Gliese 710 3D Scalar Field Oscillator

Mathematical Foundations of a 3D Function Oscillator Abstract Traditional oscillators generate sound by evaluating a one-dimensional periodic function over phase: $ s = f(\phi) $ where $\phi$ is the oscillator phase. A 3D Function Oscillator generalizes this concept by transforming phase into motion through a three-dimensional mathematical space. Rather than directly evaluating a waveform from phase, the oscillator computes a coordinate $ \mathbf{p} = (x,y,z) $ and evaluates a scalar field $ F : \mathbb{R}^3 \rightarrow \mathbb{R}. $ The resulting audio signal becomes $ s(t)=F(x(t),y(t),z(t)). $ This transforms waveform generation into the problem of navigating and sampling a multidimensional mathematical field. --- 1. Oscillator Phase Every oscillator begins with a phase accumulator. The normalized phase evolves according to $ \phi[n] ======= \phi[n-1] + \frac{f0}{Fs}, $ where $f0$ is oscillator frequency $Fs$ is sample rate The phase wraps modulo unity: $ \phi[n] \leftarrow \phi[n] \bmod 1. $ Angular phase is defined as $ \theta = 2\pi\phi. $ For a 440 Hz oscillator operating at a sample rate of 48 kHz, $ \Delta\phi ========== \frac{440}{48000} 9.1667\times10^{-3}. $ --- 2. From Phase to 3D Space Instead of $ s(\phi)=f(\phi), $ we define a trajectory through 3D space: $ \mathbf{p}(\theta) ================== \begin{bmatrix} x(\theta)\ y(\theta)\ z(\theta) \end{bmatrix}. $ A simple circular path is $ x=r\cos\theta, $ $ y=r\sin\theta, $ $ z=d\sin(\lambda\theta), $ where $r$ = radius $d$ = depth $\lambda$ = z-axis ratio Compactly, $ \mathbf{p}(\theta) ================== \begin{bmatrix} r\cos\theta\ r\sin\theta\ d\sin(\lambda\theta) \end{bmatrix}. $ --- 3. Scalar Field Evaluation A scalar field is defined as $ F:\mathbb{R}^{3}\rightarrow\mathbb{R}. $ An example field is $ F(x,y,z) ======== \sin(4x) + \cos(3y) + \sin(5z). $ The oscillator output becomes $ s = F(x,y,z). $ To maintain bounded output we apply saturation: $ s=\tanh(F(x,y,z)). $ --- 4. General Oscillator Model The complete oscillator may be expressed as $ s(t) ==== F(x(t),y(t),z(t)). $ or $ s(t)=F(\mathbf{p}(t)). $ This separates the oscillator into two independent systems: 1. Path Generator 2. Field Evaluator --- 5. Path Generators Circular Path $ x=r\cos\theta $ $ y=r\sin\theta $ $ z=0 $ Helical Path $ x=r\cos\theta $ $ y=r\sin\theta $ $ z=d\phi $ Lissajous Path $ x=\sin(a\theta+px) $ $ y=\sin(b\theta+py) $ $ z=\sin(c\theta+pz) $ Vector form: $ \mathbf{p}(\theta) ================== \begin{bmatrix} \sin(a\theta+px)\ \sin(b\theta+py)\ \sin(c\theta+pz) \end{bmatrix}. $ Parameters: $a,b,c$ = frequency ratios $px,py,pz$ = phase offsets Integer ratios create harmonic trajectories. Irrational ratios create evolving and inharmonic trajectories. --- 6. Oscillator Frequency vs Field Frequency The oscillator pitch is determined by $ \theta(t)=2\pi f0 t. $ The field may contain independent spatial frequencies: $ F(x,y,z) ======== \sin(kxx) + \cos(kyy) + \sin(kzz). $ The parameters $ kx,;ky,;kz $ control harmonic density rather than musical pitch. Thus: $f0$ controls pitch $kx,ky,kz$ control timbre --- 7. Distance Fields Distance from the origin: $ d = \sqrt{x^2+y^2+z^2}. $ Distance-based field: $ F(x,y,z) ======== \sin(kd). $ Distance from an arbitrary center: $ d = \sqrt{ (x-cx)^2 + (y-cy)^2 + (z-cz)^2 }. $ Distance fields naturally produce spherical resonances and bell-like spectra. --- 8. Dot Product Projection The dot product is $ \mathbf{a}\cdot\mathbf{b} ========================= axbx + ayby + azbz. $ Projection of a point onto a direction vector: $ v = \mathbf{p}\cdot\mathbf{d}. $ Audio output: $ s = \sin(kv). $ Rotating $\mathbf{d}$ changes the spectral perspective through the field. --- 9. Coordinate Warping Warping transforms coordinates before field evaluation. $ \mathbf{p}' =========== W(\mathbf{p}). $ Example: $ x' == x+A\sin(3y+z) $ $ y' == y+A\cos(4z+x) $ $ z' == z+A\sin(5x+y) $ The oscillator becomes $ s = F(\mathbf{p}'). $ Coordinate warping is analogous to domain warping in procedural graphics. --- 10. Function Composition Define $ A=\sin(3x) $ $ B=\cos(4y) $ Then $ F(x,y,z) ======== \sin(A+B+z). $ Expanded: $ F(x,y,z) ======== \sin \left( \sin(3x) + \cos(4y) + z \right). $ This introduces FM-like and nonlinear behavior. --- 11. Rotation Operators Rotation about the z-axis: $ Rz(\alpha) =========== \begin{bmatrix} \cos\alpha & -\sin\alpha & 0\ \sin\alpha & \cos\alpha & 0\ 0 & 0 & 1 \end{bmatrix}. $ Transformation: $ \mathbf{p}' =========== Rz(\alpha)\mathbf{p}. $ Rotation changes traversal orientation without altering field topology. --- 12. Vector Fields A vector field is defined as $ \mathbf{V} : \mathbb{R}^3 \rightarrow \mathbb{R}^3. $ Example: $ \mathbf{V}(x,y,z) ================= \begin{bmatrix} \sin y\ \cos z\ \sin x \end{bmatrix}. $ Particle evolution: $ \mathbf{p}{n+1} ================ \mathbf{p}n + h\mathbf{V}(\mathbf{p}n). $ T…