harmonic distortion explained

what happens when a signal clips

send a pure 100 Hz sine wave into a saturator and watch the output on a spectrum analyzer. where there was a single vertical line, there are now five, ten, twenty. new frequencies at 200 Hz, 300 Hz, 400 Hz, 500 Hz and beyond, each one fainter than the last. harmonic distortion explained in one image: the saturator created frequencies that were not in the original signal.

stage 1 introduced saturation as harmonic generation. this stage explains the mechanics: what determines which harmonics appear, how loud they are, and why digital systems need special care to do it right.

every saturator works by reshaping the input waveform. the shape of the output depends on how the device handles signal peaks. in analog circuits, tubes, tape, and transformers each have a physical limit. as the input approaches that limit, the output stops following it linearly. the peaks get rounded off, and that rounding is what generates harmonics.

the softness of the rounding matters. a gentle curve (soft clipping) rounds the peaks gradually, producing a few low-order harmonics that decay quickly in level. an abrupt cutoff (hard clipping) slices the peaks flat, creating dense harmonics across a wide bandwidth. the waveform tells the story: a sine wave with gently rounded peaks still looks like a sine wave. a sine wave with its top sliced flat looks like a square wave, and a square wave is nothing but odd harmonics stacked up.[^1]

the symmetry rule

stage 1 introduced even and odd harmonics and which analog topologies favor each. here is the deeper question: why does the shape of the curve determine which harmonics appear?

the answer is symmetry.

when a transfer function treats the positive and negative halves of the waveform identically (symmetric clipping), the output waveform looks the same flipped upside down. mathematically, this means only odd-numbered terms survive in the Fourier series. the 3rd harmonic, the 5th, the 7th. the even harmonics cancel out because the positive and negative halves produce them in opposite phase.

when a transfer function treats the two halves differently (asymmetric clipping), the cancellation breaks. the positive half clips harder than the negative, or the curve has a DC offset. the waveform is no longer symmetric, and even-numbered harmonics appear: 2nd, 4th, 6th.

this is why tube circuits produce even harmonics. a vacuum tube has inherently different behavior for positive and negative grid voltages. the bias point sits off-center on the transfer curve, so the positive and negative signal halves are shaped differently. tape’s magnetic hysteresis, by contrast, is roughly symmetric. the magnetization curve looks about the same in both directions. so tape saturation emphasizes odd harmonics.[^2]

in practice, no real circuit is perfectly symmetric or perfectly asymmetric. every device produces a mix of both even and odd harmonics. the character comes from the ratio. tube-like circuits lean toward even. tape-like and hard clipping lean toward odd. transformer cores produce both due to a combination of symmetric hysteresis and asymmetric core saturation.

how drive controls the recipe

the drive knob on a saturator is not a volume control. it is a gain stage before the transfer function, and it determines how far into the nonlinear region your signal reaches.

at low drive, the signal stays in the center of the transfer curve where the relationship between input and output is nearly linear. a straight line generates no harmonics. the output is almost identical to the input, with perhaps a faint 2nd or 3rd harmonic at -40 dB or below. inaudible.

increase the drive and the signal begins to enter the curved region where the transfer function bends. this is where the character emerges. the peaks of the waveform get gently rounded, and the first few harmonics become audible. the 2nd and 3rd harmonic appear at -18 to -24 dB below the fundamental. subtle, warm, present.

push the drive further and the signal hits the flat region at the top and bottom of the curve. the peaks are no longer just rounded, they are compressed against the ceiling. more harmonics appear at higher levels. the 5th, 7th, 9th become audible. the decay from one harmonic to the next gets slower, meaning the harmonic density increases. this is where saturation becomes distortion.

the “knee” of the curve, how abruptly it transitions from linear to saturated, controls the onset. a soft knee introduces harmonics gradually as drive increases. a hard knee creates an abrupt transition: quiet at one setting, harmonically dense at the next. soft knees feel musical. hard knees feel aggressive.

aliasing: when harmonics go wrong

everything described so far works perfectly in the analog domain. in digital, there is a catch.

a digital audio system has a maximum frequency it can represent: the Nyquist frequency, which is half the sample rate. at 44.1 kHz, the Nyquist limit is 22,050 Hz. any frequency above this cannot exist in the digital signal.

