Generalized keyboards, regular temperaments, and MOS

I've been looking at some of the different ways to map different temperaments and MOS scales to a generalized keyboard (and if you can map a temperament, you can also use it for Fokker periodicity blocks). I think it would be helpful to have a common way to refer to these different keyboard layouts, and Erv Wilson's terminology (as for instance in "Keyboard Schemata from the Scale-Tree", http://www.anaphoria.com/key.PDF) seems as good as any.

So for instance, the "2/5 keyboard" has a generator of 2 steps of 5-ET, or 2 steps of a 5-note MOS (which has a pattern of large - small - large - small - small or small - large - small - large - large steps). This could be tuned as a pentatonic D - F - G - Bb - C - D, or as D - Ev - Gv - G# - B - D with alternating 1.5-step and minor third intervals. But alternatively, the generator could be 3 steps of an 8-note MOS (as in a semisixths scale). If you look at the chart on page 2 of Wilson's paper you can see that the octave is either 2, 3, or 5 rows away from the root (depending on which axis you follow). So you might expect that scales having 2, 3, or 5 steps of one size (and a particular number of steps of a different size) would be most easily adaptable to this keyboard, e.g.

...E..A..D diatonic (5+2)
.D..G..C 5 large steps, 2 small steps
..F..Bb

.E#.G#.B..D hanson (4+3)
..D..F..Ab.Cb 4 large steps, 3 small steps

But scales with different numbers of steps are less convenient, such as this 15-note hanson scale (4+11 steps).

.E..G..Bb.Db
.....E#.G#.B..D
......D..F..Ab.Cb
..........D#.F#.A..C

So the 2/5 keyboard has a considerable amount of flexibility in the way the key mapping can be assigned, but other keyboard layouts are more convenient for some purposes. For hanson temperament (and others such as myna which share many of the same characteristic MOS scales), a keyboard with the octave four rows away from the root (such as the 2/7 keyboard) would be more appropriate.

May I point out one additional feature of generalized keyboards?
Because each instance of any harmonic configuration shares the same
fingering (when the generalized keyboard is viewed as wrapping around
itself), or, as Wilson puts it, template, generalized keyboard
mapping is a superb model of both harmonic structures, their
transpositions and inversions, and of voice leading. Any
keyboardist is well-aware, intuitively at least, of the usefulness of
keyboard figuring in composing parsimonious (efficient) voice
leadings; with a generalized keyboard, one gets both this and an
added degree of structural transparency due to the identical
templates. There have been numerous models proposed for mapping
harmonic activity, from lattices to orbifolds, but two-dimensional
generalized keyboards seem particularly efficient in this regard.

Daniel Wolf
Frankfurt

May I point out one additional feature of generalized keyboards? Because each instance of any harmonic configuration shares the same fingering (when the generalized keyboard is viewed as wrapping around itself), or, as Wilson puts it, template, genralized keyboard mapping is a superb model of both harmonic structures, their transpositions and inversions, and of voice leading. Any keyboardist is well-aware, intuitively at least, of the usefulness of keyboard figuring in composing parsimonious voice leadings; with a generalized keyboard, one gets both this and an added degree of structural transparency due to the identical templates. There have been numerous models proposed for mapping harmonic activity, from lattices to orbifolds, but two-dimensional generalized keyboards seem particularly efficient.

Daniel Wolf
Frankfurt