Syllabus
1 – Introduction to the Course Unit (1 Lesson)
Presentation of the course unit.
Explanation of the course structure and assessment methods.
Basic concepts of music and mathematics.
Assessment of students' prior knowledge.
2 – Characteristics of Musical Sound (1 Lesson)
The four characteristics of musical sound: duration, pitch, intensity, and timbre.
Mathematical characterisation of duration, pitch, and intensity.
Musical notation and its relation to mathematics.
ISO16 standard for the frequency of the A4 note (440 Hz).
Frequency table for equal temperament tuning.
3 – Construction of Musical Scales and Modular Arithmetic (2 Lessons)
Construction of scales using rational fractions and irrational ratios.
The Pythagorean scale and the equal temperament scale with 53 tones.
The tuning of Mozart's violin and its relationship with the 53-tone scale.
Matrices of major and minor scales (natural, harmonic, melodic).
Application of modular arithmetic to minor scales.
Analysis of Størmer’s theorem.
Use of superparticular fractions of natural numbers.
4 – Scales and Superparticular Fractions (1 Lesson)
Construction of musical scales in the Renaissance according to Zarlino.
Use of superparticular fractions in scale construction.
5 – The Wave Equation, Fourier Series, and Harmonics (2 Lessons)
Introduction to the wave equation.
Methods of solution: separation of variables and Fourier method.
Relationship between the wave equation and musical instruments.
Conclusion of the study of the wave equation.
Analysis of real-world examples and study of harmonics through Fourier series.
6 – Parameters of Musical Sound and Mathematical Functions (1 Lesson)
Relationship between mathematical functions and sound parameters.
Creation and auditory perception of audible functions.
Study of the syntonic and Pythagorean commas.
Analysis of sound spectra of real instruments.
7 – Musical Temperaments (1 Lesson)
Study of regular and irregular temperaments.
Meantone temperaments based on 1/4-comma and 1/6-comma systems.
Introduction to Eitz notation.
8 – Digital Music and Synthesis I (Professor Gonçalo Figueira, 2 Lessons)
Sound: analogue versus digital.
Concepts of sampling and the Nyquist theorem.
Introduction to audio and MIDI files.