Run the signal through an allpass filter and listen to just the output. You will hear nothing different. Every frequency passes through at equal gain. If you put an allpass filter in your signal chain and look at the frequency response on a spectrum analyzer, you will see a flat line.
And yet cascade four of these invisible filters in series, mix their output back with the dry signal, add a slow LFO, and you get one of the most recognizable sounds in recorded music.
That gap between "technically does nothing" and "perceptibly sweeping notches across the spectrum" is what phasers are built on. Understanding it changes how you think about modulation effects in general.
What an Allpass Filter Actually Does
A first-order allpass filter has the continuous-time transfer function H(s) = (s - ωb) / (s + ωb), where ωb is the break frequency. The key property of this transfer function is that its magnitude is 1 at every frequency: |H(jω)| = 1 for all ω. The filter passes everything.
What it does change is phase. The phase response is:
Θ(ω) = π - 2 · arctan(ω / ωb)At DC (ω = 0), phase = π (a full 180-degree inversion). At the break frequency ωb, phase = π/2 (90 degrees). As frequency approaches infinity, phase approaches 0. The phase rotates from π down to 0 as you sweep from DC to high frequencies.
This is the core of every phaser: a filter that rotates phase without touching amplitude.
Where the Notches Come From
When you mix the allpass-filtered signal with the original dry signal, you are adding two copies of the same audio that are phase-shifted relative to each other. At any frequency where the allpass introduces exactly 180 degrees of phase shift, the two signals are perfectly out of phase. They cancel. You get a notch.
This is destructive interference, the same physics as noise-canceling headphones. The allpass filter positions the cancellation at specific frequencies by controlling where that 180-degree condition is met.
A single first-order allpass section has a phase of exactly π (180 degrees) only at DC. Mixed with dry, it creates a notch at 0 Hz, which is inaudible. You need multiple cascaded sections to push that cancellation into the audible range.
Cascading Sections: Why Stage Count Matters
With N cascaded first-order allpass sections (all sharing the same break frequency ωb), the total accumulated phase is:
Θ_total(ω) = N · π - 2N · arctan(ω / ωb)At DC, total phase = N·π. As frequency rises toward infinity, total phase descends toward zero. Every time the total phase crosses an odd multiple of π (that is, π, 3π, 5π...) during this descent, you get a notch in the mixed output.
For a 4-stage phaser (N = 4): total phase starts at 4π and descends. It crosses 3π and then π on the way down. Two notch conditions, two audible notches.
For an 8-stage phaser: total phase starts at 8π and crosses 7π, 5π, 3π, and π. Four notches.
The rule: N stages of first-order allpass sections produces N/2 notches in the frequency response (assuming even N).
The MXR Phase 45 uses two stages, creating one notch. The Phase 90 uses four stages, creating two notches. The name references the 90-degree phase shift each allpass section produces at its break frequency.
The LFO Makes It Sweep
The LFO in a phaser does one thing: it modulates ωb over time, which shifts where the phase-versus-frequency curve sits, which moves where the 180-degree crossing points land, which sweeps all the notches up and down simultaneously.
If a 4-stage phaser's break frequency sweeps from fb = 200 Hz to fb = 2000 Hz, you can calculate where the notches land. The two notch conditions for N=4 give notch frequencies:
f_notch1 = fb · tan(π/8) ≈ 0.41 · fb
f_notch2 = fb · tan(3π/8) ≈ 2.41 · fbSo as fb sweeps from 200 to 2000 Hz:
- Lower notch sweeps from ~83 Hz to ~830 Hz
- Upper notch sweeps from ~480 Hz to ~4800 Hz
Both notches cover the harmonically dense midrange of guitar and synth signals. This is not a coincidence: phasers became popular on guitar precisely because the sweep maps naturally onto the instrument's tonal content.
JFET vs. OTA: Two Ways to Sweep the Break Frequency
In the original Phase 90, JFETs (junction field-effect transistors) implement the variable element in each allpass stage. The LFO voltage controls the gate, which adjusts the drain-source channel resistance. Higher resistance moves ωb lower; lower resistance moves ωb higher.
