Brian Hayden on the Reuther Uniform System and other self-transposing systems

We wrote to Hayden Duet Concertina inventor Brian Hayden in November, 2005 about the Reuther Uniform System. Here is an accordion which had recently been offered on eBay, an accordion built to that system:

Accordion built to Reuther Universal System This image of the Italian-made Robotti accordion employing the Reuther Uniform Keyboard System was provided by Thomas Mayers & William Pavone of Tonawanda, NY; Thanks, guys!

Brian wrote back and commented on the similarities and differences of the patented Reuther System (and others) with respect to the patented Hayden System. Together with Brian we have edited his research and observations into the following article. Any errors still found herein should be assumed to be the responsibility of the editor!

Thank-you for your eMail about the "Reuther System".  This isn't the Hayden System; the semitone intervals between the notes are 2, 1, 1; whereas the Hayden system has 7, 2, 5.  It is probably one of the earliest self-transposing systems to be invented; being first patented in 1811 (Pat. No. 3404) by Trotter for the Piano or Organ: there are no known instruments deriving from Trotter.

It was later reinvented but in a 6 row form by Janko later in the nineteenth century (1885 Pat. No. 536), for the Piano. Quite a number of Janko instruments were made and several still exist; Rubenstein is said to have tried it and very much liked it; however it was the piano teachers who managed to kill off the system, as they made most of their money not from teaching virtuosos, but by continual repeat lessons to ordinary amateur players just to learn to play in more and more difficult keys.

The system was then reinvented for the accordion around the beginning of the last century under the name "Uniform System", at a similar time as the " Continental Chromatic Accordion System" (with intervals 3, 1, 2.). The advantages of a much narrower octave on the "C.C.", seems to have given it the edge over the "U.S." and that is the one that has become popular.  The Uniform System may be ordered from several Italian accordion makers on a one-off basis.

Self-transposing systems work by having a triangle of the keyboard touches with the musical interval at constant recurring form. 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 self-transposing 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:

F  F# G  G# A  Bb
B  C  C# D  Eb E

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 continentinal 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.

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