Fact: Large amounts of distortion, particularly high order distortion, are unpleasant.

Conclusion: There is an optimal amount of distortion which appeals to tubophiles.

Suppose: The ideal distortion did not vary with signal intensity. What if it were constant for all signal levels?

We need a transfer function which has a resulting distortion:

THD = Pdist / Ptotal = constant (independent of input amplitude)

Further suppose Pdist is entirely second harmonic, just to make this simpler. We'll start with a sinusoidal signal, on the assumption that this still works for complex signals, understanding that IMD is inevitable, and hoping that THD doesn't increase much.

Then,

THD = V2^2 / (V1^2 + V2^2)

where Vn = amplitude of nth harmonic (1 = fundamental, 2 = distortion).

For this to be constant, we need

V2 = V1 * sqrt(THD / (1 - THD))

So if we have an input,

Vi = A*sin(w*t)

we need a transfer function f(x) such that:

f(A*sin(w*t)) = V1*[sin(w*t) + sqrt(THD / (1 - THD)) * sin(2*w*t)]

Is this function linear? Well, obviously not, because it's making distortion. But then, if A = B + C, we should get V1 = (B + C) / A, which means the amplitudes are linear, but the products are not. This suggests one possible way to construct a circuit: perhaps we "100% distort" the signal (like with a squaring function biased around zero, so it contains no original signal -- it's 100% distortion), adjust it for proportional amplitude, and mix it in. The amplitude control stuff necessarily has some delay, so it's not a time-invariant system, which means solving for f(x) analytically will be a pain, if possible at all.

Will it actually SOUND better? Who knows. Some features of tube amps are their "terrible" recovery behavior, arising from distortion shifting bias voltages, which takes time to return to zero. The amplitude control in the proposed system will take time, too. Maybe it will have a more pleasing effect?

Some experimental features are appealing. The type and quantity of distortion can be varied by song and user. By using other functions, 3rd harmonic could be explored, for instance.

The system could be implemented in tubes, SS or DSP. Tubes are uniquely suited, as a variable gain amp (VGA) can be made with a dual control tube (6HZ6, 6HS8) or heptode, while an envelope detector can be made with a 6AL5, resulting in no minimum signal offset that would result from silicon diodes (which would otherwise require op-amps to make a precision rectifier circuit!).

Tim