I don't have a lot of formal music training so this may well be a dumb question, but I've been looking at this off and on for a while and I can't figure out why this particular scale doesn't seem to have a widely used name.
I'll over-explain my question a little bit to avoid confusion in case I misuse a term of art:
Musical scale families are often described by an interval "formula" or pattern that specifies the number of semitones between each note in the scale.
For example, a major scale is defined by the interval pattern "whole, whole, half, whole, whole, whole, half" where a whole step represents 2 semitones (e.g., from C to D, skipping over C#/Db) and a half step represents 1 semitone (e.g. from D to D#/Eb, with no notes in between). We can follow that interval pattern (starting at an arbitrary root note) to generate a major scale in any key.
Numerically we might write that as 2,2,1,2,2,2,1. Any sequence of numbers like this that adds up to 12 wraps back to the root note (pitch class) and repeats itself, creating a one-octave scale.
In addition to the more conventional major, minor, pentatonic, blues, etc. scales, there are a number of scales that seem more inspired by abstract mathematical symmetry, like the whole tone scale (all whole steps: 2,2,2,2,2,2) or the four tone symmetric scale (all minor-thirds: 3,3,3,3).
In particular, a repeating whole-step/half-step pattern (i.e. 2,1,2,1,2,1,2,1) is known as a "diminished" scale, and if you offset that pattern by one (i.e., 1,2,1,2,1,2,1,2) that's known as a "Half/Whole Diminished" scale, for obvious reasons.
Similarly, an "augmented" scale is defined by a minor-third-step/half-step pattern (i.e., 3,1,3,1,3,1).
Logically, you could offset that pattern by one also, creating the interval pattern 1,3,1,3,1,3.
Following the naming convention used for the diminished scale you might call this a "Half/Minor-Third Augmented" scale. But virtually nobody seems to do that. In fact I'm not able to find any commonly accepted name for the scale defined by the interval pattern 1,3,1,3,1,3. My question is: why?
I recognize that this is literally a symmetric scale - i.e., that there are only 4 distinct augmented scales (since C~E~Ab, Db~F~A, and so on) - but the same thing (or a very similar thing) is true for the diminished scale (and a bunch of others). So why are the "whole/half" / "half/whole" modes of the diminished scale relatively well known while the "half/minor-third" mode of the augmented scale seems to be so obscure?
Is there another name for the scale defined by the interval pattern 1,3,1,3,1,3? I.e., what would you call the scale in the key of C that contains the notes C, Db, E, F, Ab, A, [C]? I'm not able to find a commonly accepted name for this scale (and it seems like "half/minor-third augmented" is the obvious one, and I can't figure out why that's not used).