Greater tendoneutralisma: Difference between revisions
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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]])<sup>8</sup> and so equivalently also to ([[13/3]])/([[16/13]])<sup>7</sup>. 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}}. | 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]])<sup>8</sup> and so equivalently also to ([[13/3]])/([[16/13]])<sup>7</sup>. 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 13 will usually pick between one of these two mappings. | ||
== Temperaments == | == Temperaments == | ||
Revision as of 18:58, 29 May 2024
| Interval information |
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 octaves 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 ¢. 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 13 will usually pick between one of these two mappings.
Temperaments
Greater Tendoneutralic
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.
Subgroup: 2.3.13
Comma list: 815730721/805306368
Mapping: [⟨1 4 4], ⟨0 -8 -1]]
Optimal tuning (CTE): ~16/13 = 362.248 ¢
Optimal ET sequence: 10, 33, 43, 53, 202, 255f, 308f, 361f
Badness (Dirichlet): 3.037