I had a thought today that ended up in a bit of a rabbit hole. This is one of those posts that's probably just pseudo-academic hocus-pocus but there are lots of weird chords and scales in it and maybe there's even something to the "theory" stuff too.

The General Idea

For the purposes of this post let's say a chord or scale is inversionally symmetrical if there's a mirror symmetry line in its clockface diagram. Here are a few examples with their lines of symmetry drawn in:

Note that Lulu is also transpositionally symmetrical but the other examples aren't.

As non-examples, consider the major and minor triads, which are inversions of each other:

This post is about taking things that aren't inversionally symmetrical and forcing them to be. There are a lot more chords and scales that aren't inversionally symmetrical, and as a rule the ones that are tend to be quite exotic. So we might think of this method as a way to cultivate rare flowers from common stock. Yet the Major Scale and the Melodic Minor Scale are both inversionally symmetrical, which we can call on as a kind of legitimation if we like -- if it's good enough for them, isn't it good enough for you too?

Finding Symmetry

I think there are two obvious ways to make a non-symmetrical thing symmetrical, alongside probably an infinite number of less obvious ways. Let's start with a heuristic for checking whether a scale has inversional symmetry (I'll stop saying "chord or scale" every time from now on).

We'll start at the "1" position in a clockface diagram and at the same time move one step clockwise and one step anticlockwise. We'll check the two resulting notes are either both in the scale or both no in it. If so, we repeat the process until our two notes collide at 6. Here's an example showing that the Mixolydian b6 scale (a mode of Melodic Minor) is inversionally symmetrical:

Note that you need to do this check for every starting position from 1 to 5 inclusive because this procedure will only detect inversional symmetry about the line through the starting-note and the one opposite it on the clockface (i.e. a tritone away). And we need to check the in-between positions too, to catch examples like Lulu where the symmetry line goes between two notes, as in this example of the Major 7 chord, whose inversional symmetry is only apparent is you start in the position in between the 7 and 1:

Doing this "check on both sides of the mirror" for every position is a very inefficient but very explicit algorithm for checking whether a scale is inversionally symmetrical to itself. The reason for developing it isn't to do the check, though -- that can be done much more easily -- but as a step towards forcing symmetry.

Incidentally, this is not just a paper exercise -- literally playing scales this way produces contrary motion that has a special magic when the scale is inversionally symmetrical, since then the two melodies move against each other in perfect inversion.

Minimal-Error Transpositions

Suppose we have a particular scale that isn't inversionally symmetrical, such as the pentatonic known as Kumoi:

If you start the "matching either side of the mirror" process at the 1 you'll find three pairs of notes that don't match -- b2 and 7, 2 and b7, 4 and 5. But if you start at the b3 you only find one non-matching pair, b6 and b7. And this corresponds to something you can probably see much more easily than put into words -- the mirror symmetry of Kumoi is broken, but only in one place:

If only the green note (the b6) were in the scale, it would be inversionally symmetrical -- there's nothing to stop us from adding it, of course. Or we could make it symmetrical another way: if we removed that note's mirror image, the b7. These are techniques we can use to force symmetry.

In order to do this sensibly we need to find that mirror line through b3 and 6 in Kumoi that gives us the fewest "errors" in the symmetry. That's quite easy for our pattern-recognising brains to do by eye. It's also quite easy to programme a computer to do it; the details are boring and we don't have to worry much about efficiency so I won't go into it here. For Kumoi, checking all possibilities gives us two mirror lines that each have the minimum number of errors (1, in this case):

I'll call these "minimum error transpositions". We could take an interest in the other options too, but these are the ones that allow us to use the least amount of force to turn non-symmetrical things symmetrical.

The Two Forcings

Once you have your non-symmetrical scale and its minimum error transpositions, it's pretty obvious what to do. For each of them you can either add the missing note (green in the diagrams above) or remove its mirror image.

If we add the note, we end up with the smallest inversionally symmetrical scale that contains the scale we started with. If the remove the note, we get the largest inversionally symmetrical scale that our starting scale contains.

Here's how that goes with the example of Kumoi we looked at in the previous section:

None of the results looks very familiar -- I did promise we'd be growing some weird plants by this method. Going left to right and identifying 1 with the note C we have:

• Db Major Scale without its 2
• Forte 4-8, a "contracted Lulu"
• Something monstrous
• Gb Maj 7 b5

I don't want to claim these four things have anything much in common but if I did it would be this relationship with Kumoi, which from that perspective would look extremely mysterious. Instead I want to think of these as four ways to think about Kumoi, if one were interested in that. Going down the "Remove" paths gives us subsets of Kumoi that may be interesting; the "Add" paths indicate notes that can be added to Kumoi to turn it into something else -- maybe something somehow more "complete".

Heptatonic Examples

To finish I'll show some examples of heptatonics that can be forced by removing a single note. All the heptatonics I tried had at least one minimal error transposition requiring only one note to be removed.

The resulting six-note scales are, I think, mostly unknown. There's something special about their sounds, especially when the axis of mirror symmetry is given a kind of tonal gravity, with the music tending to move away from and toward it in identical ways regardless of the up/down direction of travel.

Finally I present Salagam, the heptatonic that tries to be inversionally symmetric in so many ways but fails each time -- this is the only one I've found with four different minimum error transpositions:

I don't know whether any of this means anything or I'm just shuffling symbols around. I do think there's something audibly interesting about inversionally symmetric scales and I intend to have a look at the hexatonics specifically because you can find them in lots of common heptatonics and find pentatonics in them and that seems like maybe a fruitful way to look at things for a while. We'll see if it turns out to be any more than idle speculation...