Miscellaneous 7-limit temperaments: Difference between revisions

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Below are listed some [[7-limit]] [[rank-3 ~~temperament|rank-3]] [[regular temperament|~~temperament]]s that do not belong to some other temperament collection, the majority of which are [[restriction]]s to the 7-limit of temperaments that emerge more fully in higher [[prime limit|limits]] or [[subgroup]]s; they are sorted by [[TE logflat badness]]. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:

Below are listed some [[7-limit]] [[rank-3 temperament]]s that do not belong to some other temperament collection, the majority of which are [[restriction]]s to the 7-limit of temperaments that emerge more fully in higher [[prime limit|limits]] or [[subgroup]]s; they are sorted by [[TE logflat badness]]. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:

* Individual [[temperament families and clans]]

* [[Very low accuracy temperaments]]

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[[Badness]] (Sintel): 0.661

== Canopic a.k.a. mirkwai ==

: ''For extensions, see [[Swetismic temperaments #Indra]].''

Canopic, a.k.a. mirkwai, tempers out the [[canopic comma]] in the 7-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807

: Mapping generators: ~2, ~3, ~10/7

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0944{{c}}

: [[error map]]: {{val| -0.000 -0.172 -0.493 +0.900 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0945{{c}}

: error map: {{val| 0.000 -0.172 -0.493 +0.900 }}

[[Minimax tuning]]:

* [[7-odd-limit]]

: {{monzo list| 1 0 0 0 | 0 4/7 -4/7 5/7 | 0 -3/7 3/7 5/7 | 0 0 0 1 }}

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7

* [[9-odd-limit]]

: {{monzo list| 1 0 0 0 | 0 8/11 -4/11 5/11 | 0 -6/11 3/11 10/11 | 0 0 0 1 }}

: eigenmonzo (unchanged-interval) basis: 2.9/5.7

[[Badness]] (Sintel): 1.51

[[Projection pair]]s: <code>5 84375/16807 7 16875/2401</code> to 2.3.7/5

== Greenwoodmic ==

Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval|primes]] [[3/1|3]] and [[5/1|5]] should be tuned flat, and [[7/1|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic.

In contrast to [[sensamagic]], where two [[9/7]]'s stack to [[5/3]], here two 9/7's stack to [[8/5]]. As such, greenwoodmic induces [[essentially tempered chord]]s in the [[9-odd-limit]]. An obvious 11-limit extension then equates [[5/4]] with [[11/9]] and equates 9/7 with [[14/11]], tempering out [[45/44]] as well as [[99/98]] using the identity 405/392 = (45/44)⋅(99/98).

[[Subgroup]]: 2.3.5.7

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 {{Technical data page}}
-Below are listed some [[7-limit]] [[rank-3 temperament|rank-3]] [[regular temperament|temperament]]s that do not belong to some other temperament collection, the majority of which are [[restriction]]s to the 7-limit of temperaments that emerge more fully in higher [[prime limit|limits]] or [[subgroup]]s; they are sorted by [[TE logflat badness]]. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:
+Below are listed some [[7-limit]] [[rank-3 temperament]]s that do not belong to some other temperament collection, the majority of which are [[restriction]]s to the 7-limit of temperaments that emerge more fully in higher [[prime limit|limits]] or [[subgroup]]s; they are sorted by [[TE logflat badness]]. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:
 * Individual [[temperament families and clans]]
 * [[Very low accuracy temperaments]]
@@ Line 44: / Line 44: @@
 [[Badness]] (Sintel): 0.661
+== Canopic a.k.a. mirkwai ==
+: ''For extensions, see [[Swetismic temperaments #Indra]].''
+Canopic, a.k.a. mirkwai, tempers out the [[canopic comma]] in the 7-limit.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 16875/16807
+{{Mapping|legend=1| 1 0 -5 -4 | 0 1 3 3 | 0 0 5 4 }}
+: Mapping generators: ~2, ~3, ~10/7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0944{{c}}
+: [[error map]]: {{val| -0.000 -0.172 -0.493 +0.900 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0945{{c}}
+: error map: {{val| 0.000 -0.172 -0.493 +0.900 }}
+[[Minimax tuning]]:
+* [[7-odd-limit]]
+: {{monzo list| 1 0 0 0 | 0 4/7 -4/7 5/7 | 0 -3/7 3/7 5/7 | 0 0 0 1 }}
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7
+* [[9-odd-limit]]
+: {{monzo list| 1 0 0 0 | 0 8/11 -4/11 5/11 | 0 -6/11 3/11 10/11 | 0 0 0 1 }}
+: eigenmonzo (unchanged-interval) basis: 2.9/5.7
+{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }}
+[[Badness]] (Sintel): 1.51
+[[Projection pair]]s: <code>5 84375/16807 7 16875/2401</code> to 2.3.7/5
 == Greenwoodmic ==
 Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval|primes]] [[3/1|3]] and [[5/1|5]] should be tuned flat, and [[7/1|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic.
+In contrast to [[sensamagic]], where two [[9/7]]'s stack to [[5/3]], here two 9/7's stack to [[8/5]]. As such, greenwoodmic induces [[essentially tempered chord]]s in the [[9-odd-limit]]. An obvious 11-limit extension then equates [[5/4]] with [[11/9]] and equates 9/7 with [[14/11]], tempering out [[45/44]] as well as [[99/98]] using the identity 405/392 = (45/44)⋅(99/98).
 [[Subgroup]]: 2.3.5.7