Here's something I found on the web a few years ago: the isomorphic keyboard, which
presents each given tonal interval with the same simple, consistent geometric shape wherever it occurs.
See the patterns?
- The vertical axis = octaves;
- The horizontal axis = major seconds;
- Up-and-right hexagonal axis = perfect fifths; and
- Up-and-left hexagonal axis = perfect fourths.
A minor second is up-and-left-then-two-left (e.g., from B2 to C3).
The major scale, in any key, is tonic, right, right, up-and-left-then-two-left, right, right, right, up-and-left-then-two-left. That is, it's (from tonic) M2, M2, m2, M2, M2, M2, m2. Doesn't matter what pitch the tonic is, the major pattern on the keyboard is always the same. The other modes of the diatonic scale all have the same circular pattern, just starting on a different place in the circle (of course).
Check out the diatonic triads. All of the major triads have the same shape. Look at C-E-G, F-A-C, & G-B-D -- they all have the same shape, of an upward-pointing triangle. Same within a given key, same across all keys. Minor triads all have the shape of a downward-pointing triangle. The diminished fifth, B-D-F, is a single diagonal line (of two stacked minor thirds) slicing right through the diatonic scale.
What's going on here?
The whole note-layout is defined by just two intervals: the up-and-right hexagonal axis of perfect fifths, and the vertical axis of octaves. All other intervals are defined by a combination of movements along these two axes. For example, a major second is two perfect fifths up-and-right, minus an octave; a major third is four perfect fifths up-and-right, minus two octaves; and so on. All of the notes' locations can thus be derived from a combination of perfect fifths and octaves. Music theorists will recognize this as being the means by which the Pythagorean and meantone tunings are derived (and also 12-tone equal temperament, if the perfect fifth is tempered to 700 cents). In fact, this keyboard note-layout presents every tonal interval consistently across a very wide range of tunings, including the inharmonic indigenous tunings of Indonesia, Thailand, and Mandinka Africa. Thus, the keyboard's note-layout captures, in its geometry, something deeply fundamental about the structure of music. The keyboard is thus a form of tonnetz, or tonal lattice, as used in neo-Riemann analysis, and as implicated in research into music cognition.
I don't know about you, but I think that that this keyboard is pretty cool. It makes the underlying consistentcy of music's patterns quite obvious, don't you think? Could this be a useful tool in music education?