University of Virginia
Physics Department

## The Mathematics of Music

A Physical Science Activity

2003 Virginia SOLs

• PS.1
• PS.8

Objectives

Students will

• determine the frequencies of the notes of a musical scale;
• examine the differences and ratios between these notes;
• determine the mathematical patterns used in musical notes.

Motivation for Learning

Demonstration: The Music in our Life

Materials

• Electronic keyboard (available in toy stores)

Procedure

The main purpose of this motivation is to show students that sound waves from music are related to each other. The students will investigate this relationship further in their experiment.

• Focus on one particular octave for these steps. An octave is a given set of keys, as shown here from 1 - 12. These keys are labeled musically as C, C#, D, Eb, E, F, F#, G, G#, A, A#, B (keys 1 - 12 respectively) and returning to C one octave higher after key 12.

• Play the keys alternating between tone 1 and 3, then 1 and 5, then 1 and 6, 1 and 8, 1 and 10 and then 1 and 12, then 1 and the one right next to twelve, which is starting the next octave. This last combination (from C to C) will sound different than all the other combinations you hit, because they will seem like they are the same "type" of note, but at a higher pitch. The students will notice this phenomenon they have always taken for granted and think about why it is the case.

Background Information

The musical scale used in music originated with the ancient Greeks. Originally there were seven primary notes, which anyone who has sung do-re-mi-fa-so-la-ti-do is familiar with. The last "do" is the first pitch in the next scale, therefore going back to the first note takes 8 steps hence they call it an octave (octa means 8 in Latin). Over time, five more "in-between" notes (the black keys) were added to the scale. This 12-note scale is called the chromatic scale, and is what we are familiar with on today's modern keyboard, as shown in the figure.

Musical scales are tied closely to mathematics. An interesting pattern emerges when the ratios of frequencies of a given scale are calculated. Additionally, the frequencies of the two notes that sound good together usually have a special mathematical relationship (as 1 and the one to the right of 12 did in this demonstration).

In this activity, if the room used is noisy, the microphone may occasionally pick up background noise and give a frequency far off the expected value. These values should be discarded as outliers and the measurement retaken.

Sample Data: This is sample data that we found. Different tone generators or instruments will be tuned slightly differently. The general concept should be the same though.

Data Table 1

 Key Frequency (Hz) Df (Hz) Difference to Previous Note Ratio to Previous Note # Note 1 C4 261.72 ------ ------ 2 C4# 277.28 15.56 1.06 3 D4 293.98 16.70 1.06 4 E4b 311.25 17.27 1.06 5 E4 331.77 20.52 1.07 6 F4 351.74 19.97 1.06 7 F4# 370.70 18.96 1.05 8 G4 393.32 22.62 1.06 9 A4b 415.84 22.52 1.06 10 A4 440.76 24.92 1.06 11 B4b 466.76 26.00 1.06 12 B4 494.86 28.10 1.06 13 C5 524.77 29.91 1.06 17 E5 658.00 ------ ------ 20 G5 782.00 ------ ------ 25 C6 1043.00 ------ ------

Data Table 2

 Key Ratio to C4 (Decimal) Ratio to C4 (Fraction) # Note 1 C4 1.0 1/1 3 D4 1.12 9/8 5 E4 1.27 5/4 6 F4 1.34 4/3 8 G4 1.50 3/2 10 A4 1.68 5/3 12 B4 1.89 15/8 13 C5 2.01 2/1 17 E5 2.51 5/2 20 G5 2.99 3/1 25 C6 3.99 4/1

1. There doesn't seem to be any pattern in the subtraction of frequencies.
2. There is a pattern in that every ratio of the note and the earlier note's frequency is the same ratio that should be about 1.06.
3. There is an interesting relationship that emerges with this one. Focusing on just the white keys, the notes have simple whole-number ratios to the first note:

D: = 1 + 1/8 = 9/8
A: 1 + 2/3 = 5/3
E: = 1 + 1/4 = 5/4
B: 1 + 7/8 = 15/8
F: = 1 + 1/3 = 4/3
C: 2 = 2/1
G: = 1 + 1/2 = 3/2

4. Yes! They are integer multiples of each other. This means that one of the wave cycles has a frequency twice as much in the case of 2/1, or three times as much in the case of 3/1.

### Student Activity

Materials

• Electronic keyboard, computer tone generator, or other musical instrument such as a flute or a trumpet
• CBL system
• TI-83 calculator with PHYSICS software program installed
• CBL microphone

Procedure

1. Connect the Vernier microphone to the CH1 input on the CBL. Use the black link cable to connect the CBL unit to the calculator. Firmly press in the cable ends.
2. Turn on the calculator and the CBL unit. Start the PHYSICS program, select SETUP PROBES. At prompt, hit ENTER for one probe.
3. At the next prompt, press 4 or arrow down to select MICROPHONE and press ENTER. Press ENTER again.
4. At the SELECT MICROPHONES screen, press ENTER to select CBL.
5. At COLLECTION MODE screen, press 3 or arrow down to FREQUENCY and press ENTER.
6. If you are using an electronic keyboard, tune the keyboard to a flute sound, as it has a much nicer sinusoidal curve and can be detected easily by the microphone. If you are using a computer generated tone, set the instrument to the one that gives the clearest data. For the Intelligent Organ we suggest using the piccolo sound because it gives the clearest frequencies. Also if using the Intelligent Organ, select octave 4 at the top of the keyboard and then the keyboard will have the same keys as the keyboards in the diagrams of the activity.
7. You will be measuring the frequency of various notes on the keyboard. For every note labeled on the diagram below (keys 1-12,17, 20, 25), record the frequency of the sound displayed on the calculator. After each measurement, hit ENTER, and then select yes to measure another note.

8. For keys 2 - 13, calculate the difference in frequency between each note and the frequency of key 1. For example the difference in frequency between key 2 and key 1 is Df = Frequency 2 - Frequency 1. Fill this into Data Table 1.
9. Again for keys 2 - 13, calculate the ratio of each frequency and the one before e.g. freq 2 / freq 1, freq 7 / freq 6, etc. Fill this into Data Table 1.
10. Data Table 1

 Key Frequency (Hz) Df (Hz) Difference to Previous Note Ratio to Previous Note # Note 1 C4 2 C4# 3 D4 4 E4b 5 E4 6 F4 7 F4# 8 G4 9 A4b 10 A4 11 B4b 12 B4 13 C5 17 E5 20 G5 25 C6

11. Now, to fill out Data Table 2, you are going to explore the special relationships between just the white keys on a piano.
12. Each of these white keys has a mathematical relationship to the first key of our octave, C4. First find the decimal ratio of the frequency of each key in Data Table 2 to C4.
13. Then make a guess to the fraction that the decimal approximates. (HINT: 1/8 = .125) Fill the fraction column out in Data Table 2.

Data Table 2
14.  Key Ratio to C4 (Decimal) Ratio to C4 (Fraction) # Note 1 C4 3 D4 5 E4 6 F4 8 G4 10 A4 12 B4 13 C5 17 E5 20 G5 25 C6

Students with Special Needs

All students should be able to participate in this activity.

Click here for further information on laboratories with students with special needs.

Assessment

1. Does it look like there is any pattern in the subtraction of the frequency and the one behind it? If so, what is that pattern?

2. Does there appear to be a pattern in the Ratio to Previous Note column of Data Table 1? If so, what is it?

3. Does there seem to be a pattern in the ratio of each key to the first key? (HINT: 1/8 = .125)

4. When the different C note frequencies (13,25) are divided, do we get something special in the mathematics? (remember, they certainly sound special!). What does this mean about their frequencies?