz).

In this investigation you will investigate the math behind music and uncover what makes two notes consonant or dissonant as well as explore how noise cancelling headphones work.

We want to get an idea of what frequency each note in the chromatic scale has, but before we do that let’s get a little background on the chromatic scale. The chromatic scale consists of $12$12 notes that are all one semitone (half tone) apart. On a piano one full octave in the chromatic scale is represented by $7$7 white keys: A, B, C, D, E, F, and G. As you can see, there are twelve white and black keys in each octave: A, A# of Bb, B, C, C# or Db, D, D# or Eb, E, F, F# or Gb, G, and G# or Ab. Each black key represents a half step away from the notes on either side of it so it can be labeled as both a sharp (#) of the note on the white key before it or a flat (b) of the white key after it.

- Now let’s calculate the frequencies of notes in certain octaves of the chromatic scale beginning with the note A. In the first octave we will work with A has a frequency of $110$110 Hz. This information can be used to find the frequencies of each of the successive notes in the second octave by plugging into the formula: $F=Ar^k$
`F`=`A``r``k`where $F$`F`is the frequency of the note, $A=110$`A`=110 Hz, $r=2^{\frac{1}{12}}$`r`=2112, and $k$`k`is the number representing the note’s position in the chromatic scale starting at zero and ending at 11 (A= note #0, A# or Bb =note #1, B=note #2, C= note #3, etc.). Calculate the frequency for each of the 12 notes in this octave. Record the values you find in the table below. - Find frequencies for all $12$12 notes in the octave above as well. Use the same formula as you used when calculating frequencies for the original octave but now the position of the first note, A, will be $12$12 ($k=12$
`k`=12).

- What do you notice about the frequencies of the note A in the first octave you looked at and the note A in the second octave you looked at? Make an observation.
- Does your observation hold true for all of the frequencies of the notes separated by exactly one octave (ex. D from the original octave and D from the next octave)?
- Create one sine equation for each note in both of the octaves that will describe the way the sound wave for that note will look. The sine equation for each note should be of the form $f(x)=M\sin(2\pi Px)$
`f`(`x`)=`M``s``i``n`(2π`P``x`) where $M$`M`is the volume of the note measured in decibels, and $P$`P`is the frequency of the note measured in hertz. You can refer to this website to check out what might be a reasonable decibel reading for playing music. Once you have chosen a decibel reading use the same one for every note’s sine equation in both octaves. - Based on your previous observations, how should the graphs of A from the original octave, and A from the octave above differ in appearance?
- Look at your equation for C in the original octave. How could you represent this in degrees instead of radians?
- Look at your equation for B in the original octave. Create an equation that would describe the same note being played 10 decibels louder than the original volume. How does the graph of this equation compare the the graph of the original equation? How do you know?
- Use two different colored pens or markers to graph the equations representing C and G from the original octave on the same graph. Be sure to label the curves and the axes. You can use your graphing app or graphing calculator to determine what may be the best window size for your graph before actually graphing it. Your window should be small enough so that you are zoomed in enough to clearly see the pattern.
- What window (range of y values and x values on your graph) did you end up using? Why?
- Compare the periods of the two equations.
- Do the crests of the two waves ever overlap?
- If you said the crests of the two waves do overlap, how often does this occur? Does it occur at a regular interval? Do they overlap in the same way every time?
- Play the two notes together on a real or virtual piano. Do the notes sound consonant or dissonant?

- Use two different colored pens or markers to graph the equations representing C and F# from the higher octave on the same graph. Be sure to label the curves and the axes. You can use your graphing app or graphing calculator to determine what may be the best window size for your graph before actually graphing it. Your window should be small enough so that you are zoomed in enough to clearly see the pattern.
- What window (range of y values and x values on your graph) did you end up using? Why?
- Compare the periods of the two equations.
- Do the crests of the two waves ever overlap?
- If you said the crests of the two waves do overlap, how often does this occur? Does it occur at a regular interval? Do they overlap in the same way every time?
- Play the two notes together on a real or virtual piano. Do the notes sound consonant or dissonant?

- Based on your answers to number 7 and number 8, what criteria can you use to identify two notes as being either consonant or dissonant by comparing their graphs?
- Compare and contrast your answer to number 9 with a friend.

Have you ever wondered how sound cancelling headphones work? They first determine what kind of noises that you are hearing, and then emits an additional noise that will cancel out those sounds. The sine wave of the noise that the headphones create will have peaks that align with the troughs of the incoming outside noise and troughs aligned with the peaks of the incoming outside noise. When these two sounds are played at once the destructive interference will create a diminished waveform that produces no noise. For example, below are two graphs of the note A from the original octave. The first graph is the original curve that describes this A, and the second is the graph of the note A after a transformation.

When the two graphs of the note A from above are graphed together it can be seen that the peaks of the original graph match up with the troughs of the transformed graph so these waves will cancel out.

For this section you can work with a friend.

For the following notes find the equation of the curve that the headphones would produce to cancel out the note. Graph both the original equation of the curve and the equation of the curve that will cancel it out on the same piece of paper. Use different colored pens or markers to represent the two equations.

- B original octave
- F# higher octave
- E higher octave
- Db original octave

As you know, not every noise you hear in daily life is one solid note played alone. Often times at least two notes are played at once. When two sounds are played at the same time and their peaks and troughs are not in exactly opposite positions (which would produce noise cancelation), their equations are added together to produce a new wave. For example if I were to play A from the original octave and A from the higher octave at the same time it would produce a sound wave described by the equation $y=80\sin(2\times110x)+80\sin(2\times220x)$`y`=80`s``i``n`(2×110`x`)+80`s``i``n`(2×220`x`) and have the graph:

- On your graphing application or graphing calculator, graph the notes C and A from the original octave being played at the same time. Draw a sketch of the result.
- Clearly the addition does not produce a clear sine wave. What similarities do you notice between the graph of the two notes played at once and a normal sine curve?
- Can you come up with the equation that the headphones would emit to cancel out the combined sound of the notes C and A that you have just graphed?

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions