Isomorphism & diatonic set theory

There are lots of isomorphic note-layouts -- for example, the Bosanquet, Fokker, Janko, Wesley, Chromatic Button Accordion (B-system and C-system), and Wicki.

JIMS uses the Wicki note-layout for a variety of reasons that are beyond the scope of this post.

The Wicki note-layout is proving to have some interesting mathematical properties. For example, consider any well-formed scale constructed by stacking N tempered perfect fifths and subtracting octaves (an "alpha-reduced beta-chain," where alpha is the octave and beta is the tempered perfect fifth), and N is the "cardinality" of the scale (that is, the number of notes in the scale).

The Wicki note-layout appears to be unique in that such well-formed scales are always tightly packed together on the keyboard, with no "holes" between the notes of the scale.

For example, consider the well-formed scale of cardinality 5 (pentatonic). It's notes [Do Re Mi So La] form a single tight group that (a) has no "holes" in it, and (b) is symmetrical around Re.

The well-formed scale of cardinality 7 (diatonic) is likewise tightly grouped and centered.

So is the well-formed scale of cardinality 12 (chromatic). Notice that both Le and Si are included, which is redundant; they represent the same note in the 12-tone well-formed scale, whether in 12-tone equal temperament tuning or not. I've just included both in the drawing for symmetry. The chromatic scale is the only well-formed scale with even cardinality (well, among those scales with cardinality less than or equal to 19, anyway), which is kinda messing with my head a bit.

And so on, for the well-formed syntonic scale of cardinality 17:

...and 19:

...and 21:

...and so on, ad infinitum.

To put it another way, the Wicki note-layout appears to be unique in that, to increase the cardinality of the syntonic scales playable on a Wicki note-layout, all one needs to do is add more notes to the left & right edges of the note-layout.

The other isomorphic note-layouts do not share this property. Their design intermingles scale notes and non-scale notes. As a result, they do not present the same pattern of notes for well-formed scales of all cardinalities.

By way of comparison, consider the Chromatic Button Accordion's C system note-layout (CBA-C), shown at right.

The CBA-C layout works fine for the chromatic scale, but if you wanted to use it exclusively for the pentatonic or diatonic scales, the note-layout would be full of holes. Alternatively put, neither the pentatonic nor diatonic note-sets map to compact, contiguous button-sets in the CBA-C note-layout.

Likewise, look at the line of "semi-tones" running up-and-rightwardly from C on the CBA-C note-layout. If one wanted to put the Db and C# on separate buttons there's no room. There's only one button-space between C and D; if has to serve for both Db and C#. The CBA-C note-layout does not have a clean "edge" to which the Gb could be added, as the Wicki note-layout does. As a rule of thumb, any note-layout with a contiguous line of "semitone"-controlling buttons has the chromatic scale "baked in," because the "semitone" is only a meaningful concept in chromatic scale (i.e., in the well-formed scale of cardinality 12). In scales of cardinality higher than 12, there is no "semitone." There are augmented unisons and there are minor seconds, but there are no semitones.

Now, look back at the patterns that well-formed scales make on the Wicki note-layout. These patterns all share three characteristics:

(a) They have no "holes" between the notes of a scale of given cardinality.

(b) They are symmetrical around Re.

(c)  All of their notes fall on adjacent rows, with one row being one button/note wider than the other (including the chromatic/12, because I included both Le and Si, which is cheating, just a little).

On the other hand, one can see (using the scale chooser on the interactive keyboard below) that non-well-formed scales, such as the Neapolitan, Melodic, Harmonic Major, Marmonic Minor, and Double Harmonic Minor, do not share all of these characteristics.

This suggests that there is some common element that is shared by (a) the definition of well-formedness and (b) the definition of the Wicki note-layout. I do not yet know what that common element is, but it's pretty obvious that it's in there somewhere.  (I think that it has something to do with the fact that on the Wicki note-layout, the "beta-stack" corresponds directly to one hexagonal line of note-controlling buttons, and the "alpha-stack" corresponds directly to a second, semi-perpendicular line.  But I'm not sure.)

If you can shed any light on this common element, please don't hesitate to let me know.  :-)