Nelinda

The nelinda is a hypothetical 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).