Begin the lesson by having students answer the pre-activity questions that appear on the Pre‑Activity and Summary Questions
activity sheet. 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.
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 = 21/12.
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; visit the Software Needed page for details on downloading QuickTime):
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 arn.
each term from the formula arn
for various values of n (rather
than from the approximation of the preceding term in the sequence).
use the ENTER key. Enter 220 on a calculator, and hit ENTER. Then,
multiply by 21/12 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
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
- 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 activity sheet can be used for review.
You can use this set of Selected Solutions to review the activity sheet with students.
Questions 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.]
students actively involved in the lesson?
the lesson pull in students interested in music who otherwise might have
been uninterested with just math calculations?
the lesson allow non‑music students to fully participate?
the skill level of students sufficient to focus on the big concepts rather
than just math calculations?
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