Mathematical Foundations

The mathematics underlying Quiver’s design and DSP algorithms.

Category Theory

Quivers

A quiver \( Q = (V, E, s, t) \) consists of:

• \( V \): Set of vertices (objects)
• \( E \): Set of edges (arrows/morphisms)
• \( s: E \to V \): Source function
• \( t: E \to V \): Target function

In Quiver:

• Vertices = Modules
• Edges = Patch cables
• Source/Target = Output/Input ports

The Free Category

Given a quiver \( Q \), the free category \( \text{Path}(Q) \) has:

• Objects: Same as \( Q \)’s vertices
• Morphisms: Paths (sequences of composable arrows)
• Composition: Path concatenation

This is what patch.compile() computes.

Arrow Laws

For arrows \( f: A \to B \), \( g: B \to C \), \( h: C \to D \):

Identity: \[ \text{id}_B \circ f = f \circ \text{id}_A = f \]

Associativity: \[ (h \circ g) \circ f = h \circ (g \circ f) \]

First/Second: \[ \text{first}(f) = f \times \text{id} \] \[ \text{second}(f) = \text{id} \times f \]

Digital Signal Processing

Sampling Theory

Nyquist-Shannon Theorem: A signal can be perfectly reconstructed if sampled at rate \( f_s > 2f_{max} \).

At 44.1 kHz: \( f_{max} = 22.05 \) kHz

Z-Transform

The z-transform converts discrete signals to the z-domain:

\[ X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \]

Unit delay: \( z^{-1} \) (one sample delay)

Transfer Functions

Lowpass filter (1-pole): \[ H(z) = \frac{1-p}{1-pz^{-1}} \]

Where \( p = e^{-2\pi f_c / f_s} \)

State-Variable Filter: \[ \begin{aligned} \text{LP} &= \text{LP}_{n-1} + f \cdot \text{BP}_{n-1} \\ \text{HP} &= \text{input} - \text{LP} - q \cdot \text{BP}_{n-1} \\ \text{BP} &= f \cdot \text{HP} + \text{BP}_{n-1} \end{aligned} \]

Waveform Mathematics

Sine Wave

\[ x(t) = A \sin(2\pi f t + \phi) \]

Sawtooth (Band-Limited)

Fourier series: \[ x(t) = \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \sin(2\pi k f t) \]

Square Wave

\[ x(t) = \frac{4}{\pi} \sum_{k=1,3,5,…}^{\infty} \frac{1}{k} \sin(2\pi k f t) \]

Only odd harmonics!

Triangle Wave

\[ x(t) = \frac{8}{\pi^2} \sum_{k=1,3,5,…}^{\infty} \frac{(-1)^{(k-1)/2}}{k^2} \sin(2\pi k f t) \]

Envelope Mathematics

Exponential Segments

Attack (charging capacitor): \[ v(t) = V_{max} (1 - e^{-t/\tau}) \]

Decay/Release (discharging): \[ v(t) = V_{start} \cdot e^{-t/\tau} \]

Time constant \( \tau \): time to reach \( 1 - 1/e \approx 63.2% \)

RC Time Constant

\[ \tau = RC \]

For envelope times: \( \tau = \text{time} / \ln(1000) \approx \text{time} / 6.9 \)

FM Synthesis

Basic FM Equation

\[ y(t) = A \sin(2\pi f_c t + I \sin(2\pi f_m t)) \]

• \( f_c \): Carrier frequency
• \( f_m \): Modulator frequency
• \( I \): Modulation index

Sidebands

FM produces sidebands at: \[ f_c \pm n \cdot f_m \quad (n = 1, 2, 3, …) \]

Number of significant sidebands ≈ \( I + 1 \)

Bessel Functions

Amplitude of each sideband given by Bessel functions: \[ A_n = J_n(I) \]

Filter Response

Pole-Zero Form

\[ H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} \]

Cutoff Frequency

For bilinear transform: \[ \omega_d = \frac{2}{T} \tan\left(\frac{\omega_a T}{2}\right) \]

Resonance (Q)

\[ Q = \frac{f_0}{\Delta f} \]

Where \( \Delta f \) is bandwidth at -3dB.

High Q → narrow peak → self-oscillation

Analog Modeling

Thermal Noise

\[ V_n = \sqrt{4kTRB} \]

• \( k \): Boltzmann constant
• \( T \): Temperature (K)
• \( R \): Resistance
• \( B \): Bandwidth

Saturation Functions

Tanh (soft): \[ y = \tanh(x \cdot \text{drive}) \]

Polynomial (3rd order): \[ y = x - \frac{x^3}{3} \]

Asymmetric: \[ y = \tanh(a \cdot x^+) - \tanh(b \cdot x^-) \]

V/Oct System

Pitch to Frequency

\[ f = f_0 \cdot 2^V \]

\( f_0 = 261.63 \) Hz (C4) at 0V

Frequency to Pitch

\[ V = \log_2\left(\frac{f}{f_0}\right) \]

Semitone

\[ \Delta V = \frac{1}{12} \text{ V} \approx 83.33 \text{ mV} \]

Cent

\[ \Delta V = \frac{1}{1200} \text{ V} \approx 0.833 \text{ mV} \]

SIMD Mathematics

Vectorized Operations

For 4-wide SIMD: \[ [a_1, a_2, a_3, a_4] + [b_1, b_2, b_3, b_4] = [a_1+b_1, a_2+b_2, a_3+b_3, a_4+b_4] \]

Single instruction, multiple data.

Block Processing

Process \( N \) samples per function call:

• Reduces function call overhead by factor of \( N \)
• Enables vectorization
• Improves cache locality

References

• Smith, J.O. Mathematics of the Discrete Fourier Transform
• Välimäki, V. Discrete-Time Synthesis of the Sawtooth Waveform
• Mac Lane, S. Categories for the Working Mathematician
• Chowning, J. The Synthesis of Complex Audio Spectra by Means of FM