Symmetrical 5-chords and 7-chords

In an earlier "Curiosities" post (link), I discussed the symmetry of the major scale: starting on scale degree ^2, the inversion function, "I", produces all the same notes. Here is the first example again:

Only a small percentage of scales/pitch-sets/pc-sets in the 12-tone equal-tempered system are capable of this. Here are the best known. The chromatic scale is obvious; it has all 12 notes in a half-step sequence: of course, inversion will create the same ones. The whole-tone scale is pretty much the same, but with whole steps. The diminished scale is a little different: it's not symmetrical on a note but between notes. At (d) I have drawn a line between C4 and C#4 and then applied inversion from C4 down and C#4 up. In fact, the diminished scale is so structured that you can do this by drawing a line between any two adjacent notes of the scale.

Thus there are two sorts of scale or set symmetry: (1) from a note, as in the chromatic scale, major scale, and whole-tone scale; (2) from between notes, as in the diminished scale. (The latter can also be understood in terms of note pairs, or C4-C#4 together here.)

For a small number of sets/scales/chords, you can apply these within: I call this "internal symmetry," where you can generate a set/scale/chord from a sequence half its size. It's not common, either, but among the 5-note sets that have it is the major dominant ninth chord:

Btw, you can't find symmetry (1) with the entire V9 chord from its root--if you try it with the G9 above, that is, invert from G4 downward, you'll get an F9 instead, see (a) below. That's because the axis of symmetry is, you guessed it, scale degree ^2, just like the major scale figure at the top of this post--it's D5 at (b) below.

If we take a "scale-size chunk" of seven notes, in this case the pc complement of C: V9--that is, the seven notes in the chromatic scale that aren't in this V9--you can see that the resulting scale is also internally symmetrical. It also happens to be what is often called the "melodic minor scale," here C# minor.

Here are seven additional 7-note sets, beginning with two that have "type 1" symmetry like the major scale. These are followed by five with internal symmetry.

What this table does is to begin to put both 5-34, the pc set of the major dominant ninth, and 7-34, the ascending minor scale and the complement of V9, into a bit more general pitch-design context.

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* The literature on scales and sets is understandably large. As I have explained a number of times, I am retired and don't feel the need to tot up the usual scholarly citations. I can say (a) that very little if anything I've said above is new, but that's not the point--it is to gather and discuss information in terms of the blog's topic; (b) if you want to explore more, the literature on set theory, pitch-class sets, and symmetry or symmetrical relations is the best place to go. Joe Straus's atonal theory textbook is the standard. Of online sites, I particularly like this one: "A Brief Introduction to Pitch-Class Set Analysis" (link) from Mount Alison University in Sackville, New Brunswick. It was made more than 20 years ago but is still available, informative, and easy to use.