Lesser tendoneutralisma: Difference between revisions

(8 intermediate revisions by 3 users not shown)

Line 2:

| Ratio = 70368744177664/69894255367443

| Name = lesser tendoneutralisma

| Color name = Sasa-quadtrithu comma

| Comma = yes

}}

The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])12 / ([[3/2]]) / ([[2/1]])3 and so equivalently also to [[16/13]])11 / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]].

The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])12 / ([[3/2]]) / ([[2/1]])3 and so equivalently also to ([[16/13]])11 / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater 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]].)

== Temperaments ==

=== Lesser Tendoneutralic ===

Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence).

Line 14:

Line 16:

[[Comma list]]: {{monzo| 46 -1 (0 0 0) -12 }}

: sval mapping generators: 2, ~13

[[Optimal tuning]] ([[CTE]]): ~13/8 = 841.503{{cent}}

Line 20:

Line 23:

{{Optimal ET sequence|legend=1| 10, 47, 57, 67, 77, 164, 241, 405, 646f }}

[[Badness]] (~~Dirichlet~~): 3.930

[[Badness]] (Sintel): 3.930

== See also ==

* [[Greater tendoneutralisma]]

* [[512/507]]

* [[Tridecapyth comma]]

[[Category:Commas with unknown etymology]]

@@ Line 2: / Line 2: @@
 | Ratio = 70368744177664/69894255367443
 | Name = lesser tendoneutralisma
+| Color name = Sasa-quadtrithu comma
 | Comma = yes
 }}
-The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to [[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]].
+The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to ([[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater 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]])<sup>2</sup>/([[3/2]]) = [[512/507]].)
 == Temperaments ==
 === Lesser Tendoneutralic ===
+{{ See also | Restles }}
 Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence).
@@ Line 14: / Line 16: @@
 [[Comma list]]: {{monzo| 46 -1 (0 0 0) -12 }}
-{{Mapping|legend=1| 1 10 3 | 0 -12 1 }}
+{{Mapping|legend=2| 1 10 3 | 0 -12 1 }}
+: sval mapping generators: 2, ~13
 [[Optimal tuning]] ([[CTE]]): ~13/8 = 841.503{{cent}}
@@ Line 20: / Line 23: @@
 {{Optimal ET sequence|legend=1| 10, 47, 57, 67, 77, 164, 241, 405, 646f }}
-[[Badness]] (Dirichlet): 3.930
+[[Badness]] (Sintel): 3.930
 == See also ==
 * [[Greater tendoneutralisma]]
+* [[512/507]]
 * [[Tridecapyth comma]]
+[[Category:Commas with unknown etymology]]