but harmonics do not care about sample rates. a waveshaper generating the 5th harmonic of a 5 kHz input creates energy at 25 kHz, which is above Nyquist. in analog, that energy simply exists as ultrasonic content you cannot hear. in digital, it folds back. 25 kHz minus 22,050 Hz equals 2,950 Hz. the 5th harmonic appears at 2.95 kHz instead of 25 kHz.

that aliased frequency is not an integer multiple of the input. it is inharmonic: musically unrelated to the source. it sounds metallic, brittle, and cold, the opposite of what saturation should do.[^3]

higher harmonics fold further into the audible range. the 7th harmonic of 5 kHz would be 35 kHz, folding to 9,050 Hz. the 9th at 45 kHz folds to 850 Hz. each aliased component sits at a wrong frequency, and together they create a dense cloud of inharmonic content that makes digital saturation sound harsh and artificial.

two solutions exist. oversampling runs the waveshaper at 2x or 4x the original sample rate, pushing the Nyquist limit up so that the generated harmonics stay below it. after waveshaping, a low-pass filter removes the ultrasonic content, and the signal is downsampled back to the original rate. the harmonics that would have aliased are filtered out before they can fold.

ADAA (antiderivative anti-aliasing) takes a different approach: instead of running at a higher rate, it mathematically integrates the transfer function and computes the average value between consecutive samples. this smooths out the between-sample behavior that causes aliasing, achieving comparable suppression at lower computational cost.[^4]

intermodulation: the chord problem

everything so far has assumed a single frequency going into the saturator. music is more complex than that.

when two frequencies enter a nonlinear system simultaneously, the output includes not just their individual harmonics but also sum and difference products. a 200 Hz note and a 300 Hz note through a saturator generate 500 Hz (200 + 300), 100 Hz (300 - 200), 400 Hz (200 x 2), 600 Hz (300 x 2), and many more. these intermodulation products are not harmonics of either input. they are new frequencies at mathematical combinations of the inputs.

for simple intervals (octaves, fifths), many intermodulation products land on musically related frequencies. 200 Hz and 400 Hz (an octave) produce sum and difference tones at 200, 600, 800 Hz, all harmonics of 200 Hz. this is why octaves through saturation sound clean.

for complex intervals and chords, the intermodulation products land on non-harmonic frequencies. a minor second (200 Hz and 212 Hz) produces a difference tone at 12 Hz and sum products that fall between the harmonics of either note. the result is audible beating, roughness, and mud.

this is why heavy saturation works on bass lines and solo vocals but destroys chords. a bass guitar playing single notes benefits from the added harmonics. a piano playing dense voicings through the same saturation level sounds like mud, because the intermodulation products from dozens of simultaneous frequencies create a dense cloud of inharmonic content.

it is also why mix bus saturation must be subtle. a full mix has hundreds of simultaneous frequencies. even gentle saturation generates intermodulation products from all of them. at low drive, these products are below the noise floor. push the drive and they become audible as a loss of clarity: the mix sounds denser but less defined.

harmonic profiles by source

different sources benefit from different harmonic treatments. the goal is always the same: add harmonics that complement the source without competing with other elements in the mix.

vocals

vocal saturation is most effective at low levels. the 2nd and 3rd harmonics at -18 to -24 dB below the fundamental add body and presence without changing the perceived tone. this is the “finished studio” quality that subtle saturation provides. higher harmonics (5th and above) add grit, which works for aggressive styles but competes with cymbals and high-frequency reverb in a dense mix.

bass

bass saturation serves a specific purpose: audibility on small speakers. the fundamental of a bass guitar at 40-80 Hz disappears on laptop speakers, earbuds, and phone speakers. the 2nd harmonic at 80-160 Hz and 3rd at 120-240 Hz carry through. saturation generates exactly these harmonics, making the bass audible without boosting the sub frequencies that small speakers cannot reproduce.

drums

kick drum benefits from 2nd and 3rd harmonics for attack definition and punch. snare benefits from odd harmonics for crack and snap. overheads and cymbals need caution: they already have dense high-frequency content, and saturation adds more energy in a region where the mix is often already crowded.

