Greater tendoneutralisma: Difference between revisions

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| Ratio = 815730721/805306368

| Name = greater tendoneutralisma

| Color name = Laquadbitho comma

| Comma = yes

}}

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 cannot 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]].)

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.

== Temperaments ==

Although the comma is similar in size to something like 81/80, the corresponding temperament 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 cannot 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]].

=== Greater Tendoneutralic ===

Tempering out 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. ~~This temperament~~ is ~~related to~~ [[~~submajor~~ (temperament)|submajor]]~~, which extends it to~~ the full [[13-limit]].

Tempering out 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. [[Buzzardsmic clan #Demibuzzard|Demibuzzard]] is an extension of this which maps primes 5 and 7; [[Submajor (temperament)|submajor]] and [[interpental]] map prime 11 and thus the full [[13-limit]].

[[Subgroup]]: 2.3.13

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{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255f, 308f, 361f }}

[[Badness]] (~~Dirichlet~~): 3.037

[[Badness]] (Sintel): 3.037

== See also ==

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* [[512/507]]

* [[Tridecapyth comma]]

[[Category:Commas with unknown etymology]]

@@ Line 2: / Line 2: @@
 | Ratio = 815730721/805306368
 | Name = greater tendoneutralisma
+| Color name = Laquadbitho comma
 | Comma = yes
 }}
-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 cannot 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]])<sup>2</sup>/([[3/2]]) = [[512/507]].)
+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>.
 == Temperaments ==
+Although the comma is similar in size to something like 81/80, the corresponding temperament 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 cannot 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]])<sup>2</sup>/([[3/2]]) = [[512/507]].
 === Greater Tendoneutralic ===
-Tempering out 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. This temperament is related to [[submajor (temperament)|submajor]], which extends it to the full [[13-limit]].
+Tempering out 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. [[Buzzardsmic clan #Demibuzzard|Demibuzzard]] is an extension of this which maps primes 5 and 7; [[Submajor (temperament)|submajor]] and [[interpental]] map prime 11 and thus the full [[13-limit]].
 [[Subgroup]]: 2.3.13
@@ Line 20: / Line 23: @@
 {{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255f, 308f, 361f }}
-[[Badness]] (Dirichlet): 3.037
+[[Badness]] (Sintel): 3.037
 == See also ==
@@ Line 26: / Line 29: @@
 * [[512/507]]
 * [[Tridecapyth comma]]
+[[Category:Commas with unknown etymology]]