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 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.

Neutral third generator (split fifth)

In neutral third scales, the chromatic semitone is split in half. Unfortunately there is no good way to do this while preserving the up-and-to-the-right diatonic scale shape, but it's simple with the down-and-to-the-right diatonic scale shape. Using notation where ^ = (1/2)# and v = (1/2)b, the result looks like this:

C  D  E  F# G# A# B# cx
Cv Dv Ev F^ G^ A^ B^ c#^
Cb Db Eb F  G  A  B  c# d# e# fx
         Fv Gv Av Bv c^ d^ e^ f#^
         Fb Gb Ab Bb c  d  e  f# g#
                     cv dv ev f^ g^
                     cb db eb f  g

One drawback of this is the frets would be quite close together, for example in 31edo they would be only one 31edo step.

In a particular EDO, of course, there will be enharmonic equivalences, so there will be unisons between different strings. For example, 17edo is perhaps the simplest reasonable one which has a neutral third (sorry, 10edo!):

D^ F  G  A  B  dv ev f^ g^ a^ c' d'
D  E  Gv Av Bv c^ d^ f  g  a  b  dv'
Dv Ev F^ G^ A^ c  d  e  gv av bv c^'
C^ D^ F  G  A  B  dv ev f^ g^ a^ c'
C  D  E  Gv Av Bv c^ d^ f  g  a  b
Cv Dv Ev F^ G^ A^ c  d  e  gv av bv

Because of the particularly simple enharmonic relationships in 17edo, the up-and-to-the-right shape of the diatonic scale is restored in this one (because E^ = F).

The version of this one for 24edo happens to be exactly the same as the standard 12edo layout with a new fret in between each pair of 24edo ones.

D  E  F# G# Bb c  d  e  f# g# bb
Dv Ev F^ G^ A^ B^ dv ev f^ g^ a^
Db Eb F  G  A  B  c# eb f  g  a
C^ D^ Fv Gv Av Bv c^ d^ e^ gv av
C  D  E  F# Ab Bb c  d  e  f# g#
Cv Dv Ev F^ G^ A^ cv dv ev f^ g^

An alternative layout to the above (for any neutral thirds tuning) preserves neither shape of the diatonic scale but uses a new one where E-F and B-C wrap back to the left. It has the advantage of having frets that are a neutral second apart, which is actually even wider than in the standard 12edo Harpejji:

      db eb f  g  a  b
      cv dv ev f^ g^ a^
Gb Ab Bb c  d  e  f# g#
Fv Gv Av Bv c^ d^ e^ f#^
Eb F  G  A  B  c# d# e#
Dv Ev F^ G^ A^ B^ c#^
C  D  E  F# G# A# B#

The 24edo version of that looks like this:

A  B  c# eb f  g  a  b  c#'
G^ A^ B^ dv ev f^ g^ a^ b^ dv'
F# G# Bb c  d  e  f# g# bb c' d'
Fv Gv Av Bv c^ d^ e^ gv av bv c^'
Eb F  G  A  B  c# eb f  g  a  b
Dv Ev F^ G^ A^ B^ dv ev f^ g^ a^
C  D  E  F# Ab Bb c  d  e  f# g#

In this case the frets of a single string turn out to be exactly 8edo.