No-threes subgroup temperaments: Difference between revisions

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=== Rainy ===

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]].)

Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])3 (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])5 (the 2.7-limit [[cloudy comma]], which is similar to the 2.5-limit lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)

A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.

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 === Rainy ===
-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]])<sup>3</sup> = [[128/125]] (the 2.3.5 diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (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]].)
+Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-limit [[cloudy comma]], which is similar to the 2.5-limit lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)
 A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.