The range
Study of the relationship between musical notes and frequencies
Author:
Title 4
Learning objectives :
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.
Concepts covered
Chromatic range; Musical frequencies; Geometric progression; Octave and interval; Harmonic relationships
What students will do :
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.
What is required :
Smartphone with the FizziQ application; 'Range' and 'Octaves' recordings from the sound library; FizziQ experience notebook; Calculator to analyze frequency ratios
Scientific background :
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.