There are many conceivable ways to map 81edo onto the onto the Lumatone keyboard. Only one, however, agrees with the Standard Lumatone mapping for Pythagorean. (Note that the 81b sharp fifth is shared with 27edo, thereby making three mutually exclusive rings of fifths.)
Meantone
Diatonic
The patent val diatonic mapping would be a perfectly respectable Meantone mapping on a hypothetical XL-sized Lumatone having at least 405 keys in the space of five octaves. Since meantone is a highly efficient temperament in general, and 81 is the optimal patent val for golden meantone, the full gamut is not necessary to play the best chords and a pedal that shifts the pitch by a single edostep would be an effective way to enable you to play in all keys.
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Chromatic
Bryan Deister has demonstrated the m-chromatic (7L 5s with 8:5 step ratio) mapping for 81edo in ''microtonal improvisation in 81edo (2025). The rightward generator 8\81 functions as ~15/14 and ~16/15; six of them make the alternate sharp fifth. The down-right generator 5\81 functions as ~23/22 and ~24/23; two of them make an Alpharabian tendoneutral second (~12/11); four of them make the near-just octave-reduced 19th harmonic ~19/16; eleven of them make a near-just classic minor sixth ~8/5. The patent fifth 47\81 is four rightward generators plus three down-right generators. All that said, the patent fifth itself works perfectly well as a generator of the 7L 5s scale, as well as for the entire layout, as befits (rank-2) Meantone, so that the use of dual generators is a matter of convience rather than a requirement. The range is just over three octaves, with no missed notes, and the octaves slant down moderately.
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Porcupine
The 7L 1s scale covers 4/5ths of the notes and makes the 5th & 11th harmonics very easy to play, although it uses the second best approximation to the 3rd and 7th harmonics.
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Lithium
The Lithium scale divides the octave in three to take advantage of the efficiency of meantone in a different way, giving you a near equal scale with the 3L 6s mos covering 8/9ths of the gamut and the 9L 3s covering all of it with the same efficiency as the chromatic one, with octaves closer to horizontal.
3L 6s
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9L 3s
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Other Mappings
If you want maximum range with no concern for good temperaments, the 3L 7s mos generated by a submajor third of 25/81 is the most efficient one that can hit all the notes in the middle octaves.
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