# 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
The patch itself is three modules plus a fixed `Offset` that parks the filter cutoff at a musical spot:
<div class="quiver-explorable" data-viz="patchgraph">
<script type="application/json">
{
"modules": [
{"id": "vco", "label": "VCO", "x": 0, "y": 0,
"outputs": [{"name": "saw", "kind": "audio"}]},
{"id": "cutoff", "label": "CUTOFF (Offset)", "x": 0, "y": 2.2,
"outputs": [{"name": "out", "kind": "cv"}]},
{"id": "vcf", "label": "VCF (SVF)", "x": 1, "y": 0,
"inputs": [{"name": "in", "kind": "audio"}, {"name": "cutoff", "kind": "cv"}],
"outputs": [{"name": "lp", "kind": "audio"}]},
{"id": "output", "label": "OUTPUT", "x": 2, "y": 0,
"inputs": [{"name": "left", "kind": "audio"}]}
],
"cables": [
{"from": "vco.saw", "to": "vcf.in", "kind": "audio"},
{"from": "cutoff.out", "to": "vcf.cutoff", "kind": "cv"},
{"from": "vcf.lp", "to": "output.left", "kind": "audio"}
],
"caption": "tutorial_subtractive: a fixed Offset sets the SVF cutoff; the lowpass output goes straight out."
}
</script>
</div>
*Hear the filter carve these exact harmonics away in [Sculpting the Spectrum](../explorables/filters.md).*
```rust,ignore
{{#include ../../../examples/tutorial_subtractive.rs}}
```
### Listen to It
Run `cargo run --example tutorial_subtractive` and the example writes
`target/tutorial_subtractive.wav`—open it in any audio player to hear the
filter shape the raw sawtooth.
## 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 |
---
Next: [Envelope Shaping](./envelope-shaping.md)