No-threes subgroup temperaments: Difference between revisions

* Jubilic → [[Jubilismic clan #Jubilic]]

{{See also| Chromatic pairs #Shoe }}

{{See also| Llywelyn clan #Llywelyn aka shoe }}

{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}

Subgroup: 2.5.7.11

{{Optimal ET sequence|legend=1| 16, 21, 37 }}

Subgroup: 2.5.7.11.13

{{Optimal ET sequence|legend=1| 16, 21, 37 }}

Subgroup: 2.5.7.11.13.17

{{Optimal ET sequence|legend=1| 16, 21, 37 }}

{{See also| Hemimean clan #Didacus }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

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]].)

[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents

{{See also| Quince clan #Mercy }}

{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}

[[Subgroup]]: 2.5.7.13

{{Optimal ET sequence|legend=1| 10, 21, 31}}

[[Subgroup]]: 2.5.7.13.17

{{Optimal ET sequence|legend=1| 10, 21, 31}}

[[Subgroup]]: 2.5.7.13.17.19

{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}

{{See also| Magic family #Quadrimage }}

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Subgroup: 2.5.7.11

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  

{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}

Subgroup: 2.5.7.11.13

{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}

Subgroup: 2.5.7.11.13.17

{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}

Subgroup: 2.5.7.11.13.17.19

Optimal tuning (POTE): ~19/14 = 529.003¢

Described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.

{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...

Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.

{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...

Subgroup: 2.5.7.11

{{Optimal ET sequence|legend=1|264, 733}}, ...

Subgroup: 2.5.7.11.13

{{Optimal ET sequence|legend=1|1525, 1789}}, ...

{{Main|Bastille}}

Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Pure Bastille|pure bastille]].

{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...

== Pure onzonic ==