The piano keyboard is surprisingly poplar for how awkward it is. It puts all the keys into effectively a single row, which means you often need to move your fingers a very long way:

A four foot single-row keyboard is kind of nuts. We could get it down to 3.5 feet or 3 feet by going from piano key spacing (23mm) to accordion (20mm, or 18mm for a "ladies model") but that's still pretty big. [1] Putting in multiple rows for multiple octaves, however, can adramatically increase how many notes are under your fingers:

It turns out there are quite a few systems for laying out notes onto a hexagonal grid like this, so here's a survey of which instruments use which systems. I'm only looking at isomorphic keyboards, which are key-independent.

Chromatic Button Accordion

Chromatic button accordions have two main layouts, B-system (Moscow system) and C-system (Western system):

B-type

         D   F   Ab  B   D   F   Ab  B
      F#  A   C   Eb  F#  A   C   Eb
    E   G   Bb  Db  E   G   Bb  Db
  D   F   Ab  B   D   F   Ab  B
C   Eb  F#  A   C   Eb  F#  A
             [hand]

C-type

        E   G   Bb  Db  E   G   Bb  Db
      Eb  F#  A   C   Eb  F#  A   C
    D   F   Ab  B   D   F   Ab  B
  Db  E   G   Bb  Db  E   G   Bb
C   Eb  F#  A   C   Eb  F#  A
             [hand]

Both of them have increments of 1.5 steps horizontally, half steps along one diagonal and whole steps along the other diagonal. To tell them apart from looking, a B-system keyboard will have the river of black keys angled up to the right while the C-system will have it angled up to the left. Only three rows are necessary to get all the notes, but by repeating the first two rows fingering options are much improved. On the C-type pitch goes up as you move to the right, rising an octave every four buttons, and the B-type is the mirror image.

(I'm wondering what A-type was? Maybe something key-dependent?)

Accordions using these layouts are often just called "chromatic button accordions", but specific regional types include the Bayan and Schrammel.

Wicki-Hayden

The Wicki-Hayden layout was traditionally used for concertinas, though it wasn't one of the most popular layouts, and has recently been a popular choice for people designing new instruments.

  G   A   B   Db  Eb  F   G
C   D   E   F#  Ab  Bb  C
  G   A   B   Db  Eb  F   G
C   D   E   F#  Ab  Bb  C
  G   A   B   Db  Eb  F   G
C   D   E   F#  Ab  Bb  C
         [hand]

It has whole steps along each row, fourths on one diagonal and fifths on the other. Pitch goes up as you move toward the top, rising an octave every two rows.

Harmonic Table

The harmonic table layout is another one popular among people making electronic instruments, and was also used on the Harmonetta.

        C   E   Ab  C   E   Ab  C
      Eb  G   B   Eb  G   B   Eb
    F#  Bb  D   F#  Bb  D   F#
  A   Db  F   A   Db  F   A
C   E   Ab  C   E   Ab  C
          [hand]

It has 2 steps along each row, 1.5 steps along one diagonal, and fifths along the other diagonal. Pitch goes up as you move toward the right, rising one octave every three keys. Often it's rotated so that the fifths run vertically instead.

Janko

The Janko layout was designed for pianos and is:

  Db  Eb  F   G   A   B   D
C   D   E   F#  Ab  Bb  C
  Db  Eb  F   G   A   B   D
C   D   E   F#  Ab  Bb  C
          [hand]

It runs whole-steps along each row and each diagonal is a half-step. Pitch goes up as you move to the right, rising one octave every six keys, which means it's not much of a compactness improvement over a regular piano.

Fifths

I don't see anyone proposing making something laid out like a violin or mandolin but it seems pretty practical:

          B   C   Db  D   Eb  E   F
        E   F   F#  G   Ab  A   Bb
      A   Bb  B   C   Db  D   Eb
    D   Eb  E   F   F#  G   Ab
  G   Ab  A   Bb  B   C   Db
C   Db  D   Eb  E   F   F#

Pitch goes up as you move up and to the right, with half steps along the rows, and fifths up on the right diagonal. The left diagonal is the flat fifth, which isn't especially useful musically, but is half an octave in a way that gets you an interesting pattern.

The main advantages is this would be easy to switch to from violin/mandolin, plus we already know they work well for people in practice. They're equivalent to playing with no open strings, though, which takes some getting used to, and is more awkward, which suggests something simliar built around fourths (like a bass) would work better:

          Db  D   Eb  E   F
        Ab  A   Bb  B   C
      Eb  E   F   F#  G
    Bb  B   C   Db  D
  F   F#  G   Ab  A
C   Db  D   Eb  E

Octaves

And, just for completeness, let's take this to the extreme of just stacking the octaves:

        C   Db  D   Eb  E   F   F#  G   Ab  A   Bb  B
      C   Db  D   Eb  E   F   F#  G   Ab  A   Bb  B
    C   Db  D   Eb  E   F   F#  G   Ab  A   Bb  B
  C   Db  D   Eb  E   F   F#  G   Ab  A   Bb  B
C   Db  D   Eb  E   F   F#  G   Ab  A   Bb  B

Pitch rises to the right and up, with an octave every row. I don't think this would actually be a good layout, I just want to compare it to the others in the section below.

Summary

In terms of getting more notes in a smaller space these different options trade off width and length. Here's a table showing how many rows it takes to move an octave:

Piano	7
Janko	6
Chromatic Accordion	4
Harmonic Table	3
Fourths	2.7 (cross-angle)
Wicki-Hayden	2 (cross-angle)
Fifths	1.91 (cross-angle)
Octaves	0.96 (cross-angle)

All of these except the Janko dramatically reduce how much space it takes to move octaves because they make much better use of being two dimensional. On the other hand, the octaves layout probably goes too far in that direction, which makes me wonder which other ones do as well. The most widely used layout here for real musicians playing keys/buttons is, I think, the chromatic accordion. But wide use only tells you that something is workable; as the piano demonstrates it doesn't show that it's good! Still, if I were going to put effort into learning one of these systems I think I would learn that one.

(As far as I can tell, you can't actually buy something like this to play with. What I want is a set of velocity-sensitive keys laid out in a hex grid to play with, though I could consider something like a breath controller plus the right hand button setup from a midi accordion like the Roland FR-7xb. The Axis-49 looks like what I want, except it's discontinued and I can't find used ones anywhere.)

Update 2018-05-15: This came up at work today, and after saying how sad I was that they weren't available I looked again and found a used Axis-49. Ordered! There are two more if you're interested in getting one too. It's harmonic table by default, but I'll need to decide what layout I want to use.

[1] Unfortunately keyboards with narrower keys are all either very expensive or junky. [1] I think part of what's going on is that physically small instruments are a way for manufacturers to save money, so people making nice instruments and looking for nice instruments have "full sized keys" as a checkbox. That, and experienced players being very used to the standard spacing and not interested in switching.