There are many conceivable ways to map 36edo onto the onto the Lumatone keyboard. However, it has 3 mutually-exclusive rings of fifths, so the Standard Lumatone mapping for Pythagorean is not one of them. Since it is highly composite, many other mappings will also fail to cover the whole gamut. (This includes both the second and third best alternative fifths.)
If you want an evenly distributed heptatonic scale that gives easy access to the perfect fifth, you instead need to use the Squirrel mapping.
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Liese (Triton)
The Liese mapping also provides a heptatonic scale which reaches the fifth in three steps, and keeps octaves horizontal, but is very lopsided.
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Slendric
However, since 36edo works best as a 7-limit system, Slendric mappings are probably preferable, since it most efficiently takes advantage of the very accurate 3 and 7 while also spanning the widest range that allows access to the full gamut at the same time. Slendric mappings use the very accurate (just slightly sharp) septimal whole tone ~8/7 as the generator, and three of these make the very accurate (just slightly flat) fifth ~3/2 (the gamelisma 1029/1024 being tempered out). This achieves a range a bit over six octaves with no missing notes and some repeated notes to mitigate vertical wraparounds.
1L 4s (8:7 step ratio, small reverse chroma)
The 1L 4s (8:7 step ratio) small reverse chroma mapping uses a small soft scale to get the most level (just slightly rising) octaves, but spreads out the notes of the standard diatonic scale (although accessing them is still intuitive although rotated), although the reverse chroma may be counterintuitive for many users.
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5L 1s (7:1 step ratio, large reverse chroma, down-right-divided octave)
The 5L 1s (7:1 step ratio, upside-down scale) large reverse chroma mapping uses a slightly larger but very hard scale to compact the standard diatonic scale into two strings of single-step movements (although still rotated from the custmoary orientation), while also making offset strings of 6edo into compact down-right key sequences, thereby dividing each octave into six parts, which may be useful for playing divided-octave temperaments. However, this makes fingerings for many chords somewhat awkward while putting octaves all over the place.
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5L 1s (7:1 step ratio, large forward chroma, up-divided octave)
A 5L 1s (7:1 step ratio, large forward chroma) version of the Slendric mapping uses a large chroma (thus making a very hard version of this scale), and achieves octaves that only slope up moderately (but more than in the 1L 4s mapping) while still having the notes of the standard diatonic scale still easily accessible, although still rotated from their customary orientation. Offset strings of 6edo are now compact upwards key sequences, which may still be useful for playing divided-octave temperaments, although more likely to pass through a vertical wrapround than in the reverse-chroma 5L 1s mapping. Bryan Deister has demonstrated this mapping in 36edo jam (2025)].
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Baladic
Another good option is the 4L 2sBaladic mapping, which divides the octave in two to add easily accessible ratios of 13 & 17 to slendrics 2.3.7 subgroup and keeps octaves near horizontal, while keeping the less well tuned ratios of 5 and 11 distant from one-another.
