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@maru shared this effect so you can build on it and use it in your DAW.

AvailableEffect

Chebyshev

Choose the harmonic produced by the distortion.

About

A stereo waveshaper whose one big knob is a harmonic number. Instead of asking "how much distortion?", Chebyshev asks "which harmonic?", and turns what you feed it into that harmonic.

The idea

Chebyshev polynomials have a tidy property: put a pure sine through the polynomial of order n and what comes out is the nth harmonic of that sine. Order 2 gives the octave above, order 3 the octave and a fifth, order 4 two octaves up. So moving the Order knob does not make the effect stronger, it makes it sound like a different interval.

That holds cleanly only for a sine. Real material has many partials, and each one is transformed at once, so on chords and rich sounds the higher orders turn into dense, bright, metallic grit rather than a neat transposition. Single sustained notes are where the harmonic character is easiest to hear.

Controls

  • Order (1 to 100) picks the polynomial, and with it the harmonic. Order 1 passes the signal unchanged. Low even orders (2, 4) add octaves and thicken; low odd orders (3, 5) add fifths and a hollow, reedy colour; high orders become bright inharmonic sheen.
  • Mix blends the shaped signal with the dry one. High orders remove most of the original fundamental, so keeping some dry signal is usually what makes the result sound coloured rather than replaced.

The panel draws the shaping curve and the blended result for the current settings, so you can see how many times a high order folds the waveform.

Getting started

Chebyshev is an audio effect for any stereo source. A clean, sustained guitar or synth note shows the effect most clearly. Try the presets:

  • Octave Thicken: order 2 blended in, a musical way to fatten a bass.
  • Fifth Ghost: order 3, a fifth nobody played, kept low as colour.
  • Hollow Square: order 5 well up in the mix, reedy like a clarinet.
  • Two Octaves: order 4 fully wet; the fundamental all but disappears.
  • Metal Sheen: order 80 kept quiet, bright grit on top of the source.
  • Default: order 50 at half mix.

High orders are steep curves that would normally alias badly; Chebyshev processes them at four times the sample rate with filtering to keep the upper half of the range usable.

How it works.

The Chebyshev polynomial of the first kind has one property that makes it useful here: Tn(cos θ) equals cos(nθ). Push a pure sine through the order-n polynomial and what emerges is the nth harmonic, alone and at full level. That is why Order reads as a harmonic number rather than as an amount, it is not choosing how hard to distort, it is choosing what to produce.

Even and odd orders behave differently, and the difference is audible before it is theoretical. An even polynomial is symmetric about the vertical axis, so it maps positive and negative input to the same output and rectifies the waveform, the fundamental largely disappears and the result sits an octave up. An odd one preserves the sign of its input, so the waveform stays symmetric, the fundamental survives underneath, and the added harmonic reads as colour on the original note rather than as a replacement for it.

Real input is not a pure sine, and the clean harmonic identity does not survive it. Every partial in a chord goes through the same curve, and their sums and differences come with it, so what you hear at high orders is a dense inharmonic colouring rather than a tidy transposition. Single sustained notes are where the property is legible.

A high-order curve is extremely steep, order 50 crosses zero fifty times across the input range, which is exactly the shape that aliases badly at base rate. The browser preview runs the waveshaper at 4x oversampling for that reason, which is what makes the upper half of the order range usable rather than a wash of foldover.

What would you make with Chebyshev?

Describe what you want to make. Build it with the Studio agent.

Goes well with.

Building block

Drive

Add harmonics with a waveshaper.

Building block

Crush

Bit reduction and sample-rate reduction for digital grit.

Prototype

FX Mini

Drive, tone, and bounded feedback delay.