This is equivalent to setting the EDO × cents-error boundary at 686¢, the tempered perfect fifth of 7edo. (This will never result in a case that's exactly on the line, for the same reason there is no EDO that exactly represents the 2:3 perfect fifth.) This looks like a plausible answer to the question, and the best besides just listing some specific ones to include.Dave Keenan wrote: ↑Mon Jul 13, 2020 9:56 amBut I have a better criterion. I realised that what makes a second-best fifth worth considering for notation, is when it is not much worse than the best fifth. A convenient cutoff is where the error in the second-best fifth is no more than 4/3 of the error in the best fifth.
Putting it another way: For a b-ET to be included, the fractional part of its number of EDO steps in a pure (2:3) fifth, frac(log2(3/2)×EDO), must be between 3/7 and 4/7.
That includes all those I listed earlier, except 37b, which is OK with me. I realise this gives an infinite number of b-ETs, but that's OK too. As you might guess, it adds on average one b-ET for every 7-EDOs. Up to 72-EDO it includes only 6b, 11b, 18b (9), 23b, 30b, 35b (5), 42b (7), 47b, 59b, 64b (32), 71b.
Honestly, "noble-mediant" sounds better anyway. My choice of "phidiant" was ... something I came up with for a reason I can't tell anymore.