Last time I discussed symmetry and introduced the 7 (out of 38) 7-note sets/scales/chords that have it, besides the chromatic set and the major scale. Here are the corresponding 5-note sets, again in the interest of contextualizing the pitch content of 5-34, which is the major dominant ninth chord.
All are "internally symmetrical" as defined in the previous post, and all are "completely symmetrical" by (1) some single note, or (2) the gap between two adjacent notes, as defined in the previous post. Though it's just another way of conceiving rule 2, a rule (3) makes a "double axis"--two notes at the center. Here are examples: type (1): 5-15 & 5-22; type (2): 5-34; type (3) 5-8.
To expand the contextualizing even more (and, granted, to wander a bit off-topic for this blog), here are some comments on these sets. Traits of most are clear, but a couple are more complex. The diminished triad frames 5-8 and 5Z12. 5Z17 as shown in the first example above is five notes of a C# minor scale (if C4 is B#3), 5-33 is most of the whole-tone scale, and 5-34 is the major dominant ninth. 5-15 has four notes of a whole-tone scale, here C4-D4-F#4-G#4 (if C is B#, this is the "French +6"), but also three chromatic notes and a quartal/quintal chord as G#-C#-F# or F#-C#-G#. 5-22 seems like a mash-up of three triads: C major, c# minor, and C+. And 5Z37 tucks a chromatic fragment D#-E-F into the middle of an augmented triad. The point of interest is that the properties of each of these sheds light on the 7-note complement.
Here are two additional ways to think about the 5-note sets: as written and voiced chords, and in brief musical passages. First, the 5-8 below shows something not so obvious in the scalar version: this is a D9 with both 9 and b9.
Using the voicings of the chords above, here are short musical examples incorporating them (and where I can manage it, including V9 chords).
