Ragismic microtemperaments: Difference between revisions

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Since {{nowrap|(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 {{nowrap| 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, 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:

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, 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]]

* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]

* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]

* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]

* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]

* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]

* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]

* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]

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== Deca ==

: ''For 5-limit version, see [[~~10th~~-~~octave~~ temperaments #Neon]].''

: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Neon]].''

Deca has a period of 1/10 octave and tempers out the neon comma {{monzo| 21 60 -50 }} in the 5-limit, the [[linus comma]]{{monzo| 11 -10 -10 10 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.

Deca has a period of 1/10 octave and tempers out the [[neon comma]] ({{monzo| 21 60 -50 }}) in the 5-limit, the [[linus comma]] ({{monzo| 11 -10 -10 10 }}) and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.

[[Subgroup]]: 2.3.5.7

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== Aluminium ==

: ''For the 5-limit version, see [[~~13th~~-~~octave~~ temperaments #Aluminium]].''

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Aluminium]].''

Aluminium ~~was named by [[Eliora]] in 2023 after the 13th element. It~~ tempers out ~~the~~ {{monzo| 92 -39 -13 }} ~~comma, which~~ sets [[135/128]] ~~interval to be equal~~ to 1/~~13th~~ of the ~~octave~~.

Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit and sets [[135/128]] to 1/13 of an [[octave]]. It was named by [[Eliora]] in 2023 after the 13th element.

[[Subgroup]]: 2.3.5.7

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Badness (Sintel): 1.18

== ~~Countritonic~~ ==

== Ragitritonic ==

: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic ~~(5-limit)~~]].''

: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''

~~Countritonic (''co-un-tritonic'')~~ may be described as the {{nowrap| 53 & ~~422~~ }} temperament, ~~generated by an octave-reduced 91st~~ harmonic ~~or subharmonic in the 13~~-~~limit~~. It was named by [[Flora Canou]] in ~~2023~~ as a ~~variant~~ of ~~[[countriton]]~~.

Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.

Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.

[[Subgroup]]: 2.3.5.7

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Line 1,190:

Badness (Sintel): 1.14

~~== Counterorson ==~~

Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}~~

~~{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}~~

~~: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}

~~: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}

~~: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}~~

~~{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}~~

~~[[Badness]] (Sintel): 7.92~~

~~== Oviminor ==~~

~~: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''~~

Oviminor was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}~~

~~{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}~~

~~: mapping generators: ~2, ~5/3~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}

~~: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}

~~: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}~~

~~{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}~~

~~[[Badness]] (Sintel): 14.7~~

== Octoid ==

: {{Main| Octoid }}

: ''For the 5-limit version, see [[~~8th~~-~~octave~~ temperaments #Octoid]].''

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].

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== Trideci ==

: ''For the 5-limit version, see [[~~13th~~-~~octave~~ temperaments #Tridecatonic]].''

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''

The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic ~~temperament~~]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").

The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic]] temperament, but with the ragisma (4375/4374) rather than the octagar comma (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").

[[Subgroup]]: 2.3.5.7

@@ Line 4: / Line 4: @@
 Since {{nowrap|(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 {{nowrap| 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, 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:
+Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, 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]]
 * ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
-* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
+* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
-* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
+* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
 * [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
 * [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
@@ Line 802: / Line 802: @@
 == Deca ==
-: ''For 5-limit version, see [[10th-octave temperaments #Neon]].''
+: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Neon]].''
-Deca has a period of 1/10 octave and tempers out the neon comma {{monzo| 21 60 -50 }} in the 5-limit, the [[linus comma]]{{monzo| 11 -10 -10 10 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.
+Deca has a period of 1/10 octave and tempers out the [[neon comma]] ({{monzo| 21 60 -50 }}) in the 5-limit, the [[linus comma]] ({{monzo| 11 -10 -10 10 }}) and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 919: / Line 919: @@
 == Aluminium ==
-: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Aluminium]].''
-Aluminium was named by [[Eliora]] in 2023 after the 13th element. It tempers out the {{monzo| 92 -39 -13 }} comma, which sets [[135/128]] interval to be equal to 1/13th of the octave.
+Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit and sets [[135/128]] to 1/13 of an [[octave]]. It was named by [[Eliora]] in 2023 after the 13th element.
 [[Subgroup]]: 2.3.5.7
@@ Line 970: / Line 970: @@
 Badness (Sintel): 1.18
-== Countritonic ==
+== Ragitritonic ==
-: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic (5-limit)]].''
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
-Countritonic (''co-un-tritonic'') may be described as the {{nowrap| 53 & 422 }} temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit. It was named by [[Flora Canou]] in 2023 as a variant of [[countriton]].
+Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.
+Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,188: / Line 1,190: @@
 Badness (Sintel): 1.14
-== Counterorson ==
-Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
-{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
-: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}
-: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
-: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}
-{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
-[[Badness]] (Sintel): 7.92
-== Oviminor ==
-: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''
-Oviminor was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
-{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
-: mapping generators: ~2, ~5/3
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
-: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
-: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}
-{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
-[[Badness]] (Sintel): 14.7
 == Octoid ==
 : {{Main| Octoid }}
-: ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''
 The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].
@@ Line 1,865: / Line 1,825: @@
 == Trideci ==
-: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''
-The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").
+The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic]] temperament, but with the ragisma (4375/4374) rather than the octagar comma (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").
 [[Subgroup]]: 2.3.5.7