Harpejji

The Harpejji (see Wikipedia) is a tapping stringed instrument which is a physically constrained to be an isomorphic keyboard. It is a successor to the StarrBoard which was created by John Starrett.

The standard layout for a Harpejji is like this, with the open strings tuned in whole tones of 12edo, so that the seventh string is an octave above the first.

E  F# G# Bb c  d  e  f# g# bb c' d'
Eb F  G  A  B  c# eb f  g  a  b  c#'
D  E  F# G# Bb c  d  e  f# g# bb c'
C# Eb F  G  A  B  c# eb f  g  a  b
C  D  E  F# G# Bb c  d  e  f# g# bb

Note that the diatonic scale appears in two different shapes in this layout: an up-and-to-the-right shape:

                              c' d'
                     f  g  a  b 
               c  d  e 
      F  G  A  B 
C  D  E

and also a down-and-to-the-right shape, which is less compact:

E
   F  G  A  B
               c  d  e
                        f  g  a  b

The same general idea for a layout can also be applied to any chain-of-fifths EDO, but only one of these diatonic scale shapes will be preserved. Also, the shapes that unisons between different strings take will be different. (For a rank-2 chain-of-fifths tuning which is not an EDO, there will be no unisons on different strings at all.)

For example, a 19edo Harpejji layout (based on preserving the up-and-to-the-right shape of the diatonic scale) might look like this:

F  G  A  B  c# d# e# gb ab bb c' d'
E  F# G# A# cb db eb f  g  a  b  c#'
D# E# Gb Ab Bb c  d  e  f# g# a# cb'
Db Eb F  G  A  B  c# d# e# gb ab bb
C  D  E  F# G# A# cb db eb f  g  a

To interpret this, keep in mind that C-D is 3 steps of 19edo, E-F is 2 steps, and # and b modify pitches by 1 step. Thus there is a consistent spacing of 3 steps between adjacent strings (columns), and adjacent frets (rows) are 2 steps apart. (In contrast to 12edo, G#-Ab is not a unison but E#-Fb is.)

As you can see, the up-and-to-the-right (meantone) diatonic scale has exactly the same shape on the fretboard, and the main difference compared to the 12edo layout is that unisons are no longer a knights-move away on adjacent strings (2 up and 1 left). Instead, in 19edo the unisons are 3 up and 2 left. This means the down-and-to-the-right shape doesn't work the same way anymore, since from E you have to skip over E# to get to F.

An alternative layout for 19edo, based on preserving the shape of the down-and-to-the-right shape of the diatonic scale, looks like this:

Eb F  G  A  B  c# d# f# gb ab bb c'
D# E# Gb Ab Bb c  d  e  f# g# a# cb
D  E  F# G# A# cb db eb f  g  a  b
Db Eb F  G  A  B  c# d# e# gb ab bb
C# D# E# Gb Ab Bb c  d  e  f# g# a#
C  D  E  F# G# A# cb db eb f  g  a
Cb Db Eb F  G  A  B  c# d# e# gb ab

While the previous 19edo layout had successive frets spaced at a distance of 2 19edo steps from each other, this layout instead has every 19edo step as its own fret. This could be a disadvantage in that the spaces between frets might become too small for the fingers in the high register.