Subtractive synthesis begins with a relatively complex sound and shapes it by **removing or attenuating frequencies** through filtering. While filters are often introduced as technical processing tools, they can also be understood as **compositional gestures**. A change in the spectrum can create a sense of movement, contrast, tension, and release. A **low-pass filter** allows lower frequencies to pass while reducing higher frequencies, often making a sound darker or softer. A **high-pass filter** removes lower frequencies and emphasizes higher-frequency content, often making the sound thinner or brighter. A **band-pass filter** isolates a region of the spectrum, focusing attention on a particular frequency range. The **cutoff frequency** determines the frequency around which the filtering occurs, while **resonance** emphasizes frequencies near the cutoff. Moving the cutoff over time creates a **filter sweep**, producing a continuously changing spectrum. These changes can be heard as gestures rather than simply technical adjustments. For example, a movement from **bright → dark → bright** can function compositionally like **tension → release → tension**. The formal movement comes from changes in the spectrum rather than changes in pitch or harmony. A spectrum can remain **static**, with the filter settings staying relatively constant, or **evolve over time**, creating a sense of development. A single sound can therefore become the basis for an entire formal structure through filtering alone. ## Mathematical Representation A complex sound can be represented as a combination of sinusoidal frequency components: $ x(t)=\sum_{k=1}^{N} A_k\cos(2\pi f_k t+\phi_k) $ In subtractive synthesis, filtering changes the amplitude of these frequency components: $ Y(f)=H(f)X(f) $ where $X(f)$ is the input spectrum, $H(f)$ is the filter response, and $Y(f)$ is the resulting spectrum. ### Low-Pass Filter $ H_{LP}(f)= \begin{cases} 1, & |f|\leq f_c\\ 0, & |f|>f_c \end{cases} $ ### High-Pass Filter $ H_{HP}(f)= \begin{cases} 0, & |f|<f_c\\ 1, & |f|\geq f_c \end{cases} $ ### Band-Pass Filter $ H_{BP}(f)= \begin{cases} 1, & f_1\leq |f|\leq f_2\\ 0, & \text{otherwise} \end{cases} $ The bandwidth is: $ BW=f_2-f_1 $