2025-05-15

instruments

models of sound-generating structures

fundamental sounding elements

the most fundamental sound generator is the sine wave, since any periodic or aperiodic digital sound can be represented as a sum of sinusoids with varying frequencies, amplitudes, and initial phase offsets. in a digital system, this additive representation is complete: any sound that can be defined as a sequence of discrete samples can also be synthesized exactly using this method.

noise presents a particularly demanding case for additive synthesis. accurately reproducing white noise requires a dense collection of frequency components with randomized amplitudes and initial phases. while this is entirely feasible in a digital system, it is both computationally inefficient and tends to be artistically and technically unmanageable. the parameter space is unstructured, and small changes can lead to large, unintuitive variations in the result.

for this reason, noise is implemented as a separate generator using subtractive synthesis via filtering. this avoids the overhead and unpredictability of additive construction, but introduces its own limitations. because perfect filters are not achievable in a discrete system, any real filter introduces spectral and temporal distortion, especially under modulation or time-localized shaping.

elements and parameters

sine
  amplitude: continuous(t)
  frequency: discrete(t)
noise
  amplitude: continuous(t)
  frequency: discrete(t)
  bandwidth: continuous(t)
sound: sine/noise

structure

to construct an instrument, we begin by defining its constituent parts. each part represents a distinct sonic layer, typically composed of one or more partials.

a partial generally corresponds to a single pitch or pitch-like element contributing to the instruments overall sound.

each part can take one of two forms:

  • a flat list of configured sine or noise events
  • a nested, hierarchical structure resembling a sub-instrument

the instrument ultimately exposes a unified interface. this interface typically generalizes key parameters, most importantly amplitude and frequency, over all partials, as these are the most musically relevant controls for shaping and composition.

overview of elements and parameters

partial
  base: sound
  decorator: sound ...
parts
instrument
  amplitude: continuous(t)
  frequency: discrete(t)
  custom ...

building timbre

timbre is shaped by the structured variation of amplitude, frequency, and spectral density over time. designing timbre involves selecting and organizing partials, usually sine tones or filtered noise, with distinct temporal and spectral roles.

frequency distributions

  • harmonic - f(n) = n·f₀, for string and air columns
  • odd-harmonics – f(n) = (2n+1)·f₀, clarinet-like
  • even-harmonics – f(n) = 2n·f₀, less common alone
  • nth-harmonics – arbitrary skipping: f(n) = k·n·f₀
  • all-harmonics – full harmonic series, all integral multiples
  • prime-indexed – use primes: f(pₙ)·f₀ (non-repeating)
  • cumulative – increasing intervals: f(n) = f(n–1) + d(n)
  • decumulative – decreasing intervals: f(n) = f(n–1) – d(n)
  • geometric – exponential spacing: f(n) = f₀·rⁿ, stretched spectra
  • random / pseudo-random – for bell-like or percussive timbres
  • logarithmic – used to simulate inharmonic resonances
  • modular series – e.g. f(n) = f₀ + mod(n, k)·Δ, tiled partial sets
  • mirror-symmetric – rising then falling: f(n) = |n – c|·f₀
  • fixed sets – user-defined, e.g. 3:5:7 ratios or detuned triads
  • clustered – multiple partials within a narrow band

amplitude distributions

  • linear – steady decay, e.g. 1 - k·n
  • exponential – sharper roll-off, e.g. a·e^(–k·n)
  • bessel – mimics radial modes in circular membranes
  • gaussian – bell-curve centered distribution
  • logarithmic / logistic – saturating or slow-rise decay profiles
  • power law – e.g. 1/n^p, common in natural resonances

temporal composition

amplitude time shapes (per partial or group)

  • adsr – common envelope with attack/decay/sustain/release stages
  • zero-to-peak – 0 → a, no sustain (plucked, percussive)
  • peak-to-zero – a → 0, fading entries
  • peak-flat-zero – bowed notes, breath or energy-controlled tones
  • overdriven swell – fast rise, overshoot, settle
  • inverse swell – strong attack followed by rapid decay
  • cyclical – periodic amplitude modulation: tremolo, flutter
  • jittered – amplitude perturbed per frame or window

frequency time shapes

  • glide / portamento – smooth frequency transitions
  • pitch envelope – discrete jumps or curves
  • chirp – linear or exponential rise/fall in frequency
  • microfluctuation – slow drift (tape warble), random walk
  • vibrato – periodic frequency modulation
  • beating – intentional offset between near-identical partials

event-level temporal control

  • staggered entry – not all partials start simultaneously
  • delayed attack – late onset for distant partials
  • layer phasing – offset or modulated start times for richer motion
  • pulse trains – sequences of short bursts (e.g. machine or motor imitation)
  • transient injection – short-time bright onset layer (used for realism or punch)

layering and decor

decorators

decorators are partials added around a center partial. they can be fixed or relative, and serve multiple spectral or temporal roles:

