Brian Hayden has had interviews about his thoughts on alternate keyboards posted. I've summarized them below.
Ken Rushton. This was cribbed from "A Chat with Brian Hayden" and "on the Reuther Uniform System and other self-transposing systems"
If we take the largest sensible musical interval to be an octave (12 semitones), it might be assumed that there could be as many as 12!/3! (factorial 12 divided by 6) possible keyboards. However on closer mathematical inspection we find that this is not the case. Look at it another way. Of the 3 musical intervals in semitones, there is a greatest interval G (any whole number from 2 to 12) and the least interval L (obviously G & L cannot be the same number); the other interval is the difference between the two i.e. G - L, which we will call D. D is obviously less than G but it might be the same as L as in the case of the "Uniform System" which we were looking at. The number of possible keyboards is also reduced by a further couple of factors. If G is an even number then L cannot be an even number because D would also be even, and this would only generate the notes of a whole tone scale; and further if G is divisible by 3, L cannot be divisible by 3 or D would be divisible by 3 thereby only generating only the notes of a diminished seventh chord.
So the only possible fundamentally different consist keyboards are as follows in the order G, L, D:- (1) 2, 1, 1. (2) 3, 1, 2. (3) 4, 1, 3. (4) 5, 1, 4. (5) 5, 2, 3. (6) 6, 1, 5. (7) 7, 1, 6. (8) 7, 2, 5. (9) 7, 3, 4. Higher intervals than 7 do not generate useful keyboards, I won't list all these, with the notable exception of (24) 12, 5, 7. And of course simply a single run of semitones which I shall call - (0) 1.
Some people see the "square form" of keyboard as different from the ones that have set intervals in triangles, however I see this as a special case with right angled triangles, where one set of intervals is about to form into another. Take for example the Pitt-Taylor 1922 Pat.No. 208274 keyboard. with semitones along the rows of notes and half octaves above them, i.e. a rows of notes:
Looked at one way is G,L,D. : 6, 1, (5). or the other (7), 1, 6.
In addition mirror images and keyboards which are turned upside down are not fundamentally different. A good example of this is the B continental chromatic keyboard, which is the mirror image of the C continental chromatic keyboard.
I hadn't come across this system under that name before, however I have met it on Accordions before where it is usually called the "Uniform System". I have turned up an article by Albert Delroy from "Accordion Times" about 20 years ago, (regret I only have a photocopy of the article not the whole magazine to date it exactly). He writes that it was invented in that form for the Accordion by John H. Reuther of New York. He also mentions that it already existed as the "Austrian" system but with round buttons. The Scottish Accordionist, the late Jimmy Blair, was an exponent of this system.
Out of the mathematically possible systems, 9 have been patented, proposed, or made; including one by Wheatstone. The Wheatstone patent is 1844 Pat. No. 10041: see figures 7 & 8 of that patent. This is G.L.D. 5, 1, 4.
GLD 12, 5, 7 will be found as Wesley 2002 (US Patent # 6501011) although I did mention the possibility of this as a squashed form of mine if you read the text very carefully.
And finally, there is a free bass keyboard for the accordion Ronald Merrett, 1956 pat. no. 856926, which is GLD 7, 4, 3.