Lumatone mapping for 62edo

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There are many conceivable ways to map 62edo onto the onto the Lumatone keyboard. However, it has 2 mutually-exclusive rings of fifths, so the Standard Lumatone mapping for Pythagorean is not one of them. Due to its size, it would not quite cover the whole gamut even if it was.

Diatonic

You can use the b val, but this is extremely sharp, surpassing half comma superpyth and making major seconds near perfect 8/7's. This does mean that the better 5th is also fairly easily accessible as an augmented 4th, but the best approximation to 5/4 is a triply augmented 6th, while 11/8 is a triply diminished seventh, both of which are pretty awkward to play in a chord.

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Ekic Mappings

Since 31edo is such a well-tuned edo in general, most of the mos scales in 62edo do not improve on it, making it difficult to use the extra notes if concordant harmony is your goal. However, there are two different octatonic mappings of comparable efficiency that are clear winners if you simply want access to the full gamut with maximum range. These are the 6L 2s (TAMNAS "Ekic") scale mappings created by slicing the period in half for a myna/orwell-related temperament and a hemimeantone temperament. Both give a range of a bit over four octaves that are close to level.

Myna/Orwell-related or Nusecond-related 6L 2s (8:7 step ratio)

The first of these mappings is related to orwell, which uses as sharp septimal minor third ~7/6 as its generator, and myna, which uses a flat classic minor third ~6/5 as its generator; the temperament for this mapping splits the difference and uses a near-just tridecimal minor third ~13/11 as its generator, represented as 15\62 (one key right plus one key down-right), and splits the octave in half. It can also be interpreted as using a near-just septimal neutral second ~35/32 or a somewhat sharp undecimal neutral second ~12/11 as its generator, represented as 8\62 (one key right), related to nusecond, but again splitting the octave in half.

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Hemimeantone/Mohajira-related 6L 2s (9:4 step ratio)

The second is related to hemimeantone, using a slightly sharp tridecimal semifourth ~15/13 (represented as 13\62, one key right plus one key down-right) as its generator, making half of the (sharp) fourth ~4/3, but also splitting the octave in half. Alternatively, it can be interpreted as using a near-just undevicesimal submajor/supraneutral second ~21/19 as its generator, represented as 9\62 (one key right), related to mohajira by splitting the near-just undecimal neutral third ~11/9 in half (thus splitting the fifth ~3/2 into quarters), as well as splitting the octave in half. Bryan Deister demonstrates this mapping in 62edo improv.

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Godzilla-related rank-3 Mappings

Bryan Deister has demonstrated a Godzilla Lumatone mapping for 62edo in microtonal improvisation in 62edo (2025), with scale 5L 4s with 10:3 step ratio, which also has easy provision for dividing the near-just septimal minor seventh ~7/4 (50\62) into five equal parts each functioning as a meantone whole tone (~9/8, ~10/9, and ~19/17) (10\62 is right by one key), which is naturally useful in its own right for mixing the common practice sound in with the xenharmonic sounds. A logical implication of splitting the ~7/4 into five meantone whole tones is that the near-just classic major third ~5/4 is very easily accessible as two of these divided generators, and the only slightly flat lesser septimal tritone ~7/5 is easily accessible as three of them. As convenient as this rightward generator is, it would produce a contorted mapping if not for a second generator; the upwards generator 7\62 is convenient for this as a a tridecimal neutral second that functions as both ~12/13 and ~13/14 (the buzurgisma/dhanvantarisma 169/168 is tempered out); backing off from ~7/4 by two of these generators gives the fifth (the flat ~3/2 shared with 31edo), while backing off from ~7/5 by two of these generators gives a moderately flat classic minor third ~6/5. Octaves slant down moderately, and the range is just under 4⅓ octaves with no missed notes and no repeated notes.

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