# A4 Sine Waves and Additive Synthesis

Sine Wave Math

1. Sine waves are important to electronic music, and music in general because: Sine waves, which describes displacement from equilibrium with a single frequency, can be used to describe any disturbances in the air (“sound waves”), or vibrations. Strings, air inside wind instruments, or the vocal folds in the human throat – all these sources of musical sound can be made to vibrate in sinusoidal motion with a single frequency. More complicated sources can be described using the superposition of several sine waves in the harmonic series of different frequencies. Pitch and timbre are decided by the strongest sine wave and the relative frequencies of all waves that are added together in the mix, respectively.
2. y(x) = A*sin(2πft + ø) is used to calculate the displacement from equilibrium of a wave particle. A is the maximum displacement of the wave particle from the equilibrium (achieved when |sin(2πft + ø)| = 1), f is the frequency of the wave, which is the number of cycles per unit time, t is time/seconds, so ft is the total number of cycles travelled by the wave in time t. ø is the angle that the radius makes with the horizontal axis in the unit circle at t = 0. Assuming ø = 0, the cycle starts at an angle of 0, so the initial displacement is 0. This is not really a struggle for me as I learnt about this in high school physics.
3. We have a unit circle with radius = 1. The right triangle is formed by the angle of the radius moving counterclockwise around the unit circle. Sine is opposite over hypotenuse, in this case the hypotenuse is the radius = 1. Thus sine is simply the length of the opposite side. This side varies in length as the angular displacement increases. For example, at 45º, the length of the opposite side, thus sin of 45º, is .707. At 90º the opposite side of the right triangle “disappears” and becomes equal to the length of the unit circle. Thus the sin of 90º is 1. The Max patch demonstrates how sine wave is formed as angle of the radius moves counterclockwise around the unit circle.  4. This patch demonstrates the meaning of wave frequency. With the frequency of 1 Hz, wave particle travels a complete cycle in 1 second. When the frequency is changed to 2 Hz, wave particle travels a complete cycle in 1/2 = .5 second, meaning it travels 2 complete cycles in 1 second.
5. An example of equal musical interval would be octaves. Different frequencies that demonstrate the same note but one octave apart are in ratio of 2:1. (The same note in one octave higher are double the same note one octave lower). For example, G2’s frequency is at 100 Hz, G3’s frequency is double of G2 at 200Hz, a difference of 100 Hz from G2’s frequency, G4’s frequency is double of G3 at 400Hz, instead of 100 Hz from G3’s frequency which is at 300 Hz. Thus, equal music intervals do not mean equal differences, but equal ratios of frequencies.
6. Any two of the lower few frequencies in a harmonic series sound good together.  For example, harmonics 2 and 3 form a “perfect fifth.”  Harmonics 3 and 4 form a “perfect fourth”; 4 and 5 form a “major third,” etc. Any repeating wave shape of frequency f can be created by adding together sine waves of frequency f, 2*f, 3*f, 4*f, …  That means we can create any wave shape by using only the sine waves in the harmonic series for that pitch. So when we “mess around with ” the relationship of the partials by adding them to the sine wave with fundamental frequency, we can create harmonies and different timbres and amplitudes.
7. Sine wave possesses only a fundamental frequency, with no additional harmonic content. Triangle contains partials at the odd frequencies, with amplitudes at the inverse of the square of the harmonic number. Square wave contains odd partials with amplitudes at the inverse of the partial number. Sawtooth contains all harmonics, with amplitudes as the inverse of the harmonic number. The upper partials have increasingly greater than the fundamental wave while their amplitudes decrease due to them being the inverse of the partials.