Bryan Deister has demonstrated a 4L 4s mapping (6:5 step ratio) 44edo, in Buried Treasure - 44edo (2026) ([short clip], [short 2]). This mapping also functions as a very hard flipped superdiatonic mapping (7L 2s, 6:1 step ratio with the small steps going up instead of down-right), a flipped antidiatonic mapping (2L 5s with 7:6 step ratio and expanded small step going right + up), and as a 13L 3s (4/1-equivalent) mapping (with 6:5 step ratio, proceeding through the octave zig-zag). Right + down-right divides the octave into slices of 11\56; as an interval in its own right, this is the same as the minor third of 12edo, which functions as ~19/16 and ~25/21 (both being near-just). Down-right alone is 4\56, which is the minimal form generator for Bidia; it functions as the classic diatonic semitone ~16/15, the large septendecimal semitone ~17/16, and the small septendecimal semitone ~18/17 (which is inconsistently mapped), meaning that the charisma 256/255 and the semitonisma 289/288 are both tempered out; two of them make a rather sharp whole tone ~9/8 (which is also inconsistently mapped); three of them (passing the quarter-octave) make a sharp classic minor third ~6/5, while the afore-mentioned quarter-octave (12edo-style) minor third is about equally easy to reach; in contrast, the somewhat flat classic major third ~5/4 requires two moves rightwards plus two moves upwards; four moves right reaches the rather flat fourth ~4/3; another move right and two moves upreaches the rather sharp fifth ~3/2. The range is a bit under 4¾ complete octaves (with some extra non-contiguous notes at each end), but unlike the normal antidiatonic mapping, the octaves alternate between near/far and mid or near and far (superimposed upon an overall upwards slope).
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Pseudo-Isomorphic Pseudo-Diatonic
To get a quasi-diatonic layout with a reasonable fifth, you can shoehorn the diatonic mapping for 45edo into 44edo, with note 44 being a duplicate note 0, as Bryan Deister demonstrates in 44edo improv (Oct 2025)
Slicing the perfect fourth in half also works, but the 4L 1s mapping does not cover the whole gamut:
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Expanding this to the 5L 4s mapping solves this problem, but the scale has an 8:1 step ratio, making it very lopsided.
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Hemifourths
However, it is the Hemifourths mapping that combines the widest range that covers the full gamut with the most efficient way of reaching all prime harmonics up to 17.
