## 15.François Gaspard, Boris Wilmot & Ohme (BE)

### Sine

##### It all begins with a sine

The word sine was borrowed from the Latin noun sinus which means “curve, fold, or hollow.”
The sine function pervades many branches of mathematics as well as a large number of applications.

Take the example of a bicycle wheel that turns in a circular, counterclockwise motion on its axis.
The distance around the rim of the wheel is equal to the measurement of its central angle. In mathematics, angles are most often measured in radians. The radian measure of angle $$\alpha$$ is the ratio of length $$L$$ of the arc the angle subtends, divided by the radius of the circle, $$r$$.

When the valve stem of the wheel makes a full circle to its original position, the distance travelled is equal to 360° or $$2\pi$$ radians. Take angle $$\alpha$$ on horizontal axis $$t$$.

The notation for the height of the mobile valve stem $$y$$ on vertical $$y$$-axis, above or below the horizontal diameter, is equal to sine of $$t$$.

##### Music scale structure and Pythagoras

Pythagoras believed that everything in the world could be explained by numbers, in particular with small natural numbers. He discovered interval pitch ratios and established four main musical intervals: unison (1:1), octave (2:1), perfect fifth (3:2), perfect fourth (4:3). He then combined them.

Our 7-note modern-day musical scale (Do, Re, Mi, Fa, Sol, La, Si) was born.

##### Fourier series

In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. So any sufficiently regular periodic function $$u(t)$$ of period $$T$$ can be represented as a Fourier series of harmonics:

We assume that $$u(t)$$ is real, so that $$a_k$$ and $$b_k$$ are also real for any $$k$$.

The first harmonic is called the fundamental tone; let us denote its frequency by $$f_1$$. The other components are called overtones, and have frequencies $$\frac{n}{T} = n \times f_1$$, for any integer $$n$$.
The frequency is usually measured in Hertz: 1 Hz = 1 cycle/second.

##### Envelopes (dynamic, filters, pitch) and mathematical functions

In sound and music, an envelope describes how a sound changes over time. It may relate to elements such as amplitude (volume), filters (frequencies) or pitch.
Envelope generators, which allow users to control the different stages of a sound, are common features of synthesizers, samplers, and other electronic musical instruments. The most common kind of envelope generator has four stages: attack, decay, sustain, and release (ADSR):

• Attack is the time taken for initial run-up of level from nil to peak, beginning when the key is pressed.
• Decay is the time taken for the subsequent run down from the attack level to the designated sustain level.
• Sustain is the level during the main sequence of the sound’s duration, until the key is released.
• Release is the time taken for the level to decay from the sustain level to zero after the key is released.

While attack, decay, and release refer to time, sustain refers to level.

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