# amplitudes
defines how energy is distributed across partials in the spectrum. each form produces a distinct spectral shape - flat, tilted, peaked, or chaotic - suitable for everything from simple tones to turbulent noise-like bodies.

## overview
* `0` flat
  * description: equal amplitude for all partials
  * real-world analogy: pulse trains, square waves
  * notes: reference baseline
* `1` polynomial tilt
  * description: power-law roll-off, with optional skew across spectrum
  * real-world analogy: string harmonics (1/n), pink noise
  * notes: `a` controls decay, `b` controls skew
* `2` gaussian peak
  * description: single formant-like peak centered at arbitrary partial
  * real-world analogy: vowels, woodwinds, formant synthesis
  * notes: peak center + width
* `3` beta shape
  * description: skewed bell, u-shapes, edge emphasis
  * real-world analogy: tape noise, bass thump + hiss, asymmetrical energy
  * notes: high flexibility from beta(α, β)
* `4` shelf (sigmoid)
  * description: smooth step-like drop or rise in partials
  * real-world analogy: speaker roll-off, muted trumpet
  * notes: `a` = cutoff, `b` = steepness
* `5` logistic chaos
  * description: chaotic but continuous grain structure, with scale control
  * real-world analogy: breathy sources, granular clouds
  * notes: logistic map smoothed by window
* `6` bessel (modal)
  * description: oscillatory energy with many ripples
  * real-world analogy: bells, gamelan, metal plates
  * notes: based on bessel functions

## parameter behavior summary
* `0`
  * param `a`: (unused)
  * param `b`: (unused)
* `1`
  * param `a`: brightness (p)
  * param `b`: tilt/skew across index
* `2`
  * param `a`: peak center
  * param `b`: peak width
* `3`
  * param `a`: alpha (shape)
  * param `b`: beta (shape)
* `4`
  * param `a`: cutoff position
  * param `b`: step sharpness
* `5`
  * param `a`: chaos strength (r)
  * param `b`: smoothing window (grain size)
* `6`
  * param `a`: modal spread (x)
  * param `b`: decay factor
all parameters are bounded: `a, b ∈ [0, 1]`.

## why these were chosen
these seven families were selected to span the perceptual space of spectral shapes in a musically relevant and synthesis-efficient way:
* simple roll-offs and brightness control → polynomial tilt
* localized resonances → gaussian peak
* asymmetric or complex spectral bodies → beta shape
* spectral edges or low-pass rolloffs → shelf
* irregular dynamic motion → logistic chaos
* modal-like structured ripples → bessel
* reference → flat (no spectral shaping)
each offers unique control and timbral identity, and cannot be easily mimicked by the others-even when arbitrarily tuned.

## what is not included
* dual-formant shapes  - currently not implemented. could be added via two combined gaussians.
* exponential decay    - steep exponential drop-offs not included explicitly; tilt with `p > 5` approximates well.
* triangle / rise-fall - approximated by beta(α≈β<1); not needed as separate family.

## conclusion
with the current 7 implemented families, the system offers:
* a minimal yet rich coverage of natural and artificial spectral energy shapes
* two-dimensional control for each type
* no binary steps or discontinuities, making it ideal for purely additive synthesis
the framework remains extensible (e.g. to multi-formant distributions), but in its current form, already provides a powerful toolset for spectral instrument design.