BACH - RICERCAR a 3 from the Musical Offering, tuned into E31

[@pavelbatalov657]

Thank God! Finally!

[@ClaudiMeneghin]

Thanks to you for your interest and appreciation 😀 PS 19 and 12 are forthcoming 😉

[@Lucius_Chiaraviglio]

The flatter fifths that Bach used were quarter-comma meantone, so even though C minor gets out of the quarter-comma meantone zone for the well-tempered tuning systems of the time, it's no surprise that this fits well. Since you have 12EDO and 19EDO incoming, I am going to guess in advance that 12EDO will sound not bad (if a bit bland), since the flat side of the well-tempered tunings of the day tended to include Pythagorean and even Kirnberger fifths, but that 19EDO may sound a bit rough (although mitigated by the 19EDO fifth getting out of that uncomfortable zone between 3/2 and 112/75).

[@ClaudiMeneghin]

Well, 4 quarter-comma (flatter) fifths, together with 8 pythagorean (sharper) ones in Werckmeister 3 WT... or at least I guess so 😉

[@Lucius_Chiaraviglio]

@@Symbioticism That's why I said the flatter fifths, not all fifths. Werckmeister, Rameau, and Kirnberger well-tempered tuning systems include quarter-comma meantone fifths among the natural notes; Bach had his own temperament which is not well documented but presumably also had quarter-comma meantone fifths in this region. To make the circle of fifths close, each one also included some Pythagorean and/or Kirnberger fifths, mostly in the sharp/flat zone.

[@Lucius_Chiaraviglio]

@Symbioticism "Presumably" is not doing so much heavy lifting, since other well-tempered systems of the time regularly had fifths flattened by a quarter of a syntonic comma (or even more), including Kirnberger's; Johann Kirnberger was a student of Bach. These temperaments were designed to get pure major thirds in some regions of the circle of fifths. Doing so within a 12 note circle of fifths mathematically requires flattening some fifths at least as much as a quarter of a syntonic comma (you can also flatten some to a greater extent and combine them with less-flattened fifths). Well-tempered systems falling short of achieving pure major thirds (even including equal temperament) started to become popular after Bach's time, but weren't very popular before then (with the possible exception of some instrument consorts lacking a keyboard instrument) even though they had existed for some time. And look up Bach Temperament.

[@Lucius_Chiaraviglio]

I should mention that Some of the reconstructions of Bach's temperament (Lehman, Hill, Swich, Louie, etc.) use fifths flattened in the range 1/5 to 1/6 of a syntonic comma, so not all the way to a quarter comma, but getting close.

[@hoon_sol]

It's hilarious to "guess" that 12-TET won't sound bad; of course it won't sound bad, it's literally the objectively ideal tuning system. The 12-TET scale is literally the _only_ equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation (within one percent) _and_ contains more consonant intervals than dissonant intervals.