Isomorphic Keyboard Layouts - taken from the defunct Thummer.com website
Consider the following rectangular grid:
H stands for “horizontal,” whereas V stands for “vertical.” H and V are interval offsets from the pitch of the central square, which is labeled zero (0) since the interval between it and itself is zero (unison).
Consider the following hexagonal grid:
As you can see, it is the hexagonal equivalent of the rectangular grid above.
All isomorphic keyboard layouts can be characterized by:
a) The shape of their grid (rectangular or hexagonal)
b) The values of H and V
Hexagonal layouts are used more often than rectangular layouts, because (1) hexagons provide the densest possible packing of buttons per unit area of keyboard, and also because (2) they facilitate pressing three adjacent buttons together with a single fingertip – perfect for triads – which rectangular layouts do not.
H and V can be defined in one of three ways:
1) Number of semi-tones
2) Number of cents
3) Diatonic interval
The use of 12-tet semitones is presumed below unless I specifically state otherwise.
The Janko and Chromatic Button Accordion (CBA) B-System and C-System layouts can easily be described as hexagonal isomorphic layouts:
So can another layout, first patented Switzerland in 1896 by Kaspar Wicki, and subsequently patented in Great Britain by Brian Hayden (who did not know of Wicki's patent, as it was Swiss and in German).
Wesley patented another layout in the USA as recently as 2002:
In Wesley's layout, the wholetone row is along a diagonal – i.e., it has the shape that a minor third has on the Wicki/Hayden note-layout.
Note that if one swaps H and V in any of the above layouts, one produces a rotation of the mirror image of the same layout.
If one restricts the values of H and V to the range [-12…0…+12] chromatic semitones (a range of 25 possible values for each of H and V), there are (25*25=) 625 possible combinations. That's 625 rectangular layouts, and another 625 hexagonal layouts. Most of these are trivial, but some, such as those listed above, are musically useful.
One such useful layout is produces triads at its vertices:
It is more general to define such layouts in terms of diatonic intervals rather than chromatic semitones. For example, the Wicki/Hayden layout could more generally be defined by saying H = M2 (major second) and V = P4 (perfect fourth). Such a definition remains valid in any equally-tempered tuning which has a recognizable diatonic scale, such as 19-tone equal temperament (19-tet) and 31-tone equal temperament (31-tet). The names of the notes in the layouts do not change between such tunings; only the number of cents between them changes.
On the other hand, a layout can be defined more precisely by specifying the number of cents in H and V. Again using the Wicki/Hayden layout:
In all three tunings in the table above, H = M2 and V = P4; however, the width of those intervals (in cents) differs between tunings.
The Wicki/Hayden layout has been used in a number of concertina and bayan instruments:
Paul von Janko, inventor of the Janko keyboard layout, was a mathematician. He and/or his contemporaries are very likely to have understood the above information, and it is doubtless well-understood by today's music theoreticians.