Heptatonic scales with a minor third

Which heptatonic scales consist entirely of semitones, whole tones, and a single minor third, without having two semitones in a row?

The heptatonic scales that consist entirely of semitones and whole tones without having two semitones in a row are the diatonic and the melodic minor. Both contain two semitones and five whole tones. There are only two ways of dividing the five whole tones into two groups.

2-2-1-2-2-2-1 (diatonic: two and three whole tones)
2-1-2-2-2-2-1 (melodic minor: one and four whole tones)

Subtracting the minor third from the octave leaves nine semitones remaining. All of our candidates must contain three semitones and three whole tones (3 × 1 + 3 × 2 = 9), because the other possibilities (five semitones and two whole tones, or seven semitones and one whole tone) unavoidably have at least two semitones in a row.

If the minor third’s neighbors are both semitones (1-3-1), what remains are a semitone and three whole tones, which can have only two viable configurations: 2-1-2-2 (the harmonic minor) and 2-2-1-2 (the harmonic major). All other configurations would place a semitone on the outside, where it would abut one of the minor third’s semitone neighbors, producing two semitones in a row.

2-1-2-2-1-3-1 (harmonic minor)
2-2-1-2-1-3-1 (harmonic major)

If the minor third’s neighbors are both whole tones (2-3-2), what remains are three semitones and a whole tone, and all of their possible orders have at least two semitones in a row.

Thus the only other candidates must have a semitone and a whole tone neighboring the minor third. The semitone may come before the minor third (1-3-2), or after it (2-3-1). What remains are a pair of semitones and a pair of whole tones, and their possible orders are as follows:

1-1-2-2
2-1-1-2
2-2-1-1
1-2-2-1
1-2-1-2
2-1-2-1

The first three orders have two semitones in a row and are therefore trivially dismissed. The fourth order can also be dismissed, because placing semitones at both ends ensures that one of them will abut the minor third’s neighboring semitone, again producing two semitones in a row. Only the last two orders are viable.

To recap, we have two sets of tones. One set contains a pair of semitones and a pair of whole tones interlaced. The other set contains a whole tone, a minor third, and a semitone. Each set has only two viable orders:

Set A: 1-2-1-2, 2-1-2-1
Set B: 1-3-2, 2-3-1

To form a complete heptatonic scale, we must pick one ordering from each set. Only four combinations are possible.

1-2-1-2, 1-3-2 (A₁, B₁)
1-2-1-2, 2-3-1 (A₁, B₂)
2-1-2-1, 1-3-2 (A₂, B₁)
2-1-2-1, 2-3-1 (A₂, B₂)

The second and third possibilities have two semitones in a row, leaving only the first and last:

1-2-1-2-1-3-2
2-1-2-1-2-3-1

Both of these results exhibit the octatonic diminished scale’s characteristic pattern of alternating semitones and whole tones. Both results are reducible to prime set 7-31, which can be arrived at by removing a note from the octatonic diminished scale (8-28).

Now we can answer the original question. The heptatonic scales that consist entirely of semitones, whole tones, and a single minor third, without having two semitones in a row are 7-32 (the harmonic minor and harmonic major) and 7-31 (the octatonic diminished scale with one note removed). For clarity, the final results are shown with the minor third always between the the sixth and seventh scale degrees:

2-1-2-2-1-3-1 (harmonic minor; 7-32)
2-2-1-2-1-3-1 (harmonic major; 7-32 inverted)
1-2-1-2-1-3-2 (fourth mode of Romanian major; 7-31)
2-1-2-1-2-3-1 (third mode of Hungarian major; 7-31 inverted)