Greater tendoneutralisma: Difference between revisions

The '''greater tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of eight [[16/13]]'s minus two [[octave]]s falls short of [[4/3]]; that is, it is equal to ([[16/3]])/([[16/13]])⁸ and so equivalently also to ([[13/3]])/([[16/13]])⁷. Although the comma is similar in size to something like 81/80, it is quite accurate because the error can be split evenly over eight 16/13's, so that the pure-3's tuning (very close to [[53edo]]) has 13 off by only 2.78{{cent}}. A more accurate (lower damage) way of achieving the same (finding 3 by stacking 13's) is by tempering the [[lesser tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. (A much simpler but (relatively) much higher error way of mapping 3 for those that prefer sharp fifths is by tempering ([[16/13]])²/([[3/2]]) = [[512/507]].)

Tempering the greater tendoneutralisma in 2.3.13 leads to the highly notable 10 & 53 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[53edo]] is the tuning practically equal to the pure-3's tuning, although [[43edo]] is an interesting choice for combining this temperament with meantone and [[63edo]] is an interesting choice if you prefer slightly sharp fifths.

[[Badness]] (Dirichlet): 3.037