Nelinda

The nelinda is a hypothetical family of single-reed instruments developed by TruncatedTriangle. In contrast to the conical-bore saxophone, which produces all 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.

The 20 & 27 (with respect to the tetratave) linear temperament is a notable entry that tempers out the said comma, which we could call nelindic, and corresponds to huntington excluding every other peiod. 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). Note that 20ed4 is just 10edo, where 5/4 (and possibly also 7/4) is tuned too flat,.