Sensamagic family: Difference between revisions

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== Sensamagic ==

Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].

Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}

: error map: {{val| 0.000 +1.786 -0.769 -2.245 }}

~~~~

[[Minimax tuning]]:

* [[7-odd-limit]]

: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}

: [[~~Eigenmonzo~~ basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7

* [[9-odd-limit]]

: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}

: [[~~Eigenmonzo~~ basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5

{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}

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=== Overview to extensions ===

Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). ~~Considered~~ below ~~are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh~~.

The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.

Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.

== Undecimal sensamagic ==

Undecimal sensamagic tempers out not only [[385/384]], but [[896/891]], making itself a [[strong extension]] of [[parapyth]].

[[Subgroup]]: 2.3.5.7.11

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: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}

: error map: {{val| 0.000 +1.840 -0.683 -2.~~1554~~ +0.175 }}

: error map: {{val| 0.000 +1.840 -0.683 -2.154 +0.175 }}

~~~~

[[Minimax tuning]]:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2917{{c}}, ~9/7 = 441.1849{{c}}

: error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}

~~~~

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== Octarod ==

Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.

[[Subgroup]]: 2.3.5.7.11

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5246{{c}}, ~9/7 = 439.2798{{c}}

: error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}

~~~~

{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e }} *

{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*

<nowiki/>[[~~optimal~~ patent val]]: [[104edo|104]]

<nowiki/>*[[Optimal patent val]]: [[104edo|104]]

[[Badness]] (Sintel): 0.698

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.8402{{c}}, ~9/7 = 442.1064{{c}}

: error map: {{val| 0.000 +3.885 +3.739 +0.748 -1.638 }}

~~~~

{{Optimal ET sequence|legend=1| 19e, 22, 27e, 46, 68, 95, 141bc, 163bc }}

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* [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 703.5671{{c}}, ~9/7 = 441.2436{{c}}

: error map: {{val| 0.000 +1.612 -0.259 -2.935 -6.507 }}

~~~~

{{Optimal ET sequence|legend=1| 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e }}

[[Badness]] (Sintel): 1.31

~~== Sensigh ==~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 126/125, 169/168~~

~~Mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 | 0 0 0 0 1 0 }}~~

~~: mapping generators: ~2, ~9/7, ~11~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}

~~{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}~~

~~Badness (Sintel): 0.878~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 91/90, 126/125, 154/153, 169/168~~

~~Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}

~~{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}~~

~~Badness (Sintel): 0.917~~

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Sensamagic family| ]]

~~[[Category:Sensamagic| ]] ~~

[[Category:Rank 3]]

@@ Line 6: / Line 6: @@
 == Sensamagic ==
 {{Main| Sensamagic }}
+Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].
+Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.
 [[Subgroup]]: 2.3.5.7
@@ Line 25: / Line 29: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}
 : error map: {{val| 0.000 +1.786 -0.769 -2.245 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7424{{c}}, ~9/7 = 440.9020{{c}} -->
 [[Minimax tuning]]:
 * [[7-odd-limit]]
 : {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
-: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
 * [[9-odd-limit]]
 : {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
-: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
 {{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
@@ Line 51: / Line 54: @@
 === Overview to extensions ===
-Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
+The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.
+Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.
 == Undecimal sensamagic ==
 {{Main| Sensamagic }}
+Undecimal sensamagic tempers out not only [[385/384]], but [[896/891]], making itself a [[strong extension]] of [[parapyth]].
 [[Subgroup]]: 2.3.5.7.11
@@ Line 66: / Line 73: @@
 : [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}
-: error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}
+: error map: {{val| 0.000 +1.840 -0.683 -2.154 +0.175 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.8004{{c}}, ~9/7 = 440.9178{{c}} -->
 [[Minimax tuning]]:
@@ Line 107: / Line 113: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2917{{c}}, ~9/7 = 441.1849{{c}}
 : error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.1900{{c}}, ~9/7 = 441.1359{{c}} -->
 {{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87 }}
@@ Line 129: / Line 134: @@
 == Octarod ==
+Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 140: / Line 147: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5246{{c}}, ~9/7 = 439.2798{{c}}
 : error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0464{{c}}, ~9/7 = 439.5050{{c}} -->
-{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e }} *
+{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*
-<nowiki/>[[optimal patent val]]: [[104edo|104]]
+<nowiki/>*[[Optimal patent val]]: [[104edo|104]]
 [[Badness]] (Sintel): 0.698
@@ Line 162: / Line 168: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.8402{{c}}, ~9/7 = 442.1064{{c}}
 : error map: {{val| 0.000 +3.885 +3.739 +0.748 -1.638 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.3702{{c}}, ~9/7 = 442.1147{{c}} -->
 {{Optimal ET sequence|legend=1| 19e, 22, 27e, 46, 68, 95, 141bc, 163bc }}
@@ Line 196: / Line 201: @@
 * [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 703.5671{{c}}, ~9/7 = 441.2436{{c}}
 : error map: {{val| 0.000 +1.612 -0.259 -2.935 -6.507 }}
-<!-- * [[POTE]]: ~77/54 = 600.0000{{c}}, ~3/2 = 703.0884{{c}}, ~9/7 = 441.1060{{c}} -->
 {{Optimal ET sequence|legend=1| 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e }}
 [[Badness]] (Sintel): 1.31
-== Sensigh ==
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 126/125, 169/168
-Mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 | 0 0 0 0 1 0 }}
-: mapping generators: ~2, ~9/7, ~11
-Optimal tunings:
-* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}
-{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}
-Badness (Sintel): 0.878
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 126/125, 154/153, 169/168
-Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }}
-Optimal tunings:
-* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}
-{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}
-Badness (Sintel): 0.917
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Sensamagic family| ]] <!-- main article -->
-[[Category:Sensamagic| ]] <!-- key article -->
 [[Category:Rank 3]]