Ripple family: Difference between revisions

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The '''ripple family''' of temperaments [[tempering out|tempers out]] the [[ripple comma]], 6561/6250 = {{monzo| -1 8 -5 }}, which equates a stack of five [[27/25]]'s with [[4/3]].

The '''ripple family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ripple comma]] ([[ratio]]: 6561/6250, {{monzo|legend=1| -1 8 -5 }}), which equates a stack of five [[27/25]]'s with [[4/3]].

== Ripple ==

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== Septimal ripple ==

Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~21/16 * ~4/3 = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.

Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~(21/16)⋅(4/3) = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.

''If you are looking for the former canonical ~~extension~~ extension, see: [[#Rip]].''

''If you are looking for the former canonical extension, see: [[#Rip]].''

[[Subgroup]]: 2.3.5.7

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* [[CTE]]: ~15/14 = 101.538

: [[error map]]: {{val| 0 -11.785, -2.041, 3.651, 13.296 }}

* [[CE]]: ~15/14 = 102.319 ~~(preferred for dual-fifths 11-limit)~~

* [[CE]]: ~15/14 = 102.319

: [[error map]]: {{val| 0 -13.551 -4.868 -1.296 6.935 }}

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== Rip ==

Formerly known as ~~[[#Ripple]]~~, but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.

Formerly known as ''septimal ripple'', but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.

[[Subgroup]]: 2.3.5.7

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 {{Technical data page}}
-The '''ripple family''' of temperaments [[tempering out|tempers out]] the [[ripple comma]], 6561/6250 = {{monzo| -1 8 -5 }}, which equates a stack of five [[27/25]]'s with [[4/3]].
+The '''ripple family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ripple comma]] ([[ratio]]: 6561/6250, {{monzo|legend=1| -1 8 -5 }}), which equates a stack of five [[27/25]]'s with [[4/3]].
 == Ripple ==
@@ Line 29: / Line 29: @@
 == Septimal ripple ==
 {{ See also | Dual-fifth temperaments }}
-Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~21/16 * ~4/3 = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.
+Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~(21/16)⋅(4/3) = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.
-''If you are looking for the former canonical extension extension, see: [[#Rip]].''
+''If you are looking for the former canonical extension, see: [[#Rip]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 61: / Line 61: @@
 * [[CTE]]: ~15/14 = 101.538
 : [[error map]]: {{val| 0 -11.785, -2.041, 3.651, 13.296 }}
-* [[CE]]: ~15/14 = 102.319 (preferred for dual-fifths 11-limit)
+* [[CE]]: ~15/14 = 102.319
 : [[error map]]: {{val| 0 -13.551 -4.868 -1.296 6.935 }}
@@ Line 69: / Line 69: @@
 == Rip ==
-Formerly known as [[#Ripple]], but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.
+Formerly known as ''septimal ripple'', but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.
 [[Subgroup]]: 2.3.5.7