I was thinking about symmetrical scales last night and this morning I woke up with not one but two ideas for other kinds of symmetry a scale can have besides the usual one. Here I'll describe the first one, which I call "reflection".
Example of a Reflection
We we saw in an earlier post, a scale is best thought of as an interval map. Here's the interval map of the Major scale (I've illustrated it with the notes from the C Major scale, but the Major scale in every key has the same interval map):
C D E F G A B C 2 2 1 2 2 2 1
So we can write a major scale in shorthand as 22122221. The reflection of this scale is just that map written backwards: 1222122. This gives us the following notes:
C Db Eb F G Ab Bb C 1 2 2 2 1 2 2
This is the C Phrygian scale, which happens to be a mode of the Major scale.
We get a similar result if we try reflecting the Common Pentatonic, the Harmonic Minor and the Melodic Minor -- in each case the reflected scale is a mode of the original.
Scale Reflections and Scale Group Reflections
There are scales that are "perfect reflections", where the reflected scale is exactly the same as the original, rather than a mode of it. An obvious example would be the Whole-Tone scale, whose interval map is 222222, but we don't have to go to traditionally symmetrical scales to find examples.
Consider the Neapolitan scale, whose interval map is 1222221; the reflection is also 1222221, which is identical. Other modes in the group, though, don't exhibit this property.
We can say that a scale group (i.e. a set of all the modes of a given scale) has reflective symmetry if and only if any scale in it, when reflected, gives another scale in the same group. Additionally, we can say that a scale has reflective symmetry if and only if it is its own reflection.
Hence the Major scale group has reflective symmetry and the Neapolitan scale has reflective symmetry, but the Major scale itself does not have reflective symmetry. Clearly if a scale in a group has reflective symmetry then so does the whole group, but the converse is not true (the Major scale group is a counterexample).
Asymmetric Scale Groups
All of the common scale groups we have looked at have exhibited reflexive symmetry, which poses the question of whether all scale groups in fact have this property. It turns out that they do not.
The pentetonic Pelog scale, for instance, has the interval map 41241, and its reflection 14214 is not in its group. The same is true of the Kumoi scale, whose interval map is 14142. In fact, weirdly, the reflection of the Pelog scale is a mode of the Kumoi scale. Since a reflection of a reflection is just the scale you started with, the reflection of the Kumoi must be a mode of the Pelog, too.
In this case I would be inclined to say that the Pelog scale group is a reflection of the Kumoi scale group (or vice versa) but not to speak of symmetry, since these are quite different scales. I would say that the Pelog scale group is an asymmetric group with respect to reflection.
Reflexive and Rotational Symmetry
We might call the usual kind of scale symmetry "rotational symmetry", since it involves "rotating" the interval map and getting the same result as we started with. The usual symmetrical scales -- the Whole-Tone, the Chromatic and the Whole-Half and Half-Whole Diminished scales -- are rotationally symmetric.
They also happen to form reflexively symmetric scale groups, which raises the question of whether rotational symmetry implies reflexive symmetry. It turns out it does now.
Looking at the complete list of symmetrical scales in the Encyclopoedia we can see that the 4min + 7min scale group is not reflexively symmetrical; the same is true of the 2maj + #5maj and the 1dom + b6maj. These are uncommon scales but they are indeed rotationally but not reflexively symmetric.
I honestly have no idea at this stage whether this observation has any practical applications. We could expect some similarity in sound between scale groups that are reflections of one another, just as we can of scales that are modes of one another. Yet we know from the latter case that these similarities are not very great: the Locrian and Lydian scales have something in common in their sounds, but not a great deal.
On the figerboard the relationship is visually obvious. Here is a fingering of a scale from the Pelog group alongside one from the Kumoi group as they appear in the Encyclopoedia:
Ignoring the colour of the dots for a moment, we can see that the second pattern appears to have been rotated by 180°. The root notes (white circles with black dots) are even in the same places. Could this help you to learn the Kumoi scale group if you already knew the Pelog scale group? Possibly, although it doesn't seem very natural to me.
A final application is purely classificatory. It's this that really sparked my thinking, because I'm trying to properly organise the material on hexatonics for the new edition of the Encyclopoedia and this relationship is proving extremely useful for that.