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, which could be called a tetratave.

Operating on a single tetratave of the series gives us a 4.7.10.13 JI subgroup, of which 640/637 serves as a notable comma.

Searching in Graham Breed's temperament finder, we quickly find the 27&20 (with respect to the tetratave) linear temperament, tentatively called Nelindic, which tempers out the comma with aplomb. 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).