The Electro-Harmonix Small Stone (introduced in 1974) uses OTAs, operational transconductance amplifiers, instead. The LFO drives a control current into the OTA, which directly adjusts its transconductance and therefore the effective cutoff of each stage. The two circuits achieve the same result through different means, with subtly different nonlinear behavior that gives each pedal its characteristic tone.
In both designs, the same LFO signal drives all four stages simultaneously. The notches move in lockstep.
The Uni-Vibe: What Happens When They Don't Move in Lockstep
Fumio Mieda designed the Uni-Vibe at Shin-Ei in 1968. It was introduced in the US market in 1969 and became inseparably linked to Jimi Hendrix after his Woodstock performance that August.
The circuit topology is recognizable: four allpass stages, an LFO. But instead of JFETs or OTAs, Mieda used an optical circuit. A small incandescent bulb sits inside a light shield surrounded by four light-dependent resistors (LDRs). The LFO dims and brightens the bulb. Each LDR controls the break frequency of one allpass stage.
The critical detail: the four LDRs are positioned at slightly different distances and angles from the bulb. The light reaching each one is not identical. As the bulb's brightness changes, each LDR responds with slightly different timing and amplitude. The four allpass stages are not modulated in perfect synchrony.
The result is that the two notches do not sweep in lockstep. The lower notch moves somewhat independently from the upper one. The overall modulation has an asymmetrical, breathing quality that analog circuits with uniform modulation cannot reproduce.
This was not deliberate sophistication. It was an artifact of using an optical control mechanism where perfect uniformity is physically impossible. Hendrix, David Gilmour on Pink Floyd's "Breathe" and "Echoes," and Robin Trower all discovered that this accident produced a sound no conventional phaser could match.
Adding Feedback
The Small Stone includes a "Color" switch that introduces signal feedback into the allpass cascade. Feedback in a phaser creates resonance peaks at the frequencies between the notches, amplifying those bands while the notches deepen. This shifts the effect from a smooth, transparent sweep toward something more aggressive and nasal.
In modern digital phasers, this appears as a "Resonance" or "Feedback" knob. Increasing it brings the phaser closer to flanger territory, especially with high stage counts.
Digital Implementation
A digital allpass section uses a biquad (second-order IIR) filter. From the Audio EQ Cookbook, the coefficients for a biquad allpass at center frequency f0 and sample rate Fs are:
w0 = 2π · f0 / Fs
alpha = sin(w0) / (2Q)
b0 = 1 - alpha
b1 = -2 · cos(w0)
b2 = 1 + alpha
a0 = 1 + alpha
a1 = -2 · cos(w0)
a2 = 1 - alphaDividing all coefficients by a0 normalizes the filter. The numerator and denominator differ only in sign of the alpha term, which is the algebraic signature of the allpass property.
Modern digital phasers typically cascade 4, 8, 12, or 16 such sections. More stages produce more notches and a denser, more complex frequency response. At 12 or more stages with high feedback, the spacing between notches becomes narrow enough that the effect begins to resemble flanging, a closely related phenomenon where comb-filter notches are equally spaced in frequency rather than the logarithmically spaced pattern allpass cascades produce.
When modeling a Uni-Vibe digitally, the challenge is reproducing the asymmetrical modulation. Accurate models give each of the four stages a slightly different LFO depth and phase offset, derived from measurements of how the original LDR circuit responds to the optical modulation. Papers from the DAFx conference have explored this specifically, measuring the actual transfer characteristics of vintage units to extract those per-stage offsets.
The Paradox, Resolved
A phaser is a flat-magnitude filter followed by a summer. The flat filter alone does nothing audible. The summer creates the notches. And the LFO does nothing except move ωb in each stage, which repositions where the cancellation conditions are met.
Understanding this is useful beyond phasers. Any time you see a modulated allpass in a signal chain, you know it only has audible consequence when its output is mixed with something else. The perceptual effect is not in the filter, it is in the interference between that filter's output and the original signal. The allpass is the instrument, but the mixing is the music.