There are many conceivable ways to map 15edo 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.
Antidiatonic
It is possible to use the extremely flat 8\15 fifth of 640 ¢ to get an antidiatonic mapping. This layout preserves the location of the octave, and places the down-fifth where the fifth usually is; however, the resulting 2L 5s scale is extremely hard (5:1 step ratio), as is the 2L 7s scale that one can get by backtracking at each L step (4:1 step ratio).
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This makes fingerings for most simple chords awkward though. The mappings that organise its intervals that make it easy to find consonant chords, in order of increasing compression, are the porcupine, blackwood, and hanson mappings.
Porcupine
Porcupine is a popular temperament that 15edo supports, although the semi-expanded mapping here (1L 6s scale with 3:2 step ratio and no backtracking) spreads the octaves out more than is optimal for such a small tuning system as 15edo, thereby limiting the range to somewhat over four octaves, which slope upwards too much for convenience in splitting the keyboard into multiple manuals.
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Fully Expanded (add Augmented/Triforce)
The fully expanded 7L 1s mapping reduces the range (now just under four octaves) and increases the octave slope even further, but may be more intuitive to play, including having a forward chroma (rather than reverse as above); also, the slope is now enough upwards to permit reframing as a slight downwards slope with a 6L 3s scale (as in augmented/triforce, and gentle enough to provide the potential splitting the mapping into multiple manuals (three complete in the middle, and one partial each above and below). Bryan Deister demonstrates this mapping in 15edo (2023), although with the color scheme set for porcupine and not augmented/triforce.
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Blackwood + Augmented (Augene) + Kleismic (Hanson/Keemun/Superkleismic)
Blackwood avoids the contorsion problem of 15edo by dividing the octave into five sections. Dividing 15edo by 3 gets a 3L 3s scale on top of this. With a 3:2 step ratio, this also makes for easy division of the octave into three sections, as in augmented, specifically augene. This also functions as a kleismic (hanson/keemun/superkleismic) 4L 3s (3:1 step ratio) mapping. The range is a bit under 6 octaves, and while octaves slope up as expected for Blackwood, they do so in a way such that the vertical repeated notes cause double octaves to be nearly level.
3L 3s (right, down-right) and 4L 3s (right, up)
This version is optimized for playing the 3L 3s scale; it makes the 4L 3s scale slope upwards, although with care it is possible to use the vertically repeated notes to avoid a vertical wraparound. Bryan Deister demonstrates this mapping in GHOST – Rotary Dial (microtonal cover in 15edo) (2023).
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3L 3s (right, up) and 4L 3s (right, down-right)
This version is optimized for playing the 4L 3s scale; it makes the 3L 3s scale slope upwards, although with care it is possible to use the vertically repeated notes to avoid a vertical wraparound. Bryan Deister demonstrates this version in Waltz in 15edo (2025).
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Hanson
If hanson is most of what is needed and augmented and blackwood are not a priority, it is possible to compress the layout to a 3L 1s scale (4:3 step ratio) and increase the range to over eight octaves that are nearly level (even supporting the possibility for dual manuals). Bryan Deister demonstrates this mapping in Radiohead – Subterranean Homesick Alien [microtonal 15edo] (2024).
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