Symmetrical 5-chords

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 dominant major-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).

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 or "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 dominant major-ninth chord, 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.

More Curiosities: Major-Key Symmetries

 The topic today is another in the series of curiosities. Earlier posts were these: Ascending resolutions of the ninth: link; Extending downward: A curiosity: link; and 5-34 and its hexachords: link1, link2.

The major scale is symmetrical, that is, it has the same interval sequence going up and going down--but not on its tonic note. Instead, it is on scale degree ^2, so perhaps we should call it "Dorian symmetry." 

This is relevant to the dominant major-ninth chord as follows. If you build a ninth chord on the tonic (^1), the result is M9. Taking D4 or ^2 in C major as the axis of symmetry, as above, the inverse is d9, or m9: see the first pair under (b1) below. It follows, of course, that if you build a minor ninth chord, you'll get a M9 as the inverse: the second chord pair, which in this case happen to be the same ones: d9 and CM9. Under (b2) are additional chord pairs: the minor(flat 9) on E4 inverts around D4 to Bø(b9), and vice versa. The point of interest--the curiosity--is that the only ninth chord inverting to itself is V9: see under (b3).

Here are a few extra bits. Under (c1), ninth chords on the seven scale degrees in C major are given in the treble clef. Their inversions around the root of each chord are given in the bass clef. As expected, the only inversion that produces a diatonic chord is the one around scale degree ^2. Under (c2), all the inversions are shown with a root C3 to facilitate comparison.

Simple shifts toward IV happen by changing B to Bb, toward V by changing F to F#. These can also be understood symmetrically, as in (d1). Under (d2), the axis of symmetry and the altered notes are isolated. Under (d3), see the two dominant major-ninths that result: V9/V and V9/IV. A view through the circle of fifths is under (d4), where Bb1 is the fourth note below D4 and F#6 is the fourth note above.

Note: I show symmetry of the dominant major-ninth chord within the harmonic series in the post Harmony at the Ninth: link.