Difference Tone Lab
v12.1 · harmonic closureWolfram-derived source laws: adjacent harmonics close the complete polynomial spectrum; common-mode motion protects Δ in the multiplier.
A3
220.000 Hz
FULL CLOSURE
Tap a key
Pure multiplier
Pure multiplier
Harmonic pair designer5:4 · Δ:g = 1:1
Grid walk k = 0Shift both sources by exact g₀ steps
Full polynomial closure
g = 220.000 Hz · Δ=1g · A=4g · B=5g · Σ=9g
Every ideal memoryless polynomial product remains on an integer harmonic of Δ.
At zero continuous shift, the full nonlinear grid is exact.
Touch keyboardC3–C4
Octave 3
Sources ↔ intermod resultCenter
Source pairMultiplier
Harmonic-grid spectrum0–2.64 kHz
actual outputkΔ gridlive A / B / Δ′ / Σ′ markers
Source-motion performance padContinuous common shift plus exact common/opposed FM laws
X: CONTINUOUS COMMON SHIFT · Y: LFO DEPTH
A
B
Common-mode FM
A+m(t), B+m(t): Δ fixed; Σ receives 2m(t).
A+m(t), B+m(t): Δ fixed; Σ receives 2m(t).
LFO
Apply live source motion
Apply live source motion
Spring X
Return continuous shift to zero
Return continuous shift to zero
Intermodulation engineMultiplier · sine-exact
Envelope A 25 ms · D 220 ms · S 72% · R 550 ms
Precision and oscillator setup
A 880 Hz · B 1100 HzΔ 220 Hzsine / sine
Wolfram-derived design rules
Write the reduced source ratio as
Full nonlinear closure:
Pure sine multiplier closure:
Common-mode FM: Δ remains fixed while Σ receives twice the modulation. Opposed FM: Σ remains fixed while Δ receives twice the modulation.
B:A = p:q. Then g = Δ/(p−q), A=qg, and B=pg.
Full nonlinear closure:
p−q=1, so the sources are adjacent harmonics nΔ and (n+1)Δ.
Pure sine multiplier closure:
(p+q)/(p−q) is an integer; for reduced ratios this adds the difference-two family such as 5:3 and 7:5.
Common-mode FM: Δ remains fixed while Σ receives twice the modulation. Opposed FM: Σ remains fixed while Δ receives twice the modulation.