Beyond Diatonic

Beyond Diatonic


For chromatic-cluster-free scales that span the octave and are made entirely of semitone and whole-tone (two-semitone) steps, there are only a few possible scale types: the diatonic scale, and some useful scales that play an important role in jazz theory.

The Puzzle

Given a small pile of two types of little wooden pieces, shaped like this ...

Two wood-colored pieces: one 30º with an X; one 60º with an O. With solid black (half) balls on the ends

... which we will call “X” and “O”—and given a few simple rules:

  • You have an endless supply of dozens of each type of piece, so that is not a constraint.
  • You may not put two or more “X” pieces next to each other.
  • You must make a complete 360º ring.

How many patterns of X/O pieces, ignoring rotation, can you find, that make complete rings and satisfy the rules?

For example, here is how the diatonic scale looks at different angles:

One pattern of pieces that corresponds to the diatonic scale Another pattern of pieces that corresponds to the diatonic scale

(This counts as one pattern.)

Mathematically-minded readers may notice that this puzzle is similar to asking which of the 77 integer partitions of the number 12 correspond to one or more X/O patterns that follow the above rules.

The Answer

Hopefully the analogy to scales is obvious, with “X” pieces corresponding to semitone steps, and “O” pieces corresponding to whole-tone steps.

For scales consisting of semitones and whole-tones, only four scale types are possible (click or tap here to spoil the puzzle and see pictures):

These scales, coupled with the chromatic scale, are the foundation of jazz theory and the focus of Harmonious.

From the previous section on Evenness & Clusters, we also see that these scales are at the heart of the action:

  • The whole-tone scale is perfectly even.
  • The diatonic scale is maximally even.
  • The acoustic scale is nearly as even as the diatonic scale.
  • The octatonic (diminished) scale is also maximally even.

In fact, there are no other cluster-free eight note scales, and the only other cluster-free seven-note scales are the two seven-note subsets of the diminished scale, and the harmonic major and minor scales (more below).


Neighboring Scales

A careful look at the diagram in the section on Evenness & Clusters (reproduced above) will reveal some nearby scales that almost make the cut, and which are not just simple subsets of the above four scales. (In every case, these are marked with a red triangle or double triangle; one has a dot to indicate its limited mode-count.)

The three scales that collect up all of the red-triangle set classes and their subsets are the following:

In both cases, the presence of the minor third makes these scales less regular and more uneven or ungainly, which may partially explain why they are less common in jazz theory. Overall they are fairly even and interesting harmonically, so their modes are still worth exploring.

The red triangle represents the fact that each harmonic major or minor scale has three scales that are each a semitone voice-leading step away: one harmonic minor or major, one acoustic (melodic minor), and one diatonic (major) scale.

The Rest of the Shapes

The remaining shapes in the evenness chart are easy to explain: