Is there a theory of heptatonic scales with 1.5-step intervals?

Question

I've been alternating between reading about music theory on Wikipedia and playing around with the scales that I've learned. The "modern Western modes" are defined systematically with their constant pattern of W W H W W W H, which are just transposed all 7 possible ways for this asymmetric pattern of whole (W) and half (H) steps. All of them sound nice except Locrian, which doesn't have a perfect fifth. In mathematics, we'd call this a closure of a group, starting with one element (W W H W W W H) and an operation (permute by shifting left, with wrap-around: W H W W W H W would be next).

However, other scales sound nice that cannot possibly be generated this way because some steps are "whole-and-a-half," which I've found is notated "+" or "WH" instead of "W" or "H" (and known as an augmented second). It's important to point out that they're 7-note (heptatonic) scales, not just "gapped" scales by leaving out some notes, like a pentatonic. I ran into two of these and found their names through this StackExchange:

• Double harmonic, also known as Flamenco mode, which is H + H W H + H.
• Hungarian minor, which is W H + H H + H.

and then another by just following these pages:

• Phrygian dominant, which is H + H W H W W (differing from the Double Harmonic/Flamenco only in the 7th).

These are outside the Western system and they "sound exotic" to an American like me, but not unheard of. Those same Wikipedia articles give other names for them: Mayamalavagowla, Bhairav Raga, Byzantine scale, Arabic (Hijaz Kar), Gypsy minor, Freygish scale, Hijaz-Nahawand or Hijaz maqam, Dastgāh-e Homāyoun, Hijaz Bhairav (Basant Mukhari), and Vakulabharanam. So a lot of cultures use these scales, but not so much to the northwest of Hungary.

My question is, are scales with this 1.5 interval (+) systemized? You can rotate them around like the modern Western modes and sometimes get a perfect fifth, like this with the Double harmonic:

1. H + H W H + H (Double harmonic), has a major third and a perfect fifth
2. + H W H + H H, has both minor and major thirds and a perfect fifth
3. H W H + H H +, has both minor and major thirds and a perfect fifth
4. W H + H H + H (Hungarian minor), has a minor third and a perfect fifth
5. H + H H + H W, has a major third, but no perfect fifth
6. + H H + H W H, has both minor and major thirds, but no perfect fifth
7. H H + H W H +, has no thirds and no fifth!

This unifies the Double harmonic and Hungarian minor in a similar way that the modern Western modes unify Ionian (major) and Aeolian (minor), through group closure, throwing away the combinations that don't provide a perfect fifth. Are the other two, the ones with both major and minor thirds, used anywhere? Or is it bad for a scale to be both major and minor?

Edit: Just a few hours after posting this, I encountered another scale with a 1.5 interval (+) in an English-language Agnus Dei in Catholic Mass. (I can't copy-paste it here: it's under a license.) Anyway, it was a Dorian flat 5 ending on E, so an E Dorian with the B (5th) replaced by a B-flat (no perfect fifth, but it doesn't sound "wrong."). The pattern is W H W H + H W, which can't be accessed by a rotation of the Double harmonic above. (It has only one +, rather than two.)

It seems that there are a lot of these. My question is whether there's any system or if it's like "stamp collecting," just finding one nice example after another.

Answer 1

Something is telling me you are not an avid listener of jazz and metal, judging by your list of scales that are "outside the Western system"...

You should be very careful with Western approximations of Indian and Arabic scales. Indian and Arabic scales do not fit that snuggly into the Western 12-TET system.

Another side remark: I don't know if it's just me, but I think it's rather uncommon to write down a scale's formula without explicitly notating the root of the scale. For instance, what you wrote down as W W H W W W H, I would write down as R W W H W W W H or, alternatively, 0, 2, 4, 5, 7, 9, 11. Otherwise you are not providing a starting point to count from.

What you are referring to as "modern Western modes" I personally know as "modes of the heptatonic major scale". Similar modes exist for all scales. And you have yourself identified those as permutations, which is basically what they are. A mode is a rotational transformation of a given scale.

One of the most common "Western" scales that has both a minor and a major third is the half-whole diminished scale. You will encounter it in jazz, metal and even "modern" classical music. If you're curious about the latter, that scale, for example, is Messiaen's second mode of limited transposition. For further research, you can refer to his "La technique de mon langage musical" ("The Technique of My Musical Language"), where he provides the reader with examples of how and why he uses his modes.

Harmonic minor has the + (W+H) interval you are so interested in. You might also be interested in researching the history behind this scale and how it came about in addition to the natural and melodic minor heptatonic scales used in Western composition. The hexatonic blues minor scale has 2 such intervals. If you want a heptatonic version of that, you can add a major second to it for a heptatonic blues minor scale.

For playing around with different scales you can use Ian Ring's Amazing Scale Finder (which I have already linked to a number of times in this post) which makes use of his equally amazing database of musical scales. If you read through an article for a given scale there, it will show you a number of different names for the given scale, its various formulas, permutations and so on. It will also show you the codes and names it is given within various musical scale systematisation systems (my apologies for the tautology). I'm assuming that's something you were searching for. Hopefully, I was able to help.