Neutron Temperament

The Basic Scale, 7EDO

William Lynch gave the name "Neutron[7]" to 7edo used as a "neutral diatonic" scale. While it can function just fine as a scale on its own, implicit in this designation is that 7edo represents the starting point for a temperament with 7 periods per octave, just as 3edo represents "Augmented[3]" with 3 periods per octave.

Building a Regular Temperament

From this base, we can say that a prospective neutron temperament tempers out the difference between a stack of 7 intervals in the "greater neutral second" range and the octave. The simplest ratio that comes to mind is 11/10, and this gives us a comma of [8 0 7 0 -7⟩, or 20000000/19487171 in ratio form. Being about 6 cents sharp, it's not a bad representation. Another advantage of starting from 7 notes is that a familiar "Halberstadt-like" keyboard arrangement could be constructed for such a temperament. That's just the period, however. The generator is the hard part.

Generator Investigations

Candidate 1

One possibility is a generator around 30 cents. This generator can be chained both the positive and negative directions from the tonic 5 times without traversing the period, leading to a "complete" MOS of 77 tones, 5|5(7) in UDP notation. Example 77-tone scale here. If it doesn't pan out to anything worthy of the original name, it could alternately be called "Lucky", "Jackpot", or "Fortune" due to the abundance of 7s in its construction and structure. Although given the problems stated below, maybe that's not so apposite.

Upon study of this MOS, a curious pattern of "blind spots" emerges for certain harmonics, namely that certain pairs of intervals considered to be the "major" and "minor" of that limit have one interval tuned well but not the other. In the 5-limit, 6/5 has an very close representative, but 5/4 has to settle for one more than 8c away. Likewise in the 7-limit, where 7/6 is more out-of-tune than 9/7, although a bit less so than the 5s. 11 and 13 are tuned okay; higher harmonics aren't really represented. It is likely one would need a high complexity for an accurate tuning of 5 and 7.

<more to come; contributions such as comma ideas, different generators, other useful modes/mappings with existing generators, etc. are welcome. I very obviously don't know what I'm doing.>