The range

This activity allows students to discover the mathematical relationships between musical notes and their frequencies. It develops the understanding of the physical and mathematical foundations of music.

Why does a piano have 88 keys, and why do some note combinations sound harmonious while others clash? The answers lie in the mathematical structure of musical scales, a connection that has fascinated thinkers from Pythagoras to modern acousticians. In the Western chromatic scale, the octave is divided into twelve semitones, and the frequency ratio between consecutive semitones is always the same: the twelfth root of 2, approximately 1.05946. This means the frequencies form a geometric progression, not an arithmetic one. The note A4 is defined as 440 Hz; A5, one octave higher, is exactly 880 Hz. Between them, each of the twelve semitones multiplies the frequency by the same constant factor. This system, called equal temperament, was a revolutionary compromise adopted gradually during the 18th century, allowing musicians to play in any key without retuning. Using FizziQ's frequency meter, students can measure the frequencies of each note in the chromatic scale and discover this elegant mathematical pattern for themselves.

Learning objectives:

The student analyzes the frequencies of the twelve notes of a chromatic scale using the FizziQ frequency meter. By recording and organizing the data in a table the student discovers that the frequencies are not linearly spaced but follow a geometric progression. It also checks that the frequency doubles with each octave by comparing the same note across different octaves.

Level:

Middle school

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure the fundamental frequency of each note in a chromatic scale using FizziQ
- Discover that musical frequencies follow a geometric progression rather than an arithmetic one
- Calculate the frequency ratio between consecutive semitones and verify it equals ¹²√2 ≈ 1.05946
- Verify that the frequency doubles with each octave (2:1 ratio)
- Understand the mathematical basis of equal temperament tuning

Scientific concepts:

- Chromatic range
- Musical frequencies
- Geometric progression
- Octave and interval
- Harmonic relationships

Sensors:

- Microphone (frequency meter / fundamental frequency measurement)
- FizziQ spectrum analyzer

What is required:

- Smartphone with the FizziQ application
- 'Range' and 'Octaves' recordings from the sound library
- FizziQ experience notebook
- Calculator to analyze frequency ratios

Experimental procedure:

Open FizziQ and navigate to the Sound Library. Load the 'Range' recording containing the twelve notes of the chromatic scale.
Select the Frequency Meter (fundamental frequency) tool in FizziQ.
Play the first note of the recording and record the measured frequency in a table. Note: the first note should be close to a known reference (e.g., C4 = 261.6 Hz or A4 = 440 Hz).
Proceed through all twelve notes of the chromatic scale, recording each frequency.
Create a table with columns: note number (1-12), note name, measured frequency (Hz).
Calculate the ratio between each consecutive pair of frequencies: f(n+1)/f(n).
Verify that this ratio is approximately constant and close to 1.0595 (the twelfth root of 2).
Now load the 'Octaves' recording from the sound library, which contains the same note played across several octaves.
Measure the frequency of the same note in each successive octave.
Verify the 2:1 ratio: the frequency should exactly double from one octave to the next.
Plot a graph of frequency versus note number for the chromatic scale. The curve should be exponential (geometric progression), not linear.
Plot log(frequency) versus note number. This graph should be a straight line, confirming the geometric progression.

Expected results:

The twelve notes of the chromatic scale should show frequencies forming a clear geometric progression. Starting from A4 = 440 Hz, the sequence is approximately: 440, 466, 494, 523, 554, 587, 622, 659, 698, 740, 784, 831 Hz for the twelve semitones, with each ratio ≈ 1.0595. The octave recordings should show exact doubling: A3 ≈ 220 Hz, A4 ≈ 440 Hz, A5 ≈ 880 Hz. Measurement precision with FizziQ is typically ±1-2 Hz, sufficient to verify the geometric pattern. The plot of frequency versus note number shows the characteristic exponential curve, while log(frequency) versus note number yields a straight line with slope log(¹²√2).

Scientific questions:

- Why do musical frequencies follow a geometric progression rather than an arithmetic one?
- What does it mean physically for two notes to be an octave apart?
- Why did musicians adopt equal temperament instead of the older Pythagorean tuning?
- What mathematical operation connects the frequency of a note to the frequency of the note n semitones higher?
- Why do some frequency ratios (like 3:2 for a perfect fifth) sound consonant while others sound dissonant?
- How many semitones are in a perfect fifth? What is the corresponding frequency ratio in equal temperament?

Scientific explanations:

Western music divides the octave (the interval between two notes of the same name whose frequencies are in a 2:1 ratio) into twelve equal semitones, forming the chromatic scale. The progression of frequencies follows a geometric law: each semitone has a frequency multiplied by the twelfth root of 2 (approximately 1.059) compared to the previous one.

Thus, if f₀ is the frequency of a note, the frequency of the note located n semitones higher is: f = f₀ × (²√12)ⁿ. This organization, called equal temperament, was gradually adopted from the 18th century to allow modulation between different tones.

Previously, other tuning systems based on whole number ratios (such as the Pythagorean system) favored the purity of certain intervals over others. The international reference note is A3 (440 Hz), fixed by convention in 1939.

From this reference, all other frequencies can be calculated. For example, C3 (C3) has a frequency of approximately 261.63 Hz, and C4 is exactly twice as high (523.25 Hz).

The main musical intervals correspond to simple frequency ratios: the octave (2:1), the fifth (3:2), the fourth (4:3), etc. This relationship between simplicity of relationships and consonance (perceived harmony) has been described since Antiquity and is explained by the theory of beats and the coincidence of harmonics.

The ability of the FizziQ frequency meter to precisely measure these frequencies makes it possible to empirically explore these mathematical foundations of musical harmony.

Extension activities:

Frequently asked questions:

Q: The measured frequencies do not exactly match the theoretical values. Is this normal?
R: Yes, measurement precision of ±1-2 Hz is typical. Also, some instruments (especially pianos) use slight deviations from equal temperament called stretch tuning. The overall geometric pattern should still be clear.

Q: Why is the twelfth root of 2 the magic number?
R: Because the octave (a 2:1 frequency ratio) is divided into 12 equal semitones. If each semitone multiplies the frequency by a constant factor r, then r¹² = 2, giving r = ¹²√2.

Q: The frequency meter shows fluctuating values for some notes. How do I get a stable reading?
R: Ensure the recording is playing at a reasonable volume and that background noise is minimal. For sustained notes, the frequency should stabilize within 1-2 seconds. Take the reading during the stable portion.

Q: Can I use a real instrument instead of the recordings?
R: Absolutely. A well-tuned piano, keyboard, or xylophone will give excellent results. Ensure each note is played clearly and sustained long enough for the frequency meter to lock on.