No-threes subgroup temperaments: Difference between revisions

Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]]. Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])³ = [[128/125]] (the 2.3.5 diesis) with ([[2/1]])/([[8/7]])⁵ (the [[cloudy comma]]), which has a similar size and role to the 2.3.5 diesis so might be considered the 2.3.7 diesis, in that if you temper it, it imparts significant error on the tuning of 8/7 (so that it is now ~8.8{{cent}} sharp), similar to how tempering 128/125 imparts significant error on the tuning of 5/4 (so that it is ~13.7{{cent}} sharp). Combined with the small size of these "dieses", it implies that the locations of 5/4 and 8/7 are in some sense awkward to conceptualize due to the small but not insignificant size of the remnant with the octave, so it is might be favorable to equate 128/125 with the cloudy comma to create a general-purpose diesis. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)