# Basic Subtractive Synthesis
Subtractive synthesis is the foundation of analog synthesizers. Start with a harmonically rich waveform, then sculpt it by filtering away frequencies.
```mermaid
flowchart LR
OSC[Oscillator<br/>Rich harmonics] --> FILTER[Filter<br/>Remove harmonics]
FILTER --> AMP[Amplifier<br/>Shape volume]
AMP --> OUT[Output]
style OSC fill:#4a9eff,color:#fff
style FILTER fill:#f9a826,color:#000
style AMP fill:#50c878,color:#fff
```
## The Physics of Waveforms
Different waveforms have different harmonic content:
| **Sine** | Fundamental only | Pure, flute-like |
| **Triangle** | Odd harmonics (weak) | Soft, clarinet-like |
| **Sawtooth** | All harmonics | Bright, brassy |
| **Square** | Odd harmonics (strong) | Hollow, woody |
The mathematical representation:
**Sawtooth wave:**
$$x(t) = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \sin(2\pi k f t)$$
This infinite sum of harmonics is what gives the sawtooth its brightness.
## Building the Patch
```rust,ignore
{{#include ../../../examples/tutorial_subtractive.rs}}
```
## Understanding the Filter
The state-variable filter (SVF) in Quiver simultaneously outputs:
- **Lowpass** — removes high frequencies
- **Bandpass** — isolates a frequency band
- **Highpass** — removes low frequencies
- **Notch** — removes a specific band
```mermaid
graph TB
subgraph "SVF Outputs"
IN[Audio In] --> SVF[State Variable<br/>Filter]
SVF --> LP[Lowpass]
SVF --> BP[Bandpass]
SVF --> HP[Highpass]
SVF --> NOTCH[Notch]
end
```
### Filter Response
The lowpass filter attenuates frequencies above the cutoff:
$$H(f) = \frac{1}{\sqrt{1 + (f/f_c)^{2n}}}$$
Where $f_c$ is cutoff frequency and $n$ is filter order.
Quiver's SVF is 12dB/octave (2-pole), meaning frequencies one octave above cutoff are reduced by 12dB.
### Resonance
Resonance (Q) boosts frequencies near cutoff:
```mermaid
graph LR
subgraph "Resonance Effect"
FLAT[Low Q<br/>Flat response]
PEAK[High Q<br/>Resonant peak]
end
```
At maximum resonance, the filter self-oscillates, becoming a sine wave generator.
## Experimenting
1. **Try different waveforms**: Change `"saw"` to `"sqr"` or `"tri"`
2. **Adjust cutoff**: Lower values = darker, muffled sound
3. **Add resonance**: Creates a vowel-like quality
4. **Mix waveforms**: Combine `saw` and `sqr` for thickness
## Classic Tones
| Moog Bass | Saw | LP, low cutoff | Fat, warm |
| Oberheim Pad | Saw + Saw (detuned) | LP, med cutoff | Lush, wide |
| TB-303 Acid | Saw | LP, high resonance | Squelchy |
| CS-80 Brass | Saw | LP, following envelope | Brassy attack |
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Next: [Envelope Shaping](./envelope-shaping.md)