Scales Covered by Sus #4 or Sus b2

I've been exploring the Sus b2 and Sus #4 arpeggios a bit lately. They're inversions of each other and are an important sound in early 20th century classical music. Here are all the heptatonic scales I know of (there might be more) that can be completely covered by copies of one of these two structures.

There are some example voicings for Sus #4 and Sus b2 chords at the end of this post.

The scales I identified come in three pairs, each containing a scale and its inverse. As an aside, I've had a bit of a change of view about inversion lately. I used to think it was crazy that Forte et al considered two collections of pitches to be "essentially the same" if they were related by inversion. After all, the major and minor triads come out "essentially the same" on this measure, and isn't that completely tone deaf? But the more you deal with atonal material the more you realise the distinction between the two most common triads is actually pretty fine. There is a difference between a scale and its inversion, but it's not as huge a difference as our enculturation may have led us to believe.

And in the cases we'll look at, as it happens the two scales differ only in a single note, so they're as similar as two scales can be. This allows us to consider merging them into an eight-note scale as well. Note that inversions of scales needn't be as similar as these; I presume it's just a coincidence in this case although maybe it has something to do with the way a single three-note arpeggio can cover them (so they must be quite "regular" in some obscure sense).

Anyway, here are the three pairs. All the scales involved are Carnatic melakatas. In each case I'm only looking at modal groups, not individual scales, using one scale name to identify the whole group.

The first is Shadvidamargini (1,b2,b3,#4,5,6,b7) and its very similar inverse Ramapriya (1,b2,3,#4,5,6,b7). Shadvidamargini is covered by Sus #4 at the 1, b3 and #4; Ramapriya by the Sus b2 at the 1, #4 and 6. Note that in each case the root notes of the arpeggios form a diminished triad arpepggio.

Since they only differ by one note (b3 vs 3) you can merge these two scales into an octatonic, 1,b2,b3,3,#4,5,6,b7. This is our old friend Half-Whole Diminished so the coscale is the diminished 7 arpeggio built on the 2. There's a potentially rich little ecosystem here of Sus #4, Sus b2 and the coscale all related by sequences of minor thirds.

The second group is Vishwambari(1,b2,3,#4,5,#6,7) and its inverse, Hatakambari(1,b2,3,4,5,#6,7). In this case Vishwambari is covered by Sus #4 at the 1, 3 and b5 whereas Hatakambari is covered by Sus b2 at the 1, 3 and b7. Each of these patterns of roots can be thought of as a subset of the Whole Tone scale. This is Forte number 3-10 and probably deserves some study at some point, but not today.

The octatonic you obtain from merging them is 1,b2,3,4,#4,5,#6,7. Its coscale is 2,b3,#5,6, which is another interesting chord; this one comes from playing two tritones a semitone apart. It's Forte number 4-9 and again it's something I'd like to study more closely, since it comes up a lot in modernist composition.

The last pair is Salagam(1,b2,bb3,#4,5,b6,bb7) and Jhalavarali(1,b2,bb3,#4,5,b6,7). Salagam has a Sus #4 at the 1, b2, bb3, #4 and 5 -- a veritable feast of sus #4s! Of course, Jhalavarali has Sus b2s in the corrsponding positions: 1, b2, #4, 5 and 7.

The octatonic they cover together is 1,b2,bb3,#4,5,b6,bb7,7. Its coscale is b3,3,4,b7, an unpromising-looking structure (Forte 4-6) but one that at least is easy to find and play (it's the starting-note plus the two chromatic notes above it plus its fifth).

Rather than writing out chord scales for these, I think it's probably preferable to learn these by learning some voicings you like for Sus #4 and Sus b2 and then practicing finding them on the fly.

Here are some miscellaneous voicings to get you started, with the root at C. Many more are possible but as usual it's probably a good idea to pick a couple to focus on rather than trying to learn dozens of them at once. First Sus #4:

Then some for Sus b2: