Isomorphic Keyboards

The technical term, used (we suspect) to either scare off non-geeks or to sound intellectual is isomorphism. (What do you want to bet we'll one day hear the term in a science fiction show: "he's gone isomorphic, Jim"). Another name is self-transposing system. 

We prefer the term consistent-interval, or simply consistent, as in consistent keyboard. In such it turns out that, by design, every musical interval, for example a major third,
always travels makes the same angle and distance. Further It turns out that any consistent keyboard system always makes the same shape, a triangle, when you make a 3-note chord. This is easy both on the fingers and on the ear.

For starters, here's the rather technical Wikipedia entry: Isomorphic_keyboard

Brett Park, on his Shiverware website, offers the chance to try out any isomorphic keyboard in a simple program that runs on the iPad et al.
He has studied the designs a fair bit, and has these pocket summaries, reproduced here with permission: 
  • Gerhard (- Maj. 3rd; + Min. 3rd; Vertical) Designed specifically for the iPad and Musix, this layout attempts to balance the advantages and disadvantages of other layouts listed below. Scales are tight and chords fit naturally on the fingers.
  • Park (- Min. 3rd; + Maj. 2nd; Horizontal) A variation on the Wicki-Hayden (listed below), this layout, white notes are together making patterns easier to see, and scales can be played by sliding fingers along the screen. Alternate and jazz chords are compact, which means this pattern sounds great when button mashing.
  • Wicki-Hayden (- Perf. 4th; - Perf 5th; Horizontal) This is the classic layout used for centuries on Accordions and Concertinas. White notes are together, and the I-IV-V pattern of many many songs forms a compact triangle. Scales play linearly left-to-right (like a piano) with jumps for each semitone. Whole-tine scales are linear. Simple chords are comfortable, more complex chords are more challenging. This is also the preferred layout for "jammer" instruments.
  • Harmonic (- Maj. 3rd; - Min 3rd; Horizontal) Also called the Harmonic Table, this relatively modern layout was developed and popularized by C-Thru Music, the makers of the excellent Axis line of isomorphic MIDI keyboards. The harmonic table provides impossibly easy chords, with major and minor triads in triangles playable with a single finger. Scales are somewhat cumbersome, but as with all layouts, once learned in one key, all other keys are the same pattern.
Jim Plamondon has a succinct and easy to read analysis, but it does not detail the implications: Jim Plamondon on Isomorphic keyboards

Brian Hayden, who (re-)invented the isomorphic concept, and especially the Kaspar Wicki/Brian Hayden key layout did a deeper evaluation of isomorphic keyboards, which has been collected and edited here: Hayden on Isomorphic keyboards. It would be greatly appreciated if someone would reduce and simplify his ideas, perhaps into table form.