Nelinda
The nelinda is a conceptual family of single-reed instruments developed by TruncatedTriangle. In contrast to the conical-bore saxophone, which produces all (1n+1) harmonics, and the cylindrical-bore clarinet, which produces mostly odd-numbered (2n+1) harmonics, the nelinda has a taper opposite in direction to the saxophone (that is, wider at the mouthpiece end and narrower at the bell end) designed to highlight the 3n+1 harmonics (that is, harmonics 1, 4 ,7, 10, 13, etc.)
This implies that it will overblow not at the octave/ditave (2/1) or the twelfth/tritave (3/1) like other single-reeds, but instead at the fifteenth or double octave (4/1), giving it a wide range.
Xenharmonic Systems for Nelinda
Similar to the mutual affinity between the tritave-repeating Bohlen-Pierce scale and the clarinet, with its spectrum of odd harmonics, a tuning system specifically for a 3n+1 spectrum like the nelinda can be developed, repeating at the 4/1 ratio (or tetratave).
4:7:10:13 would serve as the basic chord for such a system, directly analogous to 4:5:6(:7) in "normal" ditave-repeating music and 3:5:7 for BP. This translates without issue to working within a 4.7.10.13 JI subgroup, of which 640/637 is a notable comma.
Searching in Graham Breed's temperament finder for said comma, we quickly find the 27&20 (with respect to the tetratave) linear temperament, which we could call Nelindic. It has an approximate 16/13 as its generator and forms MOS of 6, 7, and 13 notes for starters, the latter of which yields a good albitonic scale.
27ed4 is an okay tuning for Nelindic (especially with compression), but 47ed4 really knocks it out of the park (similar to 12ed2 vs 31ed2 for 2.3.5).
Decimophone
Unlike the saxophone, the nelinda is still viable as an instrument with its taper reversed (that is, narrower at the mouthpiece end and wider at the bell end). This will be the branch of the decimophone which highlights the 3n+2 harmonics (that is, harmonics 2, 5, 8, 11, 14 etc.)
This implies that overblowing at the major tenth, unlike other common full tubes, but instead like partial-tube trumpets, does not highlight the fundamental.
Xenharmonic Systems for Decimophone
Similar to the mutual affinity between the tritave-repeating Bohlen-Pierce scale and the clarinet, with its spectrum of odd harmonics, a tuning system specifically for a 3n+2 spectrum like the decimophone can be developed, repeating at the 5/2 ratio (or major decade).
2:5:8:11 would serve as the basic chord for such a system, directly analogous to 4:5:6(:7) in "normal" ditave-repeating music and 3:5:7 for BP. This translates without issue to working within a 2.5.8.11 JI subgroup, of which 128/125 is a notable comma.
Searching in Graham Breed's temperament finder for said comma, we quickly find the 4n (with respect to the major decade) linear temperament, which we call augmented. It is clear this is not a very interesting result, being nothing more than a less intuitively useful way of looking at 12edo.