  • harmonic overlays – small clusters above or below a harmonic
  • non-harmonic enrichments – e.g. metallic or bell-like layers
  • detuned twins – beating effects, stereo richness
  • breath shimmer – lightly jittered filtered noise
  • grit – narrow-band noise or unstable partials for realism
  • formant mask – frequency emphasis zones imitating resonance

structural groupings

groups may reflect physical or musical grouping logic.

  • center + ring – single dominant partial + concentric decorators
  • stacked clusters – multiple localized bundles (e.g. multiple registers)
  • recursive instruments – sub-instruments used as partials
  • cross-channel mirroring – spatially mirrored partials (stereo)
  • complementary layers – e.g. sine stack + filtered noise with matched envelope
  • multi-root – two or more dominant frequency centers (e.g. dyad timbre)

spectral fusion strategies

how partials blend:

  • locked envelope – shared amplitude curve across partials
  • spectral tracking – envelope varies based on frequency
  • layered time offsets – decorators enter late or early
  • alternating phase / polarity – subtle difference in attack behavior
  • envelope staggering – each partial with shifted adsr

filtered noise

noise is a core ingredient for simulating complexity, texture, and realism, especially in non-pitched or quasi-pitched contexts.

spectral shaping

  • band-pass – isolates a narrow region: hiss, breath, metal sizzle
  • low-pass / high-pass – remove upper/lower content
  • notch filtering – suppress specific bands for comb-like artifacts
  • tilt filtering – gentle slope shaping across the spectrum
  • formant filtering – boosts around fixed peaks: vowel-like, resonant body
  • multiband stacking – layered noise bands with distinct envelopes

temporal shaping

  • short bursts – used for attacks and impacts (e.g. plosives, snare transients)
  • gated – sharply on/off envelopes
  • envelope-followed – amplitude of noise modulated by sine or other partial
  • modulated bandwidth – bandwidth changes over time (tight → airy)
  • modulated center frequency – sweeping across spectrum: rustle, swish

usage styles

  • resonant breath layer – soft, high-frequency filtered noise
  • body scratch – midband friction simulation
  • metal tail – extended, shifting noise in upper-mid bands
  • non-pitched core – broadband base with mild band shaping (e.g. thunder)

detuning & instability

realistic and lively sounds benefit from imperfections and deviations. these may be fixed, drifting, or modulated.

pitch detuning

  • static offset – ±δ hz or ±cents from harmonic
  • harmonic stretch – e.g. (n + εn²)·f₀, used in piano modeling
  • random detuning – sampled from bounded distribution
  • nonuniform spread – less detuning in low partials, more in high
  • consonant dissonance – beat frequencies matched to rhythmic intent

amplitude and frequency instability

  • slow drift – tape warble, air-pressure fluctuation
  • low-rate modulation – lfo on amp or frq: tremolo/vibrato
  • random jitter – noise added to control signal per frame
  • modulated phase – e.g. slightly detuned fm-like artifacts

stereo or multichannel tricks

  • decorator shifting – panned or asymmetric offset partials
  • ear mismatch – slight differences in left/right frequency
  • per-channel drift – independent random walk per channel

miscellaneous techniques

a few techniques escape clean categorization but are essential for crafting musical or evocative timbres.

compositional contrast

  • timbre pairs – contrasting timbre a ↔ b for musical structure
  • dynamic morph – gradual change of partial configuration across time
  • timbre switching – conditional event routing (e.g. velocity-based)

symbolic mapping

  • instrumental archetypes – shape timbre around voice, string, wind templates
  • emotional affect – map parameters to tension, brightness, pressure

meta-techniques

  • partial substitution – swap partial families at runtime
  • recursive design – decorators or groups can embed new instruments
  • expression coupling – share envelope or modulator across instruments
  • spectral crossfade – morph between configurations using weighting functions