Graphs of Scales... a Sneak Peek

I've been experimenting with drawing graphs of scales for a while now, and have a few ideas on the subject; maybe even enough for an ebook one day. I'm particularly pleased with my latest batch, some selections of which I thought I'd share with you here. Consider them your Christmas present, valued reader.

I first put scales in their modal groups as I usually do, so that for example I'm interested in "all the modes of the Major Scale" as a single entity. For convenience I label this entity with a binary representation of one of the scales it contains; don't worry if you don't understand this, just consider the binary number to be a handy label for the modal group.

Now all the scales in a modal group share the same interval composition: the Major Scale, for example, contains 5 tones and 2 semitones, and so do all its modes; this is a consequence of what it means for a scale to be a mode of another scale. Interval composition makes a contribution to the way a scale sounds, so it's interesting to ask which other scales have the same interval composition. I grouped all possible modal groups together by this criterion and got 32 different groupings. Nothing new here.

For each of these 32 groupings I had some software calculate some relationships between their members. One was that two members are "alterations of each other" if I can get from one to another by sharpening or flattening one or more notes by a semitone. To help me see what was going on I had it draw some pictures: each picture contains all the modal groups that have the same interval composition, and two groups are connected by a line if they're alterations of each other.

I expected some obvious pattern to emerge but it didn't. In fact I ended up with an exotic garden of strange-looking forms. Some are little clusters of modal groups that are all disconnected; some are beautifully symmetrical; some are weird and random-looking. I don't know what this information means yet but I'm fairly sure nobody has ever seen these things looking like this before so I thought I'd show you a selection of them anyway. Merry Christmas!