Okay, here we go again:H = (1/g - 1/z) / (g - 1/z)
 = (1 - g/z) / (g² - g/z)OKAY. Now we restrict z to the unit circle and require H = -1.(1 - g∙exp(-iφ)) / (g² - g∙exp(-iφ)) = -1
1 - g∙exp(-iφ) = -g² + g∙exp(-iφ)
g² + 1 = 2g∙exp(-iφ)
exp(-iφ) = (g² + 1) / 2g
-iφ = ln [(g² + 1) / 2g]
φ = i ln [(g² + 1) / 2g]WHAT THE HELL? In order for φ to be real, the logarithm must evaluate to -i??? So(g² + 1) / 2g = exp(-i)?????????? What is wrong with me? Why can't I do this problem?

Okay, clearly I just don't know what I'm doing. Because the damn magnitude response for the above is 1/g, not 1. So I was right in the first place:H = (1 - g/z) / (g - 1/z)Again setting z = exp(-iφ) and H = -1:(1 - g∙exp(-iφ)) / (g - exp(-iφ)) = -1
1 - g∙exp(-iφ) = -g + exp(-iφ)
g + 1 = (g + 1)∙exp(-iφ)
exp(-iφ) = 1
φ = 0This is true no matter what g is. If you're getting phase inversion anywhere other than zero frequency, there's probably something wrong with your implementation. There. I'm done. Finished. Finito. Complete. No more. Goodbye.

All righty then ...

I think this is it :
rTheta = rFreq * krPi * rInvSmplRate;   
rD     = tan(rTheta);
rG     = (1.0 - rD) / (1.0 + rD);
peace y'all
pj

http://www.musicdsp.org/files/Audio-EQ-Cookbook.txt says:

w0 = 2*pi*f0/Fs
alpha = sin(w0)/(2*Q)

b0 = 1 - alpha
b1 = -2*cos(w0)
b2 = 1 + alpha
a0 = 1 + alpha
a1 = -2*cos(w0)
a2 = 1 - alpha

...as you see, with biquad allpass filters, you have an additional degree of freedom: Q, which determines how steep the slope of the phase response is, when it passes through -180 degrees

This corroborates what I was saying. You might be able to get phase inversion away from w0 = 0 with a biquad allpass, but not with a single pole and zero.

Guys!

A while back I wrote a simple VST effect called "PhaserTst" - I've already verified the formula I gave above using this test scaffold and it works as close to perfect as I need...

Currently I use a mix of low-level white noise and nine octaves of sinewaves as input to the plugin - the plugin simply combines the "phased" signal and the dry signal and then I look at the output in a spectrum analyzer for frequency dips. It was this plugin that provided the original empirical data and subsequently verified the final formula.

peace,
pj