Granular synthesis gives us a completely different conceptual approach to sound and musical form. Instead of thinking about a sound as a continuous object, we can think of it as something that can be **broken apart into extremely small pieces**. These small pieces are called **grains**. A grain may be only a few milliseconds long. By manipulating thousands of these tiny sound events, we can transform the original sound into something that may no longer resemble its source. The basic idea can be represented as: $ x(t)\rightarrow\{g_1(t),g_2(t),\ldots,g_N(t)\}\rightarrow y(t) $ Here, $x(t)$ is the original sound, $g_n(t)$ represents an individual grain, and $y(t)$ is the reconstructed granular sound. The reconstructed granular sound can therefore be thought of as: $ y(t)=\sum_{n=1}^{N}a_n\,g_n(t-\tau_n) $ where: - $g_n(t)$ = waveform of grain $n$ - $\tau_n$ = **time location** (onset) of grain $n$ - $a_n$ = amplitude of grain $n$ - $N$ = number of grains Each grain can have its own properties, allowing us to manipulate the microscopic structure of the sound. ### Grain Duration The duration of each grain can be represented as: $ d_g=t_{\text{end}}-t_{\text{start}} $ where $d_g$ is the grain duration. Short grains produce a more fragmented or granular texture: $ d_g\downarrow \quad\Rightarrow\quad \text{greater microscopic fragmentation} $ Longer grains preserve more of the original sound: $ d_g\uparrow \quad\Rightarrow\quad \text{greater continuity} $ This creates a continuum between: $ \text{continuous sound}\longleftrightarrow\text{fragmented sound} $ ### Grain Density The number of grains occurring per unit of time can be described as grain density: $ D=\frac{N}{T} $ where: - $D$ = grain density - $N$ = number of grains - $T$ = duration of the observation or reconstruction interval Increasing the density produces a denser texture: $ D\uparrow \quad\Rightarrow\quad \text{denser sound texture} $ while decreasing density can expose individual grains: $ D\downarrow \quad\Rightarrow\quad \text{sparser texture} $ Density can therefore become a compositional parameter. ### Grain Position A grain can be taken from a particular location within the original sound: $ p_n\in[0,T_x] $ where $p_n$ represents the **position** of grain $n$ within the source sound and $T_x$ is the duration of the source. The grain can then be placed at a different location in the reconstructed sound, represented by $\tau_n$. If grains are played in their original order: $ p_1<p_2<p_3<\cdots<p_N $ the listener may perceive a relatively recognizable version of the original sound. If the positions are rearranged: $ p_1,p_3,p_2,p_7,p_4,\ldots $ the temporal structure of the sound begins to fragment. ### Grain Pitch The pitch of an individual grain can also be shifted. A simple relationship between playback speed and pitch is: $ r=\frac{f_{\text{new}}}{f_{\text{original}}} $ where $r$ is the playback-rate ratio. For example: $ r=2 $ raises the frequency by an octave, while: $ r=\frac{1}{2} $ lowers it by an octave. This means that individual grains can be transposed independently of their original position. ### Grain Envelope and Windowing Each grain is usually shaped by an amplitude envelope: $ g_n(t)=w_n(t)x_n(t) $ where $w_n(t)$ controls the amplitude of the grain. A simple envelope might rise and fall: $ w(t):0\rightarrow1\rightarrow0 $ One commonly used envelope is the **Hann window**: $ w(t)=\frac{1}{2}\left(1-\cos\left(\frac{2\pi t}{T_g}\right)\right) $ where $T_g$ is the duration of the grain. The Hann window begins at zero, gradually rises to 1 in the middle, and returns to zero at the end: $ 0\rightarrow1\rightarrow0 $ The purpose of this envelope is to make the grain begin and end smoothly. Without an envelope, abruptly cutting a piece out of a sound can create sharp discontinuities, which may produce unwanted clicks or other artifacts. You can think of the window as giving the grain a **shape**: $ \boxed{ \text{grain}= \text{sound fragment}\times\text{envelope} } $ Different envelopes produce different grain shapes. For example, a rectangular window keeps the grain at full amplitude throughout its duration, while a Hann window gradually fades the grain in and out. The choice of envelope can therefore affect the character of the resulting granular sound. ### Randomness Granular synthesis can also introduce randomness into the placement and properties of grains. For example, grain position could be: $ p_n=p_n^*+\epsilon_n $ where $p_n^*$ is the intended position and $\epsilon_n$ is a random deviation. Similarly, grain timing could be expressed as: $ \tau_n=n\Delta t+\epsilon_n $ where $\Delta t$ is the regular spacing between grains. As the amount of randomness increases: $ |\epsilon_n|\uparrow \quad\Rightarrow\quad \text{greater temporal unpredictability} $ This creates a continuum between: $ \text{ordered}\longleftrightarrow\text{random} $ Randomness can therefore become another formal parameter rather than simply an effect. ### Direction Grains do not necessarily have to move through the source sound from beginning to end. Forward playback can be represented as: $ p_{n+1}>p_n $ while reverse movement can be represented as: $ p_{n+1}<p_n $ The source material can therefore be experienced: $ \text{forward}\rightarrow\text{backward} $ or through more complex patterns: $ \text{forward}\rightarrow\text{reverse}\rightarrow\text{forward} $ This creates temporal form inside the sound itself.