## Seeing Music

- Lesson

In this lesson students will calculate terms of a geometric sequence to determine frequencies of the chromatic scale. They will then compare sine waves to see and hear the trigonometry behind harmonious and dissonant note combinations.

Begin the lesson by having students answer the pre-activity questions that appear on the Pre‑Activity and Summary Questions Overhead. These could be assigned as homework or done on the day prior to the activity. The questions represent background skills that students need for the activity. If students are not able to complete these questions, some re-teaching may be necessary before attempting this lesson.

Pre-Activity and Summary Questions

This lesson allows students to consider the mathematics involved with musical scales. During various points in the activity, certain notes will need to be played. It will therefore be helpful to choose a student with a music background to assist with this lesson. The examples throughout the lesson use a keyboard as an example, but this lesson could also be conducted successfully using other types of instruments.

Distribute a copy of the Seeing Music Activity Sheet to all students. Have all students complete Questions 1‑5. Then, divide students into two groups, and allow Group 1 to work on Questions 6-7, and allow Group 2 to work on Questions 8-9. Finally, have students from both groups work together to complete Questions 10-11.

Musical notes are determined by the speed of their vibrations. This speed is known as *frequency*—which
is measured in vibrations per second, or Hertz (Hz). The notes on a
keyboard form a chromatic scale, and the frequencies of the notes form
a geometric sequence. As in any geometric sequence, multiply by *r* to get the next term in the sequence and divide by *r* to get the previous term of the sequence. The ratio that generates the chromatic scale (from lower pitched notes to higher) is *r* = 2^{1/12}.

It is likely that students have heard a chromatic scale played before, but you may wish to play one or all of the following sound clips for them (QuickTime required):

Below are some tips for making the calculations required in
Question 1 more efficient and accurate. Another approach would be to use a
spreadsheet to calculate the values directly from the formula *ar ^{n}*.

- Generate
each term from the formula
*ar*for various values of^{n}*n*(rather than from the approximation of the preceding term in the sequence). - Repeatedly
use the ENTER key. Enter 220 on a calculator, and hit ENTER. Then,
multiply by 2
^{1/12}and hit ENTER again. Repeatedly hit ENTER to generate the entire list.

Students are prompted to observe from the table in Question 1 that the frequency of a higher octave note is double the frequency of the same note an octave lower. If they are not making that observation, check their calculations to see that the frequencies are correct. If the calculations are correct but they do not make the connection, ask them to double the lower frequencies and discuss what happens. Some of the frequencies will not be exactly double that of the lower octave because of rounding. (A possible side discussion could be on when is it important to be “exact” and when rounding is acceptable. As it turns out, the chromatic scale is not exactly in tune in every key anyway, so rounding is not an issue to worry about.)

Assign Questions 6-7 to some students and Questions 8-9 to others. This gives a broader basis for student discussion. The commonality students should recognize is that regardless of the scale students are studying, the second intersection of the three sine waves comes at the end of 2 cycles of the first note. The sine waves of the notes of the triad (the first, third, and fifth notes) should have the same commonalities listed below (no matter which scale).

- Three cycles of the fifth note of the scale (i.e., the third note of the triad) fits in two cycles of the first note of the scale.
- Two-and-a-half cycles of the third note of the scale (i.e., the second note of the triad) fits in two cycles of the first note of the scale.

When students have completed the activity and have had a chance to discuss Questions 10-11 with classmates, the summary questions on the Pre‑Activity and Summary Questions Overhead can be used for review.

You can use this set of Selected Solutions to review the activity sheet with students.

- Graphing Calculators
- Electronic Keyboard or Piano
- Pre-Activity and Summary Questions
- Seeing Music Activity Sheet
- Selected Solutions

**Assessment Options**

- Listen
to student discussion. Are students making the connections between musical
sounds and sine waves? Are they recognizing that "harmonious" note
combinations have meeting points that are
*x*‑intercepts? Are they noticing that sine waves of dissonant note combinations don’t seem to resolve, either in their hearing or as common*x*‑intercepts? - Observe students calculations and tables. Are they correctly calculating terms of a geometric sequence? Can they transfer that skill to another scale (with a different starting value)?
- Have student groups present their findings about dissonance and harmony. Observe each group’s interpretation of the information and other students’ comments or questions. Are the students grasping the big ideas?

**Extensions**

- Have
students with synthesizers check their manuals and see how some of the
sounds are generated. (These are many times very complicated trig
functions, where the coefficient,
*B*in*y*= A•sin(*Bx*) is often another trig function. Although the function itself might be too complicated for students to discuss, hopefully they will realize that a variable frequency causes the pitch-changing effects of a synthesizer. - One of the possible spin-off projects would be to have interested students explore why pianos and fretted chromatic instruments cannot be perfectly tuned.

**Question for Students**

1. In terms of the period of the sine wave, describe where the graphs of the first, third, and fifth notes of the C major scale intersect. Do the sine waves intersect in the same place for the notes in the A major scale? Do you think the triads of other major scales would follow the same pattern? Explain how you would test the theory.

[The sine waves intersect at the origin and then again at the end of two cycles for the first note (which coincides with two-and-a-half cycles for the third note and three cycles for the fifth note). The same patterns occurs for both the A and C major scales and, in fact, would occur for the first, third, and fifth notes of any major scale.]

**Teacher Reflection**

- Were students actively involved in the lesson?
- Did the lesson pull in students interested in music who otherwise might have been uninterested with just math calculations?
- Did the lesson allow non‑music students to fully participate?
- Was the skill level of students sufficient to focus on the big concepts rather than just math calculations?
- Did students get bogged down in calculations and miss the big picture? If so, how can they learn to be efficient with the use of technology to focus on the concepts?

### Learning Objectives

Students will:

- Connect musical scales to frequencies generated by a geometric sequence.
- Relate sine waves to musical harmony and dissonance.

### NCTM Standards and Expectations

- Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.

- Use symbolic algebra to represent and explain mathematical relationships.

- Draw reasonable conclusions about a situation being modeled.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.