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Quadraphonic Delay and Phase Vocoder

The Quad Delay (QuadDelay in private/src/QuadDelay.hpp) is a highly complex, time-warped delay effect deeply integrated with the Theory of Time.

Time Warping and the Inverse Buffer

A traditional delay line implements the equation D(t + d) = X(t), where t is wall-clock time, d is the delay time, and X is the input signal.

The Smart Grid One implements a time-warped version of this equation. Let F(t) be the post-modulation logical time from the Theory of Time. The quad delay implements: D(F⁻¹(t + d)) = X(F⁻¹(t)) This means: “At the wall-clock time that produces the warped time t + d, output the sample that was recorded at the wall-clock time that produced the warped time t.”

Assuming F is injective (which it sometimes is, though extreme modulation can break this), this can be rewritten as: D(t) = X(F⁻¹(F(t) - d))

To implement this, the delay requires a “moveable writehead.” It must compute and store the inverse function F⁻¹.

• DelayLineMovableWriter maintains two parallel circular buffers:
  1. m_delayLine: The audio samples X(t).
  2. m_writeHeadInverse: The wall-clock time t at which each warped time F(t) occurred.
• By looking up F(t) - d in the inverse buffer, the delay can find the exact wall-clock time t_old when that warped time occurred, and then read the audio sample X(t_old).

Read/Write Head Computation (QuadDelayInputSetter)

The read and write heads are produced in QuadDelayInputSetter::Process (private/src/QuadDelay.hpp) per quad channel i.

• Loop selection: WriteTapeHead receives the processed loop-selector knob value, but only accepts changes when both old and new loops are simultaneously at top (m_top) to avoid discontinuities.
• Glue offset: WriteTapeHead owns the additive glue offset that preserves continuity across transport stops and tempo-scale changes.
  • When transport stops, glue is initialized from the current write head, then incremented each sample.
  • On master-loop-size change, glue is rescaled so absolute position continuity is maintained.
• Delay ratio quantization: at loop top, delay-time factor is quantized to one of:
  • 0.8, 2/3, 1.0, 3/4, 5/8 and stored as m_bufferFrac[i].
• Read-head speed quantization: read-head speed is quantized to the same ratio family and its negative counterparts:
  • -4, -3, -2, -3/2, -4/3, -1, -1/2, -1/4, 1/4, 1/2, 1, 4/3, 3/2, 2, 3, 4

The final head equations are:

• Write head
  • writeHead = masterUnwoundPhasor * masterLoopSamples + glue
  • stored in delayInput.m_writeHeadPosition[i]
• Read head
  • effectiveDelaySamples = (masterLoopSamples / externalLoopMultiplier(selectedLoop)) * (bufferFrac * widen)
  • readTarget = writeHead * readHeadSpeed - effectiveDelaySamples
  • readHead = wrap_mod(writeHead - resynthesisHopSamples - selectedLoopSamples, writeHead - resynthesisHopSamples, readTarget)
  • stored in delayInput.m_readHeadPosition[i]

So both heads live in the same unwound sample coordinate system. The read head is projected into the selected-loop-length region behind the write head, offset by the resynthesis hop size, with delay shaped by quantized ratio and widener.

Phase Vocoder Done Right

Because the read head is moving through the audio buffer at a variable rate (due to the time warping), a simple read would result in severe pitch shifting (like scratching a vinyl record).

To preserve the original pitch while allowing the time-warping to stretch and compress the audio, the delay uses a Phase Vocoder (Resynthesizer in private/src/Resynthesis.hpp).

• The audio is processed in overlapping grains (GrainManager).
• For each synthesis frame, the system computes two analysis frames:
  1. One at the target read position F⁻¹(F(t) - d).
  2. One exactly H (hop size) unwound samples before that position.
• By comparing the phases of these two analysis frames, the resynthesizer can accurately update the synthesis phases, preserving the pitch of the original signal regardless of the playback speed. This roughly follows the “Phase Vocoder Done Right” methodology.

Quad Delay Features

The Quad Delay supports a rich set of features:

• Synced Delay Time: Delay times are synchronized to the Theory of Time.
• Feedback > 1: Allows for self-oscillating, runaway feedback loops.
• Per-Partial Fade In (Slew Up): The resynthesizer can slew the amplitudes of individual partials, creating a smooth, blooming fade-in effect for the delayed signal.
• LP and HP Damping: Low-pass and high-pass filters in the feedback path (PostFeedbackFilter) to dampen the echoes.
• Unison and Pitch Shifting: The phase vocoder allows for independent pitch shifting of the delayed signal.
• Ping Pong: 90-degree or 180-degree quadraphonic ping-pong routing.
• Return Sends: The output of the delay can be routed directly into the Quad Reverb or the Partial Machine.

Related

• Theory of Time
• Phase Vocoder
• Quad Reverb
• Partial Machine
• DSP Overview

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