Ragismic microtemperaments: Difference between revisions

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Since (10/9)<sup>4</sup> = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)<sup>2</sup>, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]

* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]

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Badness: 0.044

== Crazy ==

: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an obvious tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}

: mapping generators: ~332150625/234881024, ~1125/1024

[[Optimal tuning]]s:

* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475

* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}

* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474

* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

[[Badness]] (Smith): 0.0394

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

Optimal tunings:

* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485

* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

Badness (Smith): 0.0170

== Orga ==

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 Since (10/9)<sup>4</sup> = 4375/4374 × 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 × (27/25)<sup>2</sup>, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
-Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
+Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
 * ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
 * ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
@@ Line 361: / Line 361: @@
 Badness: 0.044
+== Crazy ==
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
+Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an obvious tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
+{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
+: mapping generators: ~332150625/234881024, ~1125/1024
+[[Optimal tuning]]s:
+* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475
+* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
+* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474
+* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}
+{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
+[[Badness]] (Smith): 0.0394
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
+Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
+Optimal tunings:
+* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
+* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481
+{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
+Badness (Smith): 0.0170
 == Orga ==