synths

synthesizers produce harmonically rich signals by design. a saw wave already has every integer harmonic. saturation on a saw wave adds density but risks muddiness because there is no spectral space for new content. better results come from saturating simpler waveforms (sines, triangles, sub-oscillators) where there is room for harmonics to fill in, or from using saturation after a low-pass filter has removed some of the existing harmonics.

when distortion becomes damage

saturation has a sweet spot. below it, the effect is inaudible. above it, the harmonics compete with other elements and reduce clarity. three mechanisms define the upper limit.

frequency masking. the harmonics generated by one source occupy the same spectral space as fundamentals of another. saturated bass at 200-400 Hz competes directly with vocal fundamentals. saturated vocals at 4-8 kHz compete with cymbals and air. the more saturation you add, the more spectral overlap you create.

dynamic range loss. saturation compresses peaks. a little compression from gentle saturation sounds like glue. too much removes the dynamic variation that gives music life: transients flatten, quiet passages get louder relative to loud ones, and the performance loses contrast.

the “louder is better” trap. saturated signals sound louder and more dense, which the brain interprets as better. this is a psychoacoustic bias, not an aesthetic judgment. without level-matched A/B comparison, you will keep adding saturation because each increment sounds “better” until the cumulative effect is mud. always compare at matched levels.

frequently asked questions

frequently asked questions

what is the difference between harmonic and inharmonic distortion?

harmonic distortion generates new frequencies at integer multiples of the input: 2x, 3x, 4x. these are musically related to the original and tend to sound pleasant at moderate levels. inharmonic distortion generates frequencies that are not integer multiples, like intermodulation products from two input frequencies mixing. inharmonic content sounds dissonant and unmusical. aliasing in digital systems is a form of inharmonic distortion.

why do even harmonics sound warm and odd harmonics sound harsh?

even harmonics (2nd, 4th, 6th) are octave-related to the fundamental. these intervals are maximally consonant. odd harmonics (3rd, 5th, 7th) create intervals like perfect fifths and major thirds relative to their neighbors. at moderate levels they add character and presence. at higher levels, the increasing density of non-octave intervals creates a harder, more aggressive tone.

what is aliasing in digital saturation?

when a waveshaper generates harmonics above the Nyquist frequency (half your sample rate), those harmonics fold back into the audible spectrum at wrong frequencies. these aliased frequencies are not musically related to the input, so they sound harsh and metallic. oversampling and ADAA (antiderivative anti-aliasing) are the two main solutions.

how does the drive knob control harmonics?

the drive knob scales the input level before it hits the transfer function. at low drive, the signal stays in the linear center and few harmonics are generated. as drive increases, the signal pushes into the curved nonlinear region, generating more and louder harmonics. the shape of the curve determines which harmonics appear. the drive determines how many and how loud.

can too much harmonic distortion damage a mix?

yes. excessive harmonics fill the spectrum with energy that competes with other elements. bass saturation generates midrange harmonics that clash with vocals. vocal saturation generates high-frequency content that fights cymbals. the result is a dense, muddy mix where nothing has space. saturation works best when subtle: adding presence and body without becoming audible as an effect.

references

a note from the developer

this guide is built on four years of studying psychoacoustics and DSP research. reading papers, building prototypes, making mistakes, and learning from all of it. i am a solo developer in copenhagen, and i am still learning every day.

when i built KERN WARM, the insight that changed everything was individual harmonic control. most saturators use a single transfer function: one curve, one harmonic recipe, take it or leave it. Chebyshev polynomials let you set the level of each harmonic independently. the 2nd harmonic at -12 dB, the 3rd at -18 dB, the 5th at -30 dB. each character in WARM (tube, tape, transformer) is a different set of coefficients, a different recipe. the months of iterating on those coefficients, listening on headphones, comparing against analog references, adjusting by fractions of a dB, that loop was the hardest and most rewarding part of building the plugin.

if i got something wrong, missed an approach that works for you, or if you just want to share your workflow for using saturation, i genuinely want to hear from you. reach out at jonas@kernaudio.io. every piece of feedback makes these